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Self-Assessment • 20 min read

How Good Is Your Problem Solving?

Use a systematic approach..

By the Mind Tools Content Team

Good problem solving skills are fundamentally important if you're going to be successful in your career.

But problems are something that we don't particularly like.

They're time-consuming.

They muscle their way into already packed schedules.

They force us to think about an uncertain future.

And they never seem to go away!

That's why, when faced with problems, most of us try to eliminate them as quickly as possible. But have you ever chosen the easiest or most obvious solution – and then realized that you have entirely missed a much better solution? Or have you found yourself fixing just the symptoms of a problem, only for the situation to get much worse?

To be an effective problem-solver, you need to be systematic and logical in your approach. This quiz helps you assess your current approach to problem solving. By improving this, you'll make better overall decisions. And as you increase your confidence with solving problems, you'll be less likely to rush to the first solution – which may not necessarily be the best one.

Once you've completed the quiz, we'll direct you to tools and resources that can help you make the most of your problem-solving skills.

How Good Are You at Solving Problems?

Instructions.

For each statement, click the button in the column that best describes you. Please answer questions as you actually are (rather than how you think you should be), and don't worry if some questions seem to score in the 'wrong direction'. When you are finished, please click the 'Calculate My Total' button at the bottom of the test.

Answering these questions should have helped you recognize the key steps associated with effective problem solving.

This quiz is based on Dr Min Basadur's Simplexity Thinking problem-solving model. This eight-step process follows the circular pattern shown below, within which current problems are solved and new problems are identified on an ongoing basis. This assessment has not been validated and is intended for illustrative purposes only.

Below, we outline the tools and strategies you can use for each stage of the problem-solving process. Enjoy exploring these stages!

Step 1: Find the Problem (Questions 7, 12)

Some problems are very obvious, however others are not so easily identified. As part of an effective problem-solving process, you need to look actively for problems – even when things seem to be running fine. Proactive problem solving helps you avoid emergencies and allows you to be calm and in control when issues arise.

These techniques can help you do this:

PEST Analysis helps you pick up changes to your environment that you should be paying attention to. Make sure too that you're watching changes in customer needs and market dynamics, and that you're monitoring trends that are relevant to your industry.

Risk Analysis helps you identify significant business risks.

Failure Modes and Effects Analysis helps you identify possible points of failure in your business process, so that you can fix these before problems arise.

After Action Reviews help you scan recent performance to identify things that can be done better in the future.

Where you have several problems to solve, our articles on Prioritization and Pareto Analysis help you think about which ones you should focus on first.

Step 2: Find the Facts (Questions 10, 14)

After identifying a potential problem, you need information. What factors contribute to the problem? Who is involved with it? What solutions have been tried before? What do others think about the problem?

If you move forward to find a solution too quickly, you risk relying on imperfect information that's based on assumptions and limited perspectives, so make sure that you research the problem thoroughly.

Step 3: Define the Problem (Questions 3, 9)

Now that you understand the problem, define it clearly and completely. Writing a clear problem definition forces you to establish specific boundaries for the problem. This keeps the scope from growing too large, and it helps you stay focused on the main issues.

A great tool to use at this stage is CATWOE . With this process, you analyze potential problems by looking at them from six perspectives, those of its Customers; Actors (people within the organization); the Transformation, or business process; the World-view, or top-down view of what's going on; the Owner; and the wider organizational Environment. By looking at a situation from these perspectives, you can open your mind and come to a much sharper and more comprehensive definition of the problem.

Cause and Effect Analysis is another good tool to use here, as it helps you think about the many different factors that can contribute to a problem. This helps you separate the symptoms of a problem from its fundamental causes.

Step 4: Find Ideas (Questions 4, 13)

With a clear problem definition, start generating ideas for a solution. The key here is to be flexible in the way you approach a problem. You want to be able to see it from as many perspectives as possible. Looking for patterns or common elements in different parts of the problem can sometimes help. You can also use metaphors and analogies to help analyze the problem, discover similarities to other issues, and think of solutions based on those similarities.

Traditional brainstorming and reverse brainstorming are very useful here. By taking the time to generate a range of creative solutions to the problem, you'll significantly increase the likelihood that you'll find the best possible solution, not just a semi-adequate one. Where appropriate, involve people with different viewpoints to expand the volume of ideas generated.

Tip: Don't evaluate your ideas until step 5. If you do, this will limit your creativity at too early a stage.

Step 5: Select and Evaluate (Questions 6, 15)

After finding ideas, you'll have many options that must be evaluated. It's tempting at this stage to charge in and start discarding ideas immediately. However, if you do this without first determining the criteria for a good solution, you risk rejecting an alternative that has real potential.

Decide what elements are needed for a realistic and practical solution, and think about the criteria you'll use to choose between potential solutions.

Paired Comparison Analysis , Decision Matrix Analysis and Risk Analysis are useful techniques here, as are many of the specialist resources available within our Decision-Making section . Enjoy exploring these!

Step 6: Plan (Questions 1, 16)

You might think that choosing a solution is the end of a problem-solving process. In fact, it's simply the start of the next phase in problem solving: implementation. This involves lots of planning and preparation. If you haven't already developed a full Risk Analysis in the evaluation phase, do so now. It's important to know what to be prepared for as you begin to roll out your proposed solution.

The type of planning that you need to do depends on the size of the implementation project that you need to set up. For small projects, all you'll often need are Action Plans that outline who will do what, when, and how. Larger projects need more sophisticated approaches – you'll find out more about these in the article What is Project Management? And for projects that affect many other people, you'll need to think about Change Management as well.

Here, it can be useful to conduct an Impact Analysis to help you identify potential resistance as well as alert you to problems you may not have anticipated. Force Field Analysis will also help you uncover the various pressures for and against your proposed solution. Once you've done the detailed planning, it can also be useful at this stage to make a final Go/No-Go Decision , making sure that it's actually worth going ahead with the selected option.

Step 7: Sell the Idea (Questions 5, 8)

As part of the planning process, you must convince other stakeholders that your solution is the best one. You'll likely meet with resistance, so before you try to “sell” your idea, make sure you've considered all the consequences.

As you begin communicating your plan, listen to what people say, and make changes as necessary. The better the overall solution meets everyone's needs, the greater its positive impact will be! For more tips on selling your idea, read our article on Creating a Value Proposition and use our Sell Your Idea Skillbook.

Step 8: Act (Questions 2, 11)

Finally, once you've convinced your key stakeholders that your proposed solution is worth running with, you can move on to the implementation stage. This is the exciting and rewarding part of problem solving, which makes the whole process seem worthwhile.

This action stage is an end, but it's also a beginning: once you've completed your implementation, it's time to move into the next cycle of problem solving by returning to the scanning stage. By doing this, you'll continue improving your organization as you move into the future.

Problem solving is an exceptionally important workplace skill.

Being a competent and confident problem solver will create many opportunities for you. By using a well-developed model like Simplexity Thinking for solving problems, you can approach the process systematically, and be comfortable that the decisions you make are solid.

Given the unpredictable nature of problems, it's very reassuring to know that, by following a structured plan, you've done everything you can to resolve the problem to the best of your ability.

This assessment has not been validated and is intended for illustrative purposes only. It is just one of many Mind Tool quizzes that can help you to evaluate your abilities in a wide range of important career skills.

If you want to reproduce this quiz, you can purchase downloadable copies in our Store .

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Afkar Hashmi

😇 This tool is very useful for me.

over 1 year

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Succeed at Exams

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What are some strategies for studying for problem-solving exams, what are some strategies for writing problem-solving exams, what can i do after the exam.

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More help: Studying

• Book Studying Appointments
• Problem-solving exams come in a variety of formats, from multiple choice, to short answer, to long calculations.
• They frequently test your ability to apply the problem-solving skills you’ve learned in lectures, labs, and readings to new types of questions.
• Pay attention to example problems emphasized in class, the text, and assignments, especially those that appear in more than one of these places.
• Don’t assume, however, that the same or similar problems will appear on the exam. The exam will likely test your ability to apply what you’ve learned about solving problems to new types of questions, rather than your ability to memorize and regurgitate examples you’ve already seen.
• Focus on the process the instructor used for solving the problem.  Think about your own problem-solving strategies.
• Practise, practise, practise! The more problems, and more importantly, the more types of problems you solve, the better prepared you’ll be.
• Look for connections between concepts and equations and note how to choose the correct equation in complex practice problems.
• Generate your own test questions with a study group or partner. Practise answering questions within a limited time frame.
• Review previous tests, quizzes, or midterms, and figure out why you lost marks before.
• Look over the entire exam before beginning and budget your time according to how much each question is worth.  Leave enough time to read over your answers at the end of the exam.
• Before starting, find a blank page on your exam and write down equations, concepts, and constants that you memorized.
• Do the easy questions first to warm up your brain and calm exam nerves.
• Read the questions carefully and rephrase them in your own words.
• Keep track of all units. Convert values to keep the units consistent. Be aware of +/- signs.
• Clearly mark assumptions, if they are necessary, and place them at or near the beginning of the solution whenever possible.
• It can be helpful to look over your exam if you get it back to see where you have gone wrong, and what you have done well. If your instructor doesn’t routinely return exams, ask if you can see your exam to learn from your errors.
• Use this information to help you study more effectively next time.
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What Are Logical Reasoning Tests?

Common logical reasoning test publishers, 6 logical reasoning tips to prepare for logic tests in 2024, frequently asked questions (faqs), logical reasoning tests: 2024 guide for logic tests.

Updated November 10, 2023

Logical reasoning tests measure a candidate’s problem-solving and logical reasoning skills in a wide variety of ways.

These logic tests are used in recruitment, particularly when assessing graduates for entry-level positions.

Logical reasoning tests may seem daunting and somewhat alien, but they use the same reasoning processes we rely upon daily – just in more overt and abstract ways.

As with any recruitment assessment, understanding exactly what the logic test involves and practicing similar questions will help to prepare you well.

Logical reasoning tests are a form of aptitude test commonly used during the recruitment process, especially in the corporate sector; investment banks, law firms, consultancies or accountancy firms, for example.

These logical aptitude tests aim to provide insight into your mental agility.

They most often require the interpretation and manipulation of statements, numbers, shapes and patterns.

The logical tests require the ability to calmly approach an abstract problem.

Questions will test candidates in a variety of ways but you may be asked to:

• Deduce trends
• Identify patterns
• Analyze arguments
• Draw conclusions
• Interpret relationships
• Identify flaws

Your score will display how quickly you can absorb and process new information and whether you have an aptitude for problem-solving and critical thinking.

Success in these tests is important, as it displays to a prospective employer that you are a deft candidate with a well-rounded mental capacity.

The name of the logical reasoning assessment you are presented with may vary depending upon the test provider.

The assessments may be referred to as logical reasoning tests, but may also be called inductive , abstract or diagrammatic reasoning tests .

There is overlap in the logical assessment test areas of these styles of questions and the overall principle remains the same.

If you are sitting a logical reasoning test, it is prudent to find out which company will be providing the assessment so you can get more of an idea of the kinds of questions that are most likely to appear.

Prepare for Any Job Assessment Test with JobTestPrep

Common Logical Reasoning Test providers include:

Logical reasoning tests may be used during the recruitment process for any position at any level but are most common for roles that have a significant problem-solving component and require high functioning logic.

They are most often encountered at the assessment center stage of the process but can also be used at the interview stage.

The Most Types of Logical Reasoning for the Logic Test

Diagrammatic logical reasoning.

The most common type of logical reasoning test you will encounter is the diagrammatic form .

Simply put, you will be asked to answer questions based on shapes and patterns.

There will be a written question posed and then multiple-choice options to choose from for your answer.

These logic tests commonly test your abstract and inductive reasoning skills as they will assess your ability to identify patterns or trends and extrapolate or reapply this logic to select the correct diagram to complete a sequence (or identify the ones which don’t).

Verbal Logical Reasoning

Another common form of logical reasoning test is the verbal logical reasoning test.

This type of test is comprised of a series of paragraphs of text. Each will then have its own set of questions for you to work through.

Most commonly, the questions will consist of a list of statements.

You will then be required to decide which statement is the most accurate response, based on the information you have been given.

Some questions may present just one statement and require a choice between ‘Yes’, ‘No’ or ‘Cannot Say’.

These logic tests are looking at your deductive reasoning . You will be assessed on how well you can use the information given to come to logical conclusions.

This style of logical reasoning test requires active and attentive reading .

It is a good idea to ask yourself questions as you read to make sense of the scope and detail of what is covered in the text.

This will help you make judgments as to the validity of the statements in the subsequent questions more rapidly.

If you need to prepare for a number of different employment tests and want to outsmart the competition, choose a Premium Membership from JobTestPrep . You will get access to three PrepPacks of your choice, from a database that covers all the major test providers and employers and tailored profession packs.

Practice Logical Reasoning Questions with JobTestPrep

Examples of Logical Reasoning Tests

Here you can find three example questions, similar to those you may encounter when taking a logical reasoning assessment:

Logical Reasoning Example 1

Look at the diagram above. In this process, the input shapes are being transformed by the functions 'M' and 'S'. Use the diagram to determine the effect each of these functions then apply them to the input shape in the question to arrive at the correct output.

Logical Reasoning Example 2

Which of the answer figures below belong in neither Group 1 nor Group 2?

Before you write down your answers, just double-check what the question was asking you.

It may seem obvious, but it is easy to misread ‘neither’ and answer the opposite question.

After spending time deducing the way the shapes in the patterns presented are governed, it would be galling to lose points because, when under time pressure, you accidentally answered the inverted question.

Logical Reasoning Example 3

Assuming the premises in the following text are true, what can we say about the conclusion drawn?

• Some employees of the cybersecurity company only drink coffee.
• Some employees only drink tea.
• Some drink both tea and coffee depending on the day.
• Some employees bring lunches.
• Some eat their lunches at their desk, whilst drinking coffee.
• Some never eat lunch at their desks.

There are some employees of the cybersecurity company who only drink coffee and never eat lunch at their desks.

Is the conclusion:

A) True B) False C) Uncertain

Step 1 . Find Out Which Type of Test Your Prospective Employer Is Using

Before you begin to prepare for your logical reasoning test, confirm the type of test your prospective employer is using. Don’t be afraid to ask, as the term ‘logical reasoning’ can encompass many types of test.

Step 2 . Practice Diagrammatic and Verbal Logical Reasoning

It is best to practice with questions closest to those you will encounter in the assessment, so trying to narrow down the question style is prudent.

If, however, you are unable to gain any further information on the test style, be sure to practice both the diagrammatic and verbal logical reasoning style of questions.

Step 3 . Work Out the Answer Before Looking at the Answer Options

Attempt to work out what the correct answer, rule or sequence is before looking at the answer options.

This means that your logic will not be influenced by any of the possible answers.

It also ensures you double-check the logic of your answer upon viewing the options for selection.

Step 4 . Start With the Answer

If you find yourself struggling to get to the correct answer in your practice sessions, work through some of the questions backward .

Look at the correct answer and then figure out what rule/pattern/evidence led to that answer.

This way, you will get used to working through the process of logic common in the questions.

Step 5 . Practice Under Timed Conditions

When you are practicing logical reasoning questions, make sure you time yourself .

Sometimes, logical reasoning tests have strict time limits, so being aware of how long you might have for each question will help you when you are under pressure on the test day.

Step 6 . Relax

The more familiar you become with logical reasoning tests, their format and the logic behind them, the faster and more accurate you will become at answering them.

It is important to relax so that your brain can focus on correctly solving the problem.

Remember that, with adequate practice, you will be in a great position to showcase your logical reasoning skills, so be confident and you’ll be well on your way to success.

What logical reasoning tests do major companies use?

Logical reasoning tests assess a candidate's ability to solve problems logically. Problem-solving is an essential part of many roles, meaning that logical tests are often included as part of an employer's recruitment process.

There are a variety of types of logical reasoning tests, each focusing on assessing a different type of logic used in problem-solving.

The most common logic tests used by employers to determine a candidate's logical reasoning include inductive reasoning, abstract reasoning, diagrammatic reasoning, or verbal reasoning.

How can you practice for logical reasoning tests?

Practicing logical reasoning tests vital to ensure you perform to the best of your ability when sitting the actual test.

When you practice logical reasoning, find out what type of logical reasoning test you will be completing and practice questions closest to those you will be asked to sit. If in doubt, practicing diagrammatic reasoning and verbal logical reasoning tests will help you understand the format and type of questions.

When sitting practice logic tests, make sure you time yourself to simulate the time pressure you will feel when taking the test itself.

What site would you prefer for a logic test?

Employers use several different test providers to administer Logical reasoning tests . These include SHL, Revelian, Talent Q and Kenexa.

Practicing logical reasoning tests is an important part of your preparation. The Psychometric Success website and app have many practice logical assessments tests that help you understand the format and style of questioning for each type of logical reasoning test.

You can find the Psychometric Success is available on Apple and Android devices.

How do you pass a reasoning test?

Preparation is essential in ensuring you perform to the best of your ability when sitting a Logical reasoning test . Make sure you practice as many types of logical reasoning tests, including the kind that you are most likely to encounter in the actual test.

When practicing, do so under timed conditions to experience the time pressure you will face when it comes to sitting the actual test.

When faced with a question, try and work out the rule, pattern, or sequence before looking at the multiple-choice answers.

If you are struggling in the test, try and remain calm. If you are unsure of an answer, mark your best choice, then move on to the next question.

Does a logical reasoning test work for selecting the right people?

A Logical reasoning test is an objective way of assessing your logical reasoning ability when solving problems.

When used along with the information gained from all parts of the recruitment process, it provides a reliable way to predict whether candidates demonstrate the logical ability required for success in a role.

As problem-solving is a requirement for the job, logical reasoning tests are popular in the recruitment process for many graduate-level roles.

What are the logical reasoning tips to crack the logical questions ?

Preparation is vital. There are different types of Logical reasoning tests that employers use, such as diagrammatic reasoning, inductive reasoning, or verbal logical reasoning. Knowing which type of test you will be asked is important to ensure you practice the questions you are likely to be presented within the actual test.

Practicing these questions under timed conditions means you become used to how the question is formatted and how you react under time pressure. When sitting the test, try and remain calm, read the question and approach each question in a positive frame of mind.

When you start the test, make sure you are clear on the time limit for the test, this gives you the best opportunity to answer as many questions correctly as possible.

When tackling the questions, try and identify the pattern or rule before selecting your answer. Don't dwell too long on a question if you are struggling. Make an educated guess, then move on to the next question.

Which free website is best for logical reasoning test prep?

Practicing logical reasoning tests is an integral part of your preparation.

Becoming familiar with the format, style of questioning, and how you react under time pressure can help you perform to the best of your ability in the actual test.

For practice tests and free tips and advice on approaching and passing a logical reasoning test, visit the Psychometric Success website.

Which is the best app for getting free education daily for logical reasoning test?

Employers use various types of Logical reasoning tests to assess your logical reasoning and problem-solving abilities.

When practicing logical reasoning tests, it is crucial to practice questions similar to those you will encounter in the ones you will sit.

The Psychometric Success app has lots of free advice, hints, and tips to help you prepare for any logical reasoning test.

Which android app can I use to draw logical reasoning test questions?

The psychometric Success app is available on Android devices and is a valuable resource to prepare for and practice logical reasoning test questions .

There are many different logical reasoning tests, each assessing different kinds of logical reasoning and critical thinking.

When practicing logical reasoning tests, ensure you know the type of logical reasoning assessment you will be asked to sit as part of an employer's recruitment process.

What are the various logical reasoning topics I should prepare for the test?

Logical reasoning tests are assessments used by many employers for graduate-level vacancies. These tests assess your approach to problem-solving and whether you can evaluate information logically to reach a solution.

There are several logical reasoning tests, each assessing a different logical reasoning approach to problem-solving. These include diagrammatic reasoning, inductive reasoning, and abstract reasoning tests. When preparing for a logical reasoning test, it is helpful to understand what type of test you will be asked to sit.

If you are unable to determine which type of test you will be sitting it is advisable to practice diagrammatic reasoning and verbal logical reasoning tests as part of your preparation.

How are Logical Reasoning Tests Scored?

The benchmark that indicates a good score in a Logical reasoning test varies by employer. To ensure you perform at your best, showcase your logical reasoning abilities and attain a good score, it is essential to practice the tests.

When practicing, simulate test conditions by timing yourself and keeping to the recommended time duration of the test.

When sitting the test itself, remain calm, identify the rule or pattern in the information given before looking at the answer, and don't lose confidence if you find the assessment challenging.

Can you still get hired if you fail a logical reasoning aptitude test?

Logical reasoning tests are used in the recruitment process for many graduate-level roles. These tests assess a candidate's ability to solve problems logically, an essential ability required for many positions.

There are several types of logical reasoning tests that employers use. Each employer will have its own criteria for assessing and scoring a logical reasoning test.

Understanding the type of logical reasoning test, you will be asked to sit, preparing through practicing tests, and when sitting the test remaining calm can all help you showcase your aptitude for logical reasoning.

You might also be interested in these other Psychometric Success articles:

Or explore the Aptitude Tests / Test Types sections.

Problem solving test: Pre-employment screening assessment to hire the best candidates

Summary of the problem solving test.

This Problem Solving test evaluates candidates’ ability to define problems and analyze data and textual information to make correct decisions. This test helps you identify candidates who use analytical skills to evaluate and respond to complex situations.

Covered skills

Creating and adjusting schedules, interpreting data and applying logic to make decisions, prioritizing and applying order based on a given set of rules, analyzing textual and numerical information to draw conclusions, use the problem solving test to hire.

Any role that involves managing constantly shifting variables with tight deadlines. This may include administrative assistants, project managers, planners, and people working in hospitality or sales.

You can use this test for free when you sign up for a free plan

About the Problem Solving test

Effective problem-solving requires a broad skill set that enables individuals, teams, and businesses to advance towards stated objectives. It involves the ability to define a problem, to break it down into manageable parts, to develop approaches to solve the (sub)problem using creativity and analytical thinking, and to execute flawlessly.

This problem solving test allows you to identify candidates who display these abilities. The test presents candidates with typical problem-solving scenarios like scheduling on the basis of a diverse set of conditions, identifying the right sequence of actions based on a number of business rules, and drawing conclusions based on textual and numerical information.

The test requires candidates to identify the right answers to the questions in a limited amount of time. Successful candidates can quickly identify the key elements of the problem and work through the problem at speed without making mistakes. This is a great test to include to check candidates' overall analytical skills.

The test is made by a subject-matter expert

The global IT industry has benefited from Anirban’s talents for over two decades. With a flawless reputation that precedes him, Anirban has earned a status as a sought-after agile project manager and consultant. He’s worked internationally as a Senior Project Manager with companies such as Ericsson, IBM, and T-Mobile.

Anirban’s love for learning helps him keep his skills sharp. He holds an MBA and a degree in engineering, is a certified Scrum Master, and has certifications in Prince2 and ITIL.

Crafted with expert knowledge

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Our feedback mechanisms and unique algorithms allow our subject-matter experts to constantly improve their tests.

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View a sample report

The Problem Solving test will be included in a PDF report along with the other tests from your assessment. You can easily download and share this report with colleagues and candidates.

Why problem-solving is a crucial skill

We’ve all been thrown a curveball at work. An unexpected problem crops up and we need to make a plan to solve it. This is called problem-solving and it’s an important skill in most job roles because employees will encounter difficult or complex situations or problems that need to be solved.

Interestingly, unlike some career skills, problem-solving translates to both an individual’s personal and professional lives, making it crucial to potential success. But this may make it harder for recruiters to find the right candidate for a job where problem-solving skills are needed. In this case, a problem-solving test can help you find the best candidate capable of handling situations that influence business functions.

Problem-solving in the workplace

In business, problem-solving relies on a candidate’s ability to create processes that mitigate or remove obstacles that prevent the company from achieving its goals. Consequently, these issues or situations can create a gap between desired outcomes and actual results. This means that problem-solving plays an important role in how employees meet this challenge and work through it.

Roles such as project management, administrative assistance, and planning work with changing circumstances and tight deadlines on a day-to-day basis. When recruiting for these roles, finding candidates who have good problem-solving skills is crucial to their success in the role.

To gain insight into a candidate’s skill in this area, you can use a problem-solving test. Through expert formulation, a skill-specific test can help you understand a candidate’s level of proficiency. And testing your applicants before you start the interview process can highlight the candidates with the skills most relevant to the role.

A process-driven skill

In the workplace, there are important steps that can contribute to a candidate’s ability to successfully solve problems. Let’s take a look:

Identify the problem

Problem-solving begins with accurately identifying the problem. This determining factor looks at whether a candidate can find the origin and the implications of the problem. It includes:

• Differentiating between fact and opinion

• Compiling data to determine the problem

• Identifying underlying causes

• Recognizing which processes are affected• Pinpointing the process standard

By accurately identifying the problem, individuals can proceed to the next step to solve the problem.

Determine alternative solutions

Once an individual has established the source of the problem, they can determine alternative solutions. The goal of plotting solutions to the problem is to remedy it and realign it with business goals. A creative problem-solving test may identify whether an individual has the competency to determine solutions. Key competencies in seeking solutions include:

• Establishing alternative solutions that align with business goals

• Determining whether a problem needs short- or long-term solutions

• Evaluating how solutions may impact on resources; and determining if there are any barriers to implementing the solutions.

Although any problem can have multiple solutions, the simplest or fastest one may not always be the best course of action. This is where solution comparison comes into play.

Compare solutions and plot a course

Once all possible solutions are determined, it is important to compare them. This involves evaluating each solution without bias to determine the optimal solution to the problem.

Through the evaluation process, the individual should rule out options that do not align with company goals, may take too much time and/or resources, or are unrealistic in their approach.

Some considerations when determining the best solution include the likelihood of solution implementation, whether all parties involved will accept the solution, and how it fits in with business goals. Additionally, it is important to note that the goal of the optimal solution is to solve the problem without causing additional or unanticipated problems.

In essence, problem-solving is about finding solutions that cause as little disruption as possible and correcting a project’s course.

Implement the solution

The last stage in problem-solving is the implementation of the final step. This step focuses on the remedial solution and requires continuous evaluation to ensure its effective implementation. For you as a recruiter, knowing if a candidate can find a solution as well as implement it may be important to the goals of the role.

Continually evaluating the solution will give the individual insight into whether the project goals are aligned, whether all stakeholders accept the new solution and whether the outcomes are managed effectively.

Considerations for recruiters

When hiring for a role in which problem-solving skills are crucial, it may be beneficial to test a candidate’s ability to define problems and analyze data and textual information to make decisions that best serve the business. Some of the considerations for a problem-solving test include:

Schedules are living documents that need to adapt as eventualities come into play. Candidates should be able to understand what they can realistically achieve with the time and how to adjust schedules to account for variable outcomes.

Data-driven decision-making should inform a course of action before an individual commits to it. For recruiters, this means candidates should have an aptitude for aligning data with business goals and making actionable decisions.

By using prioritization rules and supporting information, candidates can determine which project tasks take priority. This system aims to optimize resources for project delivery.

Examining textual and numerical information to reveal patterns, relationships, and trends can tell the candidate what connection exists among variables. Conclusions can then be drawn from the data to gain an accurate assessment of the overall situation.

When broken down, problem-solving is a skill that relies on a variety of disciplines to achieve success. Although this skill is transferable to many job roles, determining candidates’ proficiency can be difficult, so it can be beneficial to recruiters to use a problem-solving test to review candidates’ aptitude when recruiting for a role.

Using a pre-formulated problem-solving test will enable you to quickly assess your candidates and help you recruit the best person for the role.

An assessment is the total package of tests and custom questions that you put together to evaluate your candidates. Each individual test within an assessment is designed to test something specific, such as a job skill or language. An assessment can consist of up to 5 tests and 20 custom questions. You can have candidates respond to your custom questions in several ways, such as with a personalized video.

Yes! Custom questions are great for testing candidates in your own unique way. We support the following question types: video, multiple-choice, coding, file upload, and essay. Besides adding your own custom questions, you can also create your own tests.

A video question is a specific type of custom question you can add to your assessment. Video questions let you create a question and have your candidates use their webcam to record a video response. This is an excellent way to see how a candidate would conduct themselves in a live interview, and is especially useful for sales and hospitality roles. Some good examples of things to ask for video questions would be "Why do you want to work for our company?" or "Try to sell me an item you have on your desk right now."

Besides video questions, you can also add the following types of custom questions: multiple-choice, coding, file upload, and essay. Multiple-choice lets your candidates choose from a list of answers that you provide, coding lets you create a coding problem for them to solve, file upload allows your candidates to upload a file that you request (such as a resume or portfolio), and essay allows an open-ended text response to your question. You can learn more about different custom question types here .

Yes! You can add your own logo and company color theme to your assessments. This is a great way to leave a positive and lasting brand impression on your candidates.

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TestGorilla replaces traditional resume screening with a much more reliable and efficient process, designed to find the most skilled candidates earlier and faster.

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Our cognitive ability tests allow you to test for skills that are difficult to evaluate in an interview. Check out our blog on why these tests are so useful and how to choose the best one for your assessment.

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Computational thinking, basic math calculations, mechanical reasoning, understanding instructions, attention to detail (visual), intermediate math, basic triple-digit math, basic double-digit math, attention to detail (textual), numerical reasoning.

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Exam Strategies: Problem Solving Exams

Problem-solving exams can come in various formats and include different question types such as multiple-choice, short answer, or long calculations to test your ability to apply the skills and concepts you’ve learned in a course.

Preparing for a problem-solving exam

• Practice example problems from your lecture notes, textbooks, and assignments and take note of the types of questions that appear frequently. Focus on the process used for solving the problem and how that strategy can apply to potential exam questions.
• Practice and then practice again! The more problems, and the more types of problems, you solve the better prepared you’ll be for your exam.
• Review your previous quizzes or exams to be sure you understand where you may have lost marks. Self-test with these types of questions, or study with a partner to create questions and quiz each other.

Answering problem-solving questions

• Look over the questions and budget your time according to how much each question is worth - try to leave time to review your exam at the end.
• Before beginning, jot down any equations, concepts, and constants that you may have memorized and need for later questions.
• Read each question carefully and think about what it is asking you to solve before you answer:
• Try to identify what concepts or theory the problem is based on
• Write down all the given information in bullet form
• Draw a clear diagram with labels, directions, axes, if needed or to help illustrate the problem
• Write out formulae and expressions you may need
• Show every step as you work through the problem - even if you don’t get the correct answer, you may get points for your process.
• Review your work - check your calculations, keep track of units, and check for common sense (e.g. magnitude, dimensions, etc.).

Looking for more strategies and tips? Check out MUN's Academic Success Centre online!

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Problem-solving tests and exams.

Problem-solving exams focus on your ability to apply what you know rather than simply showing the answer. There are strategies for preparing for and writing problem solving exams to maximize your success:

• When you’re studying, be sure to work in a way that tests and extends your knowledge: just having read the textbook or jumping to the solutions can give you a false sense of knowing the information.
• Study problems out of sequence: mixing problem types when you’re preparing will help you in an exam situation where questions come from different parts of the course.
• If you’re not allowed aid sheets in the exam, think about what you’d wish to have in the exam room, and work on memorizing it. For instance, in the first few minutes of an exam, you can write out a list of equations and associated instructional verbs to make sure you have them while you’re problem solving.
• Write problems and solutions out in full: double-check that you haven’t made any transcription errors, and do every step on paper so that you can double-check your answers and earn partial marks.

Helpful documents

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Other resources

While U of T has a plethora of programs and services for students contending with problem-solving courses, here are some external resources that have become online classics. They’re maintained by external authors, but our students have often found them insightful:

• How to Study Math (Paul Dawkins, Lamar University)
• Problem Solving I: Frequent Solver (Richard Zajchowski for McMaster University)
• Problem Solving II: Concepts (Richard Zajchowski for McMaster University)
• Problem Solving III: Decision Steps (Richard Zajchowski for McMaster University)

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How to Tackle Exam Questions

Learn more about how to tackle different kinds of exams and exam questions.

We cover the following topics on exam preparation on this page:

• Quantitative Questions
• Multiple Choice Questions
• Essay Questions

First, Let’s Think About De-Coding Different Types of Exam Questions

It’s helpful to understand the kinds of question that are asked on a exam, because the response you need to come up with depends on the type of question. Knowing about different types of exam questions can help you activate appropriate strategies for formulating answers and reduce exam-taking anxiety.

Exam questions generally fall into one of three categories: 1

“Green Light”

• Go right ahead!
• These are factual questions, and the answers are straight-forward. You either know the answer or you don’t; it’s right there in your head or it’s not.
• Some green light questions can be very difficult, and your ability to recall details is often tested with this typeof question.
• Study for this type of question by using recitation, making flash cards, quizzing yourself or a study partner, etc.
• If you don’t know the answer to a green light question right away, circle it and move on; often the answer will pop into your head later on during the exam.

“Yellow Light”

• These questions are more detailed than green light questions, but are based on the same idea: you either know the answer or you don’t.
• Often you’ll have to put multiple or “green light” details together.
• Similar strategies work for yellow and green questions, but with yellow light questions you’ll need to recall many ideas, concepts, formulas, etc., just to answer one question.

• These questions ask you to make inferences or apply your knowledge to new situations, which is sometimes called “critical thinking”.
• You need to know the material being covered to answer these questions at the “green light” level, but the exam question is not asking you to simply regurgitate it. You will need to take what you know and use it in ways you have not yet used it.
• This type of question sometimes flummoxes students, because they are surprised to they are being asked a question that wasn’t exactly covered in class. Remember that with red light questions you are not supposed to already know the answer. You have to come up with the answer yourself, it is not already in your head. (You will need to know the basic information, though, to be able to answer this type of question.)
• Red light questions are asked more frequently in college than in high school.
• To study for red light questions, make diagrams or concept maps that link ideas or topics from the course together. Think about how what you’re learning relates to what you’ve learned in other classes. Sit down with friends or classmates and talk about how one might use information from the class in a job setting.

See this link for a pdf of  Decoding exam questions.

How to Tackle: Problem-Solving and Quantitative Questions

Study for problem-based exams by practicing (new!) problems

As you work on the problems, remember:

• DO let yourself be stuck.* (yes, we mean that!)
• DON’T sneak a peak at the answer if you get stuck. (keep trying!)
• Check your answer only after you’ve put something–anything–down. Think partial credit, which is better than no credit if you freeze when you get stuck on hard problems on the test.

* You need to get your “stuck” muscles stronger so you know what to do on tests when you feel stuck.

Watch: LSC’s Mike Chen Shares “The Key to Problem-Solving Tests”

Taking problem-based exams

1. Understand the problem: Determine what you are supposed to find, what you need to find it, and what the unknown is (and if there is extra information). Consider whether drawing a sketch will help. Also – note each part of the question. Not answering each part is an easy way to lose points.

2. Determine a way to solve the problem: Write down all that is given or known. Draw a sketch when appropriate to show relations. Write down all relevant formulas.

3. Carry out the procedure you have devised: For numerical problems, try and estimate an answer first. This will help you to check your work later. Neat, careful work keeps you from making mistakes, and allows you to find them when you do make them (show your units!!). Additionally, when the instructor can see your work clearly, he or she may give you partial credit for what you do know, even if your ultimate answer is incorrect.

4. Check your Answers: This requires the same quality of thought originally used to solve the problem. Is your answer what you thought it would be in your original estimate? Is it a quantity that makes sense? Is your answer in the correct units? If your answer does not seem reasonable, rework the problem.

How to Tackle: Multiple Choice Questions

1. Read the stem: First, read the stem and make sure you understand what it is getting at. Look out for double negatives or other twists in wording before you consider the answer.

2. Try to come up with the correct answer: Before you look at the answer choices, try to come up with the correct answer. This will help you to rule out choices that are similar to the correct answer. Now read and consider each option carefully.

3. Look for clues in the stem: Look for clues in the stem that suggest the correct answer or rule out any choices. For example, if the stem indicates that the answer is plural you can rule out any answers that are singular. The basic rule is: the correct answer must make sense grammatically with the stem. Options which fail this exam can be ruled out.

4. Cross off any options you know are incorrect: As you rule out options cross them off with your pencil. This will help you focus on the remaining choices and eliminates the chance of returning to an item and selecting an option you had already eliminated.

5. Come back to items you were unsure of: Put a mark next to any questions you are unsure of. If you complete the entire exam with time to spare, review these questions – you will often get clues (or even answers) from other questions.

Take a look at some additional information on difficult “ Multiple Choice Tests ” (opens a PDF).

How to Tackle: Essay Questions

The best way to  prepare  for essay tests is to practice writing essays.

• Anticipate questions : Make outlines of possible essay topics using your course materials so you know you’ve got a good grasp of what might be on the test. Then recreate your outlines from memory (unless it’s an open-notes test).
• Practice writing  at least one full essay; be mindful of the time you spend practicing and think about how much time you will have during the exam. It is also important to think about  how  you are organizing the information you are including in your essay — for example, if you are asked to compare and contrast two theories as they relate to an issue, you might want to define each of them, describe the issue, and then compare and contrast them.
• If your exam is closed book,  memorize key events, facts, and names  that you will need to support your argument. If it is open-notes, then make sure you develop good outlines.

When you are  taking  essay tests:

• Manage your time  well. As with all exams, if there are multiple essay questions, be sure to look at them all at the beginning (taking note of the points each is worth), and prioritize the order you answer the questions.
• Read the directions  carefully. Ask yourself honestly: are you answering the  actual  question on the test, or the question you  want  to be on the test?  (tip: instructors know when you aren’t really answering the exact question, so make sure you are addressing the actual question and don’t just write random information that is unrelated to the question.)
• Before you write the essay,  decide on your argument  and  quickly list your supporting evidence  (it is ok to do a brain dump of all the important information that you want to include so that you have it handy when you begin writing).
• Make a quick outline  of what you are going to write to organize your thoughts and arguments.
• Write! And, make your point right away – you don’t want to get to the end of a timed essay test with your amazing argument still unmade!
• If you have time, go back and quickly  proof-read  your essay for errors.

You might want to take a look at some “ Words to Watch for in an Essay ” (opens a PDF).

References:

1 Taffy E. Raphael, Teaching Question Answer Relationships, Revisited, The Reading Teacher, Vol. 39, No. 6 (Feb., 1986), pp. 516-522.

Ellis, D. (1998). Becoming a Master Student. Houghton Mifflin: Boston

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How to Master GMAT Problem Solving

Stacey Koprince - Manhattan Prep

Stacey Koprince is an mba.com Featured Contributor and the content and curriculum lead and an instructor for premier test prep provider Manhattan Prep .

The GMAT™ exam feels like a math test, especially GMAT Problem Solving problems. They read just like textbook math problems we were given in school; the only obvious difference is that the GMAT Quant section gives us five possible answer choices.

It’s true that you have to know certain math rules and formulas and concepts, but actually, the GMAT is really not a math test. First of all, the test doesn’t care whether you can calculate the answer exactly (e.g., 42). It cares only that you pick the right answer letter (e.g., B)—and that’s not at all the same thing as saying that you have to calculate the answer exactly, as you did in school.

More than that, the GMAT test-writers are looking for you to display quantitative and critical reasoning skills (the section is literally called Quantitative Reasoning ); in other words, they really want to see whether you can think logically about quant topics. They’re not interested in testing whether you can do heavy-duty math on paper without a calculator. And here’s the best part: They build the problems accordingly and you can use that fact to make GMAT Problem Solving problems a whole lot more straightforward to solve. I’ll show you how in this article!

GMAC’s team (aka, the people who make the GMAT) gave me three random problems to work through with you. I had no say in the problems; I didn’t get to choose what I liked. Nope, these three are it, and every single one illustrates this principle: The GMAT is really a test of your quantitative reasoning skills, not your ability to be a textbook math whiz.

GMAT Quant is not a math test

Okay, let’s prove that claim I just made. Grab your phone and set the timer for 6 minutes. (If you’ve been granted 1.5x time on the GMAT, set it for 9 minutes. If you’ve been granted 2x time on the GMAT, set it for 12 minutes.)

Do the below 3 problems under real GMAT conditions:

• Do them in order. Don’t go back.
• Pick an answer before you move to the next one. (Don’t just say you’re not sure and move on. Make the guess, as you have to do on the real test.)
• Have an answer for all the problems by the time your timer dings—even if your answers are random guesses.

Problem #1: Fellows in the org

According to the table above, the number of fellows was approximately what percent of the total membership of Organization X? 

(A) 9% (B) 12% (C) 18% (D) 25% (E) 35%

Problem #2: Yolanda and Bob

One hour after Yolanda started walking from X to Y, a distance of 45 miles, Bob started walking along the same road from Y to X. If Yolanda’s walking rate was 3 miles per hour and Bob’s was 4 miles per hour, how many miles had Bob walked when they met?

(A) 24 (B) 23 (C) 22 (D) 21 (E) 19.5

Problem #3: Oil cans

Two oil cans, X and Y, are right circular cylinders, and the height and the radius of Y are each twice those of X. If the oil in can X, which is filled to capacity, sells for $2, then at the same rate, how much does the oil in can Y sell for if Y is filled to only half its capacity? 

(A) $1 (B) $2 (C) $3 (D) $4 (E) $8

Time’s up! Do you have an answer for each problem? If not, make a random guess—but do choose an answer for every problem.

You probably want me to tell you the three correct answers so you’ll know whether you got them right. But I’m not going to.

We’re going to review these in the same way that I want you to review them when you’re studying on your own—and that means *not* looking up the correct answer right away. 

• How confident are you about this problem?
• Did/do you have another idea for how to solve? Try it now.
• Were you straining to remember some rule or formula? Look it up and try again.
• Still stuck? Okay, look at the correct answer. Does knowing that give you any ideas? Push them as far as you can. 
• Stuck again? Start to read the explanation. Stop as soon as the explanation gives you a new idea. Push it as far as you can before you come back to the explanation again.

Basically, push your own thinking and learning as far as you can on your own. Use the correct answer and explanation only as a series of hints to help unstick yourself when you get stuck.

Okay, let’s dive in!

GMAT Problem Solving #1: Estimate

We’re going to use the UPS solving process: Understand, Plan, Solve. (A mathematician named George Polya  came up with this.) Use this rubric to approach any quant-based problem you ever have to figure out in your life!

The basic idea is this: Don’t just jump to solve. (That’s panic-solving! We’ve all been there. It does not end well.) Understand the info first. Come up with a plan based on what you see. Only then, solve. 

And if you don’t understand or can’t come up with a good plan? On the GMAT, bail! Pick your favorite letter and move on. UPS can help you know what to do and what not to do.

Glance at the answers. Yes, before you even read the problem! 

The answers indicate that this is a percent problem and they’re also pretty decently spread apart. One is a little less than 10% and another is a little greater than 10%, so that’s one nice split. The remaining three are a little less than 20%, exactly 25%, and about 33%, otherwise known as one-third. Those are all “benchmark,” or common, percentages, so now I know I can probably estimate to get to my answer. Excellent.

And then the problem actually includes the word approximately ! Definitely going to estimate on this one.

Start building a habit of glancing at the answers on every single Problem Solving problem during the Understand phase, before you even think about starting to solve. (And yes, I really do glance at the answers before I even read the question stem!)

Here are some examples of the types of answer-choice characteristics that indicate there’s a good chance you’ll be able to estimate at least a little:

• The answers are really spread out (e.g., 10, 100, 300, 600, 900)
• Some are positive and some are negative
• Some are less than 1 and some are greater than 1
• They’re spread out on a percent scale (0 to 100) or on a probability scale (0 to 1)—less than half, greater than half, etc.

Next, there’s a table with a bunch of categories and each category is associated with a specific number. What does the question ask?

It wants to know the Fellows as a percent of the total. That’s a fraction with fellows on the top and the total of all members on the bottom:

The Fellows category is already listed in the table. Great, that’s the numerator.

What about the total? That means adding up all the numbers in the table without a calculator or Excel. Rolling my eyes. And that’s how I know that I will not be doing “textbook math” here. Pay attention to those feelings of annoyance! There’s some other easier, faster path to take. Use your Plan phase to find it.

I need the Total. I can estimate. Look at the collection of numbers. Can you group any into pairs that will add up to “nicer” numbers—numbers that end in zeros?

Here’s one way: 

• Honorary is a tiny number compared to the rest. Ignore it. 
• Fellows are a little under 10,000 and Members are a bit over 35,000. Group them. 
• Associates are a little less than 28,000 or a little more than 2,000 away from 30,000. And Affiliates are a little over 2,000! Combine those two groups.

We’re already spilling into the solve stage on this one. Fellows and Members together are about 45,000. Associates and Affiliates together are about 30,000. Altogether, there are 75,000 members:

That goes on the bottom of the fraction. Fellows go on top. They’re about 9,200, so let’s call that 9,000. Make a note on your scratch paper that you’re underestimating —just in case you need to use that to choose your final answer. I use a down-arrow to remind myself.

How to simplify 9 out of 75? Both of those numbers are divisible by 3.

Ok, 3 out of 25: what percent is that? We normally see percentages as “out of 100.” Hmm. 

If you multiply the denominator by 4, that gets you to “out of 100.” And whatever you do to the denominator, you have to do to the numerator, so the fraction turns into 12 out of 100, or 12%.

12% is in the answers; the next closest greater value (since we slightly underestimated) is 18%. That’s too far away, so the only answer that makes sense is (B).

Notice how the numbers looked really ugly to start out, but as soon as you started estimating, they combined and simplified really nicely? It’s not just luck. The test-writers know you don’t have access to a calculator, so they’re building the problems to work out nicely if you use these types of approaches. They actually want to reward you for using the kind of quantitative reasoning that you’d want to use at work and in business school.

You can certainly solve GMAT Problem Solving problems using traditional textbook math approaches. You’ll just do a lot more work that way. And using textbook approaches won’t actually help train your brain for the kind of analytical thinking about quant that you’ll need to do in business school or in the working world.

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GMAT Problem Solving #2: Logic (and draw!) it out

One hour after Yolanda started walking from X to Y, a distance of 45 miles, Bob started walking along the same road from Y to X. If Yolanda’s walking rate was 3 miles per hour and Bob’s was 4 miles per hour, how many miles had Bob walked when they met? 

The answers are real values and on the smaller side. They’re pretty clustered, so probably won’t be estimating on this one. Four of the five are integers. I wonder whether I can work backwards on this one (i.e., just try some of the answers)?

This problem is what I call a Wall of Text—a story problem. Get ready to sketch this out. Take your time understanding the setup; if you don’t “get” the story, you’ll never find the right answer. (And if you don’t get the story, that’s your clue to guess and move on.)

There are two people, 45 miles apart, and they’re walking towards each other. Normally, I’d only write initials for the two people, but annoyingly, Yolanda shares her initial with one of the locations.

The first sentence has a critical piece of info that’s easy to gloss over: Yolanda starts first, an hour before Bob. 

It’s super annoying that they don’t start at the same time. I don’t know what to do about that yet, but I’m noting it because I want to think about that when I get to my Plan stage. Again, pay attention to whatever annoys you about the problem! That’s why I put START FIRST in all-caps on my scratch paper.

Next, Yolanda walks a little slower than Bob. Add that to your diagram.

Finally, the problem asks who walked further by the time they meet—and how far that person walked. If Yolanda and Bob had started at the same time, then I’d know Bob walked farther, since he’s walking faster, but Yolanda started first, so I can’t tell at a glance. Still annoyed by that detail.

The two people have to cover 45 miles collectively in order to meet somewhere in the middle. Glance at the answers again. There are two sets of pairs that add to 45: (A) 24 and (D) 21 and (B) 23) and (C) 22. 

On a problem like this one, the most common trap answer is going to be solving for the wrong person (in this case, Yolanda instead of Bob). So the correct answer is going to fall into one of those pairs, because then the most common trap answer will also be built into the problem. The other pair will represent some common error when solving for Bob—and then also mistakenly solving for Yolanda instead. But answer (E) 19.5 doesn’t have a pairing, so it has no built-in trap. If you have to guess, don’t guess the unpaired answer, (E).

Once I subtract the 3 miles that Yolanda walked alone, the two of them together have 42 more miles to cover before they meet. I did note the extra 3 miles she walked off to the side just in case.

Bingo. Now I know how I’m going to solve this problem, because now it’s a more straightforward rate problem.

From here, you can do the classic “write some equations and solve” approach to rates problems. But I’m going to challenge you to keep going with this Logic It Out approach we’re already using—both because it really is easier and because it’s what you would use in the real world. You’re not getting ready to take the GMAT because you want to become a math professor. You’re doing this to be able to think about quant topics in a business context. So make your GMAT studies do double-duty and get you ready for b-school (and work!) as well.

Back to Bob and Yolanda. They’re 42 miles apart and walking towards each other. Every hour, Yolanda’s going to cover 3 miles and Bob’s going to cover 4 miles, so they’re going to get 7 miles closer together. Together, they’re walking 7 miles per hour.

When two people (or cars or trains) are moving directly towards each other, you can add their rates and that will tell you the combined rate at which they’re getting closer together. (You can do the same thing if the two people are moving directly away from each other—in this case, the combined rate is how fast they’re getting farther apart.)

One more thing to note: The distance still to cover is great enough (42 miles) compared to their combined rate (just 7 mph) that Bob is going to “overcome” the 3 miles that Yolanda walked on her own first. So Bob covered a greater distance than Yolanda did. The answer is going to be one of the two greater numbers in the pairs: (A) 24 or (B) 23.

So Yolanda and Bob are getting closer together at a rate of 7 miles each hour and they have a total of 42 miles to cover until they meet. How long is it going to take them?

Divide 42 by 7. They’re going to meet each other after 6 hours on the trail. At this point, Bob has spent a total of 6 hours walking, but not Yolanda! She started first, so she spent a total of 6 + 1 = 7 hours walking. The question asks how far Bob walked: 4 miles per hour for 6 hours, or a total of (4)(6) = 24 miles. 

The correct answer is (A).

If you’d solved for Yolanda first, you’d have gotten (3 miles per hour)(7 hours) = 21 miles. That’s in the answer choices, but it’s less than half of the total distance, so she wasn’t the one who walked farther. In other words, answer (D) is a trap.

Even if you do know how to solve the problem, it’s important to have done that earlier thinking to realize that the answer must be (A) or (B). That way, when you solve for Yolanda, you won’t accidentally fall for answer (D), since Yolanda’s distance is in the answer choices.

When the problem talks about two people or two angles in a triangle or two whatevers and the problem also tells you what they add up to, the non-asked-for person/angle is almost always going to show up in the answer choices as a trap. You do the math correctly, but you accidentally solve for x when they asked you for y . We’ve all made that mistake. 

Noticing that detail earlier in your process is a great way to avoid accidentally falling for the trap answer during your Solve phase.

(Have Polya and I sold you yet on using the UPS process? I hope so.)

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GMAT Problem Solving #3: Draw it out; Do arithmetic, not algebra; Choose smart numbers

Glance at the answers. Small integers. Kind of close together, so estimation might not be in the cards, but perhaps working backwards (try the answer choices) could work, depending on how the problem itself is set up. (I don’t know yet because I haven’t actually read the problem.)

Now I’m part-way into the first sentence and see the word cylinders . Overall, I’m not a fan of geometry and I really dislike 3D geometry in particular. So as soon as I see that word, part of my brain is thinking, “If this is a hard one, I’m out.”

But I’m going to finish reading it before I decide. Let’s see. Two cylinders, and then they give me some relative info about the height and radius. They’re probably going to ask me something about volume, since the volume formula uses those measures, and scanning ahead: yep, volume.

So now I know I need to jot down the volume formula and I’m also going to draw two cylinders and label them.

I’m going to make sure I note really clearly what I’m trying to solve for. On geometry problems in particular, it’s really easy to solve for something other than the thing they asked you for. And on this one, I’m also making an extra note that the larger cylinder is only half full. I both wrote that down and drew little water lines in the cylinders to cement that fact in my brain.

This is a complex problem, so just pause for a second here. Do you understand everything they told you, including what they asked you to find? If not, this is an excellent time to pick your favorite letter and move on.

If you are going to continue, don’t jump straight to solving. Plan first. (And if you can’t come up with a good plan, that’s another reason to get out.)

The thing that’s annoying me: They keep talking about the dimensions for the two cylinders but they never provide real numbers for any of those dimensions. And boom, now I know how I’m going to solve. When they talk about something but never give you any real numbers for that thing, you’re allowed to pick your own values. Then you can do arithmetic vs. algebra—and we’re all better at working with real numbers than with variables.

My colleagues  and I call this Choosing Smart Numbers. The “Smart” part comes from thinking about what kinds of numbers would work nicely in the problem—make the math a lot less annoying to do.

We usually avoid choosing the numbers 0 or 1 when choosing smart numbers because those numbers can do funny things (e.g., multiplying with a 0 in the mix will always return 0, regardless of the other numbers involved).

And if we have to choose for more than one value, we choose different values. Finally, as I mentioned earlier, we’re looking to choose values that will work nicely in the problem. (Most of the time, this means choosing smallish values.)

Finally, before I start solving, I’m going to ask myself two things: What am I solving for and how much work do I really need to do?

I’m trying to figure out how much oil is in the larger (but only half-full!) cylinder. I know that the full capacity of the smaller cylinder costs $2 and that the oil is charged at the same rate for the larger one. So if I can figure out the relative amount of oil in the larger cylinder, I can figure out how much more (or less) it will cost. For example, if it turns out that the larger cylinder contains twice as much oil as the smaller one, then the cost will also be twice as much.

In the volume formula, the radius has to be squared while the height is only multiplied, so I want to make the radius a lower value. I’m going to choose r = 2 and h = 3.

Use those values to find the relative volumes of the two cylinders. Reminder yet again: The larger cylinder is only half full, so multiply that volume by one-half:

What’s the relative difference between the two? They both contain pi, so ignore that value. The difference is 12 to 48—if you multiply 12 by 4, you get 48.

So the money will also get multiplied by 4: Since the oil in the smaller cylinder costs $2, the oil in the larger one costs (2)(4) = $8. The correct answer is (E).

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Understand, plan, solve on GMAT Quant

Whenever you solve any GMAT Problem Solving (PS) or Data Sufficiency (DS) problem, follow the Understand, Plan, Solve process. Print out this summary and keep it by you when you’re studying:

• Glance at the answers (on PS) or the statements (on DS) and the question stem. Anything jump out—an ugly equation, a diagram, an indication that you might be able to estimate, etc?
• Read the question stem. Focus just on understanding what it’s telling you and what it’s asking you.
• Jot down what it’s asking, along with any other useful info (equations, etc.). Don’t solve! Just jot (write or sketch).
• Reflect on what you know so far. Lost? Guess and move on. But if you do understand everything, then consider what your best plan is. Can you estimate anywhere? How heavily? Can you use a real number and just do arithmetic? Is there a way to draw or logic it out? What are they really asking you? This reflection is how I realized I just needed a relative value on the Oil Cylinders problem.
• Organize your thoughts and your scratch work to get set up for the Solve stage. Maybe you need to redraw or add something to your diagram, as I did for Yolanda and Bob. Maybe you need to group the data or equations a little differently, as I did on the Membership problem.
• Don’t have a plan you feel pretty good* about? Forget it—guess and move on. (*You don’t have to feel 100% confident. But you want to feel like it’s a decent plan. If you don’t, let it go.)
• Be systematic. You’re almost there. Write your work down. Don’t try to compress steps or work more quickly than is comfortable for you. Keep your scratch paper organized.
• Don’t do more work than you have to. Estimate when you can. Keep an eye on the answers as you work. Eliminate impossible answers as you go. Stop as soon as only one answer letter is left.
• Be willing to bail. Even if you understand and have a decent plan, you still might get stuck. Don’t start trying some other plan at that point. Something’s not working with this one; guess and go spend your time on a better opportunity later in the test.

Finally, remember your overall goal here: You want to go to business school. The point is not to show how much of a mathematics scholar you are. The point is to learn how to think logically about quant topics—with, yes, some amount of actual textbook math tossed in there. 

Actively look for the Logic It Out / Draw It Out / Quick and Dirty approaches. They’ll not only save you time and stress on GMAT Problem Solving and Data Sufficiency, but they’ll also help train your brain for quant discussions in business school and in the boardroom.

Want more strategies to improve your GMAT Problem Solving skills? Sign up for Manhattan Prep’s free GMAT Starter Kit  and check out the section on Foundations of Math.

Happy studying!

She’s been teaching people to take standardized tests for more than 20 years and the GMAT is her favorite (shh, don’t tell the other tests). Her favorite teaching moment is when she sees her students’ eyes light up because they suddenly thoroughly get how to approach a particular problem.

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The Aptitude test assesses the ability to use reason to solve problems which involve rigorous and methodical thinking skills.

The assessment includes work-sample tasks such as:

Understanding numerical data in order to calculate accurate answers.

Analyzing patterns in information and evidence to arrive at correct conclusions.

Evaluating language to summarize information and make the right decisions.

Good analysts, managers, and developers all need to be adept with these analytical, problem-solving, and communication skills.

Sample public questions

After saving for years, John's parents bought him his first car for his birthday. He was so happy that they decided to keep buying him cars on future birthdays that came after the same interval of years.

Today they bought him yet another car—38 years after the first.

Here are the birthday cars.

How old was John when he got his first car from his parents?

The company's sick leave policy says:

This company’s sick leave policy applies to all our employees who have been with our company for more than six months. Our employees can take sick leave only when they want to recover from a sudden illness, accident, or injury. They can use up to 10 days of sick leave for these purposes per calendar year. Upon completion of each 12 month period of employment, employees will receive 3 additional days of sick leave for every completed 12 month period of working for the company. Unspent additional sick leave days cannot be transferred to the next 12 month period. For example, an employee who has worked for the company for 10 years would receive 30 additional days of sick leave, or 40 in total. Keep in mind that employees who become sick should either use their sick days or work from home to avoid spreading illnesses.

Fill in the blanks, with numbers, for the cases below. Enter the number 0 for cases when an employee doesn't have the right to sick leave, according to the policy.

• Emily, who has worked for us for almost half a year and still hasn't used any sick leave days, has the right to use up to __ sick leave days this year.
• Faith, who just started her 3rd year of working for us and has not taken sick leave until now, had a car accident. She has the right to use up to __ days of sick leave to recover from her injuries.
• Vanessa, who started to work for us a little over a year ago, took 2 days of sick leave last year. She was Faith's passenger in the car accident. Vanessa has the right to use up to __ days of sick leave to recover from her injuries.
• Edith, who began working for us 8 months ago, has the right to use up to __ sick leave days for a volunteer adult care program at a local hospital that lasts the whole day.
• If Gabrielle, who has worked for us for 20 months, and who took a 10-day sick leave immediately after her 6-month trial period had finished, gets ill tomorrow, she will have the right to use up to __  days of sick leave.

Select the tile below that should be placed in the unknown tile above so that all three rows above follow the same pattern.

You are having a discussion with your friend about the apps you both use.

Every app your friend uses, you also use. Spreadsheet is the app you use the most. You don't use the Calculator app at all.

With regard to what’s written above, select which of the following statements are true.

John has a bag of apples, and two brothers, Jack and Jim.

The image below shows the number of apples that John has eaten and put in the family's compost barrel.

John then splits the remainder equally between himself and each one of his brothers. Jack and Jim both eat half the apples they have been given. Together the three brothers have 4 apples left.

How many apples did John start with?

Carefully read the following excerpt from an article on paper production:

With the recovery rate of used paper for recycling approaching 70 percent in the United States and Europe, and approaching 80 percent in Japan, to keep up with our demand for paper we need to continue using fresh fiber as well as recycled, according to the premise set forward in a new report by the World Business Council for Sustainable Development (WBCSD). Detailing the reasons why fresh fiber combined with recycled is important for a single integrated wood fiber system, the report examines the “complementarity” of using both and discusses the functions of different types of fibers and the issues related to both recycling old fiber and sourcing fresh fiber. In 2012, 400 million tons of paper and paperboard were produced and consumed globally, which is double that in 1985, notes the report. As the population continues to expand and standards of living increase, this number is expected to climb by another 40 percent by 2028. While many would advocate for cutting down on paper use in the first place, in the face of global demand the most sustainable fiber may have to be the next best thing. But therein lies the rub: finding adequate quantities of said fiber will be the challenge of the global pulp and paper industry.

Which statement most effectively summarizes the text?

In an upcoming referendum, voters will be asked whether the minimum wage should be increased. Telephone polls of likely voters were conducted to predict what the result might be. All respondents were put into at least one of six categories based on profession and age: scientist, lawyer, hourly wage earner, small business owner, younger voter, older voter. The opinion polls showed: A majority of voters are in favor of keeping the current minimum wage. A majority of scientists and lawyers supported a rise in the minimum wage. Younger voters are more inclined to support a rise in the minimum wage. Older voters tend to support no change. Hourly wage earners overwhelmingly supported a rise in the minimum wage. Small business owners were evenly split on the subject.

Select all the statements that can be concluded from the above text:

An author writes an argumentative essay to persuade readers to agree with a claim about a topic. When writing an argumentative essay, it's important to establish credibility with readers to convince them that the author is trustworthy. True statements, accurate evidence, and clear logic increase an author's credibility. However, false statements, inaccurate evidence, and unclear logic make an author less credible. With lower credibility, an author is less likely to persuade readers to agree with a claim--even when it is trustworthy.

Select the statement that most effectively summarizes the above text:

Five types of tires were tested in three different driving conditions.

Consider the following table of their test scores:

If a tire type scored 4 or less in any category, it failed the test.

What is the highest average score of the tire types that passed the test?

In which year did the company make the largest profit per employee?

The charts below show the number of cars John sold last year and the profit he made per car.

Considering all car types, in which quarter did John make the largest average profit per car?

Which product line saw the largest absolute increase in income (dollar amount, not percentage) in the second half of the year compared to the first half of the year?

The charts below show the number of cars John sold last year and the profit per car he made.

How much did John earn last year?

No matter how tired Billy is, he never slows down or stops painting. In fact, it would only take Billy 4 hours to paint a fence by himself. It would take Suzy 6 hours to paint the same fence by herself.

On Friday, Billy and Suzy worked together to paint the fence, but Billy got tired after only 2 hours.

How much time did it take to paint the whole fence?

Cats are common pets throughout the world, and their worldwide population exceeds 500 million. Cats are the second most popular pet in the U.S. by number of pets owned, behind freshwater fish. Although cat ownership has commonly been associated with women, research has shown that men and women in the U.S. are equally likely to own a cat. They are ranked as the third most popular pet in the U.K. by number of pets owned, after fish and dogs, with 8 million being owned.

What can be concluded from the text above?

In the summer, Agathe wore an ill-fitting sun hat and oversized sunglasses to the beach in Greece. Both slipped off when she fell asleep in her swimsuit on the beach at 2 p.m.

What will most likely happen to Agathe?

Dan put a large bet on a horse. However, a day before the race, the horse was injured.

Select which of the following statements are true:

Germany's economy avoided falling into recession during the final three months of last year. This means that it avoided two consecutive quarters of negative GDP growth, which is the definition of a recession. This was a small, but positive, surprise for all analysts who, after the July-to-September period that featured a 0.3% decline, predicted the continuation of this negative trend. Reasons for slower growth last year include a slowdown in the global economy and a weaker car sector, with some German consumers less willing to buy new cars amid confusion over new emission standards. Joe Johnson, senior financial analyst, told the BBC that US tariffs on EU car exports, which US President Donald Trump has threatened, could have a major impact on Germany. He thinks that, if this happens, Germany might fall into recession.

It has rained continuously for 15 days. Tomorrow, an important football match is being held in an outdoor stadium.

An eagerly awaited new album has been leaked several hours before its official release. Listeners are now able to download the album for free.

How will this affect album sales?

A postcard and a stamp together cost $1.50. The postcard costs one dollar more than the stamp.

How much does the stamp cost?

A brother and sister own equal parts in a company. The minority shareholders have the remaining 12,000 shares or 30% of the company.

What is the number of shares that the sister owns?

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Skills and topics tested

• Verbal Reasoning
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• True Statement
• Logical Reasoning
• Deductive Reasoning
• Fallacy of the Undistributed Middle
• Existential Fallacy
• Modal Fallacy
• Fallacy of Exclusive Premises
• Fallacy of Division
• Fallacy of Composition
• Gambler's Fallacy
• Affirming a Disjunct
• Masked-Man Fallacy
• Numerical Reasoning
• Divide and Conquer
• Working with Time
• Linear Equations
• Table Lookup
• Dependent Events
• Probability
• Correlation and Causation
• Attribute Substitution
• Abstract Reasoning
• Duplicate Elements
• Three Horizontal Rows
• Numeric Representation
• Rotating Elements
• Exclusive Elements
• Additive Elements
• Chart Lookup
• Percentages
• Most Effective Summary
• Newspaper Excerpts
• Critical Thinking
• Understanding Rules
• Argument from Fallacy
• Affirming the Consequent
• Double Counting
• Circular Reasoning
• Illicit Major
• Conjunction Fallacy
• Arithmetic Mean
• Venn Diagram

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• Prof. Gilbert Strang

Departments

• Mathematics

As Taught In

• Linear Algebra

Learning Resource Types

Final exam problem solving.

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Analytical Reasoning Tests

• 538 questions

Analytical reasoning tests examine an individual’s ability to apply logic to solve problems. The questions vary depending on the type of analytical reasoning test you’re taking: from extracting key information from complex passages of text (verbal reasoning), to looking for patterns in a series of images (non-verbal reasoning), or using given information to draw conclusions or make predictions (inductive and deductive reasoning).

What is an analytical reasoning test?

An analytical reasoning test is a type of aptitude test that is often used by employers to assess a job candidate’s ability to think critically and solve complex problems.

As well as these skills, employers want to see evidence that you can keep calm under pressure and work quickly against the clock.

The test is nearly always timed, meaning you don’t have long to work through each question to find the correct answer.

Depending on which type of job you’re applying for, the test you take may be in the style of verbal reasoning , non-verbal reasoning , inductive reasoning or deductive reasoning . As such, it’s worth practicing as many different types of tests as you can to familiarise yourself with the questions.

The analytical reasoning test is widely used because it looks for skills that are sought after in almost every industry. It helps employers find candidates who will be quick to learn, adapt and solve problems.

Why do use analytical reasoning tests?

Employers use analytical reasoning tests to assess candidates’ ability to analyze complex information, make logical deductions, and solve problems effectively. These tests help employers evaluate candidates’ critical thinking skills, decision-making abilities, and aptitude for handling challenging situations. By administering analytical reasoning tests, employers can identify candidates who possess the cognitive abilities necessary for success in roles that require analytical thinking, such as management, finance, engineering, and data analysis. Additionally, these tests provide employers with valuable insights into candidates’ problem-solving approaches and their capacity to navigate intricate scenarios, aiding in the selection of the most suitable candidates for the job.

As applicants have to work harder and harder to make their CV stand out, an aptitude test like this is a good way of ensuring candidates possess the necessary skills.

It’s common for employers or recruiters to set the analytical reasoning test before the interview stage, so they can select candidates based on their test performance. The test therefore acts as a filter, ensuring employers get to meet the people they believe are most likely to excel.

Completing a good analytical reasoning test gives an indication that you’re a strong critical thinker who can rise to the challenge – an attractive proposition for any employer.

How do analytical reasoning tests work?

An employer will select the type of analytical reasoning test (verbal, non-verbal, inductive or deductive) based on the skills they want to examine.

Finding out exactly which type of test you’ll be taking is helpful so you can focus your preparation, but if you don’t know we recommend trying out all of the different mock tests to familiarise yourself with the individual question styles and formats.

When you take the test, you’ll normally have around one minute to answer each question – which is yet another reason to familiarise yourself with the kinds of questions you’re likely to be asked.

Here’s a brief overview of the four different test types:

• Verbal reasoning – requires you to read through long passages of text and showcase your comprehension and analysis skills by answering a series of questions on what you’ve just read.
• Non-verbal reasoning – presents you with images such as graphs, pictures and patterns, and requires you to use your logic and problem-solving skills to decipher the rule that connects the sequence.
• Inductive reasoning – equips you with certain facts or information, and then asks you to make predictions or assumptions based on that evidence.
• Deductive reasoning – will ask you to use the statements given to you to make further statements of fact.

After the test, your score will be calculated and compared to those of the other individuals who took the same test, or a normative group (which can help an employer see how well you fared compared to previous candidates).

Analytical test formats

Verbal Reasoning Tests

Verbal reasoning tests examine your ability to draw out key information from long, often complex passages of text, to form a conclusion. Very often this takes the form of questions to which you would select ‘true’, ‘false’ or ‘cannot say’ as the response.

No prior knowledge of the subject matter is required, but it is important to practice verbal reasoning tests as it can take a while to get used to the question format.

You will need to be able to distinguish between what’s fact and what’s merely being inferred when you’re reading through the passages of text. This shows an employer that you have the comprehension, logic and analytical skills they’re looking for.

Practising verbal reasoning tests before you take the one that really matters is vital if you want to showcase the best of your abilities to a potential employer. The more mock tests you take, the better you’ll get at sifting through the passages of text for evidence, quickly assimilating the information and confidently deciding what’s true, false or uncertain.

You’ll normally have around one minute to answer each question on the verbal reasoning test (although it’s always worth checking this is the case with your test when you begin). It’s important you don’t spend ages on a challenging problem, as you could end up not answering other questions that you might have easily been able to answer.

At the end, if you have time left you can always go back to anything you weren’t sure about and have another go.

The verbal reasoning test is most commonly used by employers or recruiters hiring for roles where strong communication skills are critical – which applies to most jobs, hence their popularity.

Non-Verbal Reasoning Tests

Non-verbal reasoning tests comprise graphs, tables and data, and the accompanying questions will assess how adept you are at drawing conclusions from limited information, finding connecting patterns and working quickly under considerable time pressure.

These types of analytical reasoning tests are often part of the application process for roles in industries such as finance, engineering and HR.

The best way you can prepare for a non-verbal reasoning test is to take as many mock tests as you can. After you’ve completed a test, it’s important to look back through your answers and identify your weaker areas, so you know where you need to direct your focus.

Not only will practising ensure you get quicker and better, it’ll also help you familiarise yourself with the different graphs, tables and images you’re likely to be confronted with on a non-verbal reasoning test.

As with the verbal reasoning test, you normally get around one minute to answer each question, so finding the right balance between speed and accuracy is really important – something that you’ll find a lot easier if you’ve put the practice time in beforehand.

A successful non-verbal reasoning test will prove to an employer that you have the critical thinking, reasoning and logical skills needed to cope with the demands of the job you’re applying for.

Inductive / Deductive Reasoning Tests

If you’re asked to take an inductive test or deductive test , you’re essentially being asked to show how well you can identify patterns and use your logic. Although the overall skills you’ll demonstrate are very similar, the two tests are slightly different.

Inductive reasoning test – you’ll need to identify relationships between statements, images or facts and figures, and use this analysis to show, logically, what should come next.

Deductive reasoning test – you’ll be given a statement of fact and you’ll need to use this information to deduce another factually correct statement.

These aptitude tests are most commonly used in the hiring of science, tech and IT roles, as the type of skills they seek to showcase – logical thinking, identifying patterns, problem solving and critical thinking – are all valuable in these industries.

So even if you have the type of brain that finds these kinds of problems easier than most, it’s always worth practising inductive/deductive reasoning tests beforehand to familiarise yourself with the specific style of question, and what’s required of you in a short amount of time.

Prepare yourself for leading employers

5 Free Example Analytical Reasoning Questions

Here are five example analytical questions to try out. Answers for all five are below the tests. If you need further practice, try out our full free tests.

Verbal Question 1

Statement : A derivative could be used by an airline to secure the price of oil now, which it won’t use until six months time.

Verbal Question 2

Statement : More people taking early retirement is the major contributory factor to the public sector pension deficit.

Diagrammatic Question 1

Which is the next logical image in the sequence?

Numerical Question 1

What was the ratio of the cost of a Google click in April compared to the cost of a Facebook and Yahoo click in February?

Abstract Question 1

Which of the boxes comes next in the sequence?

Verbal Question 1 : True – “to secure the price of a commodity which is to be “bought” at a future date, but at a price that is set today.”

Verbal Question 2 : Cannot tell – the passage refers to both this fact, extended life expectancy, and that the value of pension fund assets has fallen.

Diagrammatic Question 1 : There is a central figure and four figures with one in each corner: (i) The central figure firstly increases in size over a series of three, then decreases in the same fashion; (ii) The central figure changes from white, to having a dotted outline, to black; and (iii) The four figures rotate around the four corners, moving two corners at a time. So the correct answer is F.

Numerical Question 1 : Step 1. Extract the relevant figures from the graph Cost of per click in April. Google 18 cents : Facebook + Yahoo (14 + 6 = 20 cents). Step 2. Divide 20 by 18 to calculate the ratio. 20 ÷18 = 1.11 Step 3. Present as a ratio 1 : 1.11

Abstract Question 1 : Arrow changes direction from pointing up, to pointing down, with each turn. 2. Triangle moves from top left corner in an anti-clockwise direction around the frame with each turn. So the answer is B.

I could prepare for specific companies and industries. It’s been an invaluable resource.

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Analytical Reasoning Tests Tips

1 background research.

Get as much background information as possible on the test you’ll be taking from the employer or recruiter, so you know which areas to focus on.

2 Prepare with mock tests

Preparation is key – take mock tests in a quiet, distraction-free area and always make sure you go back through your answers at the end to identify any areas you need to work harder at.

3 Tips for test day

On the test day itself, make sure you have everything you need to complete the test. When you start, ensure you know roughly how long you’ve got to answer each question, as although you’ll always need to work quickly, it’s important to read the question thoroughly and ensure you’ve understood it before getting started.

4 Stay positive

Try and remain positive. The tests are designed to be challenging, since employers want to push you. If you’ve put the time and effort into practising aptitude tests, you should feel confident you’ve given yourself the best chance possible to succeed.

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Analytical reasoning 01.

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Analytical reasoning 03, improve your scores with our intelligent learning system, prepare for your analytical reasoning test.

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• 30 Numerical reasoning tests
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Reviews of our Analytical Reasoning tests

What our customers say about our Analytical Reasoning tests

Bob Gautier

United States of America

October 23, 2023

I really do not think negatively in any way about this test. It dies what it’s supposed to do, and designed to do what it does.

Andrew Smith

United Kingdom

October 05, 2023

A good range of alternating patterns, some repeat themselves on several questions, while others are one-offs.

Caramel Teoh

The seqence

I like how convenient it was to answer to question. I dislike that all the question is almost all the same

Juan Garcera

August 06, 2023

Interesting

It is a good first immersion on the complexity of analytical reasoning and a good first step to get into more demanding exercises.

Stephanie Scalzo

July 25, 2023

Find patterns, but attack each question individually

I have not had the opportunity to take a test like this in years! It was really cool to use my brain in this kind of way again and to work through each individual problem while also finding patterns throughout the test.

MemeLord 29

July 13, 2023

Understanding the sequences

I liked the fact you had to use logical thinking and process of elimination sometimes, to figure the answer

Simulation Aeronautics

July 09, 2023

Attention to detail

The shapes in the pattern have changes which require sharp attention to detail to select the next sequence.

Marco Cavallari

June 03, 2023

My 1st ever psychometric test

It was quite challenging at first, but after a while it became more and more easier to find patterns.

Elizabeth M.Calinawan

Philippines

May 31, 2023

The refreshing abstract reasoning

i like the test very much. Refreshing the next sequence, need enough time to think it over but with the time limit. Yeah, very interesting this test too. Well, when.this test refer to a real life of course anticipation in the area may prevail have a swift solution in every conce

khadijah Ansari

May 16, 2023

My brain had a hard time focusing and differentiating between them, trying to recognise a pattern was difficult.

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Critical Thinking Test: Sample Questions with Explanations (2024)

Employers value and seek candidates who demonstrate advanced critical thinking skills. They often administer critical thinking tests as part of their hiring process. Critical thinking tests can be very difficult for those who don’t prepare. A great way to start practicing is by taking our critical thinking free practice test.

What Does The Critical Thinking Test Include?

The Critical Thinking Test assesses your capacity to think critically and form logical conclusions when given written information. Critical thinking tests are generally used in job recruitment processes, in the legal sector. These tests measure the analytical critical thinking abilities of a candidate.

Why Is Critical Thinking Useful?

Critical thinking is put into action in various stages of decision-making and problem-solving tasks:

• Identify the problem
• Choose suitable information to find the solution
• Identify the assumptions that are implied and written in the text
• Form hypotheses and choose the most suitable and credible answers
• Form well-founded conclusions and determine the soundness of inferences

What is Watson Glaser Test and what Critical Thinking Skills it Measures?

The most common type of critical thinking test is the Watson-Glaser Critical Thinking Appraisal (W-GCTA). Typically used by legal and financial organizations, as well as management businesses, a Watson Glaser test is created to assess candidates’ critical thinking skills.

The test consists of 10 questions to be answered in 10 minutes approx (although there is no timer on the test itself). Our test is slightly harder than the real thing, to make it sufficiently challenging practice.

You need to get 70% correct to pass the test. Don’t forget to first check out the test techniques section further down this page beforehand.

Questions          25

Pass percentage          70%.

The test is broken down into five central areas:

• Assumptions
• Interpretation

Critical Thinking Course

• 1 BONUS Interview Prep Video Guide Buy this Course: Get full access to all lessons, practice tests and guides.

The Five Critical Thinking Skills Explained

1. recognition of assumption.

You’ll be presented with a statement. The statement is then followed by several proposed assumptions. When answering, you must work out if an assumption was made or if an assumption was not made in the statement. An assumption is a proclamation that an individual takes for granted. This section of the tests measures your ability to withhold from forming assumptions about things that are not necessarily correct.

• 1: Assumption Made
• 2: Assumption Not Made

Although the passage does state that Charlie’s fundraising team is doing its best so that the charity event can meet its goal, nowhere did it state that their team is leading the event.

2. Evaluation of Arguments

You will be presented with an argument. You will then be asked to decide whether the argument is strong or weak. An argument is considered strong if it directly connects to the statement provided, and is believed to be significant.

No, participation awards should not be given in every competition because studies have shown that this would cause the participants to put in less effort because they will get a prize no matter what the outcome is.

• 1: Strong Argument
• 2: Weak Argument

This is a strong argument as it provides evidence as to why participation awards should not be given in every competition

3. Deductions

In deduction questions, you will need to form conclusions based solely on the information provided in the question and not based on your knowledge. You will be given a small passage of information and you will need to evaluate a list of deductions made based on that passage. If the conclusion cannot be formed for the information provided, then the conclusion does not follow. The answer must be entirely founded on the statements made and not on conclusions drawn from your knowledge.

In a surprise party for Donna, Edna arrived after Felix and Gary did. Kelly arrived before Felix and Gary did.

• 1: Conclusion Follows
• 2: Conclusion Does not Follow

For questions like this, jot down the clues to help you out. Use initials as a quick reference.

K | F&G | E

Looking at the simple diagram, “K”, which stands for “Kelly,” arrived before Edna “E” did. The answer is A.

4. Interpretation

In these questions, you are given a passage of information followed by a list of possible conclusions. You will need to interpret the information in the paragraph and determine whether or not each conclusion follows, based solely on the information given.

A number of students were given the following advice:

“The use of powerful words is a technique, which makes you a better writer. Your choice of words is very important in molding the way people interaction with the article. You should use powerful words to spice up your article. Power words should be used liberally to enhance the flavor of what you write! ”

In the fourth sentence, it is stated, “Power words should be used liberally to enhance the flavor of what you write!”

Thus, if you were to write an essay, using powerful words can give more flavor to it.

5. Inferences

An inference is a conclusion made from observed or supposed facts and details. It is information that is not apparent in the information provided but rather is extracted from it. In this section, you will be provided with a passage of information about a specific scene or event. A list of possible inferences will then be given, and you will need to decide if they are ‘true’, ‘false’, ‘possibly true’, ‘possibly false’, or whether it is not possible to say based on the information provided.

With the advancement of technology, the need for more infrastructure has never been higher. According to the plan of the current U.S. Administration, it aims to put a $1 trillion investment on improving infrastructure, a portion of which will include priority projects and technologies that can strengthen its economic competitiveness such as transportation, 5G wireless communication technology, rural broadband technologies, advanced manufacturing technologies, and even artificial intelligence.

It stated that it expects to work with Congress to develop a comprehensive infrastructure package, which is expected to have a budget of $200 billion for certain priorities.

• 2: Probably True
• 3: Not Enough Information
• 4: Probably False

Although it was mentioned in the passage that the U.S. government is to allocate $200 billion on certain priorities, it did not specify if these certain priorities were for ‘transportation, 5G wireless communication technology, rural broadband technologies, advanced manufacturing technologies, and artificial intelligence’ or if the aforementioned priorities will have a different allocation.

What we can be sure of, however, is that at least a portion of the $1 trillion infrastructure budget will be used on the mentioned priorities regardless, meaning that there is a chance that $200 billion will be used on those aforementioned areas.

Improve Your Score with Prepterminal’s Critical Thinking Course

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Logical Reasoning Tests

Practice tests, solutions, and tips to help you pass employers' logical reasoning tests.

Page contents:

What is a logical reasoning test, how assessmentday can help with logical reasoning tests, logical reasoning video tutorial - part 1.

• Free example logical reasoning tests

Logical Reasoning Video Tutorial - Part 2

Different types of logical reasoning, the most common logical reasoning tests used by employers, general logical reasoning test advice, logical reasoning faqs.

Updated: 17 January 2023

A logical reasoning test is used measure a candidate’s problem solving ability. They assess the ability to come to conclusions based on logic. You are presented with a series of shapes and are required to find patterns and rules to help you find the correct answer. These tests may be encountered for any position at any level of recruitment, but they may be particularly common when recruiting for positions which require significant problem solving ability or higher use of logic.

What is an example of logical reasoning?

Here are screenshots of our logical reasoning tests to understand what an example question involves:

AssessmentDay offer numerous types of logical reasoning test which can help you perform to your best in the real thing. Practising logical reasoning tests is an ideal method of preparation as it allows you to learn from your mistakes, improving performance with every practice trial. Similarly experiencing time limits, the test layout and the overall test experience can help ease worries and anxieties about the test by familiarising yourself with them. It goes without saying that a candidate that has undertaken a logical reasoning test numerous times and seen their prior mistakes, and learned from them will be less nervous than a first time test candidate.

Free practice logical reasoning tests

Free logical reasoning test 1.

This free logical reasoning test contains 10 questions and has a time limit of 70 seconds per question .

Free Logical Reasoning Test 2

Logical reasoning test 1.

• 12 questions

Logical Reasoning Test 2

Logical reasoning test 3, logical reasoning test 4.

There are numerous types of logical reasoning test, and many of these are used interchangeably. These tests tend to be similar in their layout and methodology, but with subtle and important differences.

Survey results

We analysed a sample of logic-based tests, to find the most common terms/most popular type was: Inductive reasoning

Here is a breakdown of the most common logical ability tests:

• Inductive reasoning: - Inductive reasoning is the ability to reach general conclusion based on perceived patterns observed in specific events. Inductive logic is often used in everyday life and is therefore practical to a work place environment. In these tests candidates will be provided with a series of diagrams with an evident pattern. Candidates will need to identify the pattern in the sequence of diagrams and select the next diagram in the sequence.
• Deductive reasoning: - Deductive reasoning involves a general rule or principle that leads to a specific conclusion. These tests will evaluate and measure a candidate's ability to make logical arguments and draw sound conclusions based on provided data, as well as identify flaws in a piece of information. As a result this is a useful tool in selection procedures as this type of reasoning will be used in the workplace. This type of reasoning will often be used in verbal reasoning tests and numerical tests, and is therefore very likely to be encountered in recruitment processes.
• Abstract reasoning: - Abstract reasoning, also known as conceptual reasoning measures your lateral thinking ability. In these tests candidates will be tested on their ability to identify relationships, patterns and trends. Candidates will be provided with a series of images that follow a logical sequence or underlying rules. This may include following a rule in a sequence, identifying a code or finding a missing diagram.
• Diagrammatic reasoning: - Diagrammatic reasoning is a specific form of abstract reasoning. Tests which assess this ability will typically show a flowchart of diagrams and symbols, with an input and an output. Candidates will need to identify which inputs effect diagrams, and therefore generate a specific output based on those rules.
• Critical thinking: - Critical thinking tests are a type of verbal critical reasoning task which assesses various different types of logical reasoning in arguments, assumptions and conclusions. Typical logical abilities tested include analysing arguments, making inferences and evaluating conclusions.

Did You Know

Different test publishers use different names for their assessments. The term logical reasoning is used by TalentQ. Other companies may call their test abstract, inductive, or diagrammatic reasoning. It is good advice when being asked to sit a logical reasoning test to speak to the person who invited you and ask for a bit more detail; they may even give you a few example questions so you know what to expect.

Our 2020 study asked candidates about their logical reasoning test experience, in doing so we managed to find the most popular test publishers from our sample:

• 1. Talent Q Elements Logical Ability - the important feature of these tests is that they are adaptive. That is to say the difficulty of each question is automatically determined by your performance in the previous question. So the questions become more difficult as you progress in order to quickly find your level of logical reasoning ability. There are typically 12 questions to these TalentQ logical tests and a time limit of 75 seconds per question.
• 2. Kenexa Logical Reasoning - this test published by Kenexa is actually very similar in style to what SHL call an inductive reasoning test. They are effectively the same thing; the candidate is asked to select which diagram fits within the given series from a choice of five options. Typically Kenexa will give the candidate 20 minutes for 24 questions for their logical reasoning test.
• 3. Ravens Progressive Matricies (Ravens APM / Ravens SPM) - The grid-style of symbols each following a pattern is also used in the Ravens Progressive Matrices assessments. With Raven's logical test, there are two levels of this test: Advanced Progressive Matrices (23 questions, 42 minutes) and Standard Progressive Matrices (28 questions, 47 minutes). Our logical tests are suitable for Raven's APM-III and Raven's SPM tests, you can alter the time limit with of our tests to create a more authentic experience.

Although all tests evaluate a specific logical ability, or set of abilities, there are general strategies which can be applied to ensure maximum performance in a logical reasoning test.

Here is a list of useful tips and advice for logical reasoning tests:

• 1. Stay calm: - Logical reasoning tests of all kinds can be nerve racking, particularly ones which are time limited. As a result it is important to stay calm as to allow optimum performance during your exam. A small amount of anxiety can be a performance booster, maximise focus and therefore performance. However, serious test anxiety can severely hamper performance. Proper practice, enough sleep the night before and deep and regular breathing can all help settle your nerves, and perform to your best on the day of your test.
• 2. Research the type of test: - Learning as much about the test beforehand can help you dive straight into the test once you have received it, saving you time. Similarly after researching the test, and the logical abilities which it assesses, can help you hone these skills and ensure you demonstrate the particular aptitude required for the test, optimising your performance.
• 3. Clarify what type of test: - If an employer states that you will need to undertake a logical reasoning test, it is important to gauge what type of logical reasoning will be tested due to the broad nature of logical reasoning. Don’t be afraid to ask for clarification to identify which logical reasoning test will be used, and which logical reasoning skill will be tested as this information will be invaluable for your pre test preparation.
• 4. Figure out the answer first: - A general tip for logical reasoning tests is to figure out the correct answer/sequence/rule before looking at the multiple choices. This way once you have an idea in your head of the correct answer, you can simply pick it out. If you look at the multiple choice answers first, you will be more inclined to pick the answer which best looks like the correct answer, rather than take the time to evaluate it logically. Your logic will be subject to more bias if you base your answer on which answer seems correct on face value, instead of evaluating it using the logical skills being tested.
• For more advice on logical reasoning tests, check out our logical reasoning tips where we go through an example question and give you advice on how to pass logical tests.

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Yes, logical reasoning is a skill just like numerical reasoning which can be developed and practised. Some people will naturally be talented with logical reasoning and be able to solve logical puzzles much easier than others. Logical reasoning involves being able to solve logic puzzles and draw conclusions from patterns.

Logical reasoning is important for your ability to solve problems and generate creative ideas. It's this reason that many employers use logical reasoning tests in their application process.

The best way to practise logic skills is by using logical reasoning tests. These will provide the best practise as they directly involve all the skills needed in solving logic problems. You can also practise things like word puzzles or any kind of puzzle that requires you to identify patterns to find answers.

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Free Math Worksheets — Over 100k free practice problems on Khan Academy

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That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

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Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

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High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Addition and subtraction
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• Measurement and data
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• Place value
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• Add and subtract within 20
• Add and subtract within 100
• Add and subtract within 1,000
• Money and time
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• Intro to multiplication
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• Add decimals
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• Multi-digit multiplication and division
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• Foundations
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• Algebra foundations
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• Exponents & radicals
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• Irrational numbers
• Performing transformations
• Transformation properties and proofs
• Right triangles & trigonometry
• Non-right triangles & trigonometry (Advanced)
• Analytic geometry
• Conic sections
• Solid geometry
• Polynomial arithmetic
• Complex numbers
• Polynomial factorization
• Polynomial division
• Polynomial graphs
• Rational exponents and radicals
• Exponential models
• Transformations of functions
• Rational functions
• Trigonometric functions
• Non-right triangles & trigonometry
• Trigonometric equations and identities
• Analyzing categorical data
• Displaying and comparing quantitative data
• Summarizing quantitative data
• Modeling data distributions
• Exploring bivariate numerical data
• Study design
• Probability
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• Two-sample inference for the difference between groups
• Inference for categorical data (chi-square tests)
• Advanced regression (inference and transforming)
• Analysis of variance (ANOVA)
• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
• Normal distributions
• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
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• Limits and continuity
• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
• Applications of derivatives
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• Applications of integrals
• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
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• Integrals review
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• Applications of multivariable derivatives
• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

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(TOPS-3:E) Test of Problem Solving-3:Elementary

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About This Product

Test purpose.

The TOPS 3 Elementary assesses a school-aged child's ability to integrate semantic and linguistic knowledge with reasoning ability by way of picture stimuli and verbal responses.

Test Description

TOPS 3 Elementary focuses on the student's linguistic ability to think and reason.  Language competence is the overall indicator of how a child's language skills affect his ability to think, reason, problem solve, infer, classify, associate, predict, determine causes, sequence, and understand directions.  The TOPS 3 Elementary test questions focus on a broad range of language-based thinking skills, including clarifying, analyzing, generating solutions, evaluating, and affective thinking.

While other tests may assess students' thinking skills by tapping mathematical, spatial, or nonverbal potential, the TOPS 3 Elementary measures discrete skills that form the foundation of language-based thinking, reasoning, and problem-solving abilities.

The test is composed of 18 situations that examine six thinking tasks.  Carefully selected items and situations are relevant to most students and common across cultures and in most schools or home settings.

Although the skills tested on the TOPS 3 Elementary are necessary for developing social competence, it is not primarily a test of pragmatic or social language skills.  Rather, it should be part of a battery of tests/observations used to assess pragmatic competence.

The subtests consist of full-color photographs and questions that address critical thinking skills:

• Subtest A: Making Inferences

The student gives a logical explanation about a situation combining what he knows or can see with previous experiences and background information.  The ability to infer is critical for success in the classroom, academics, and social development.

• Subtest B: Sequencing

The student determines and explains logical, everyday sequences of events.  This skill is critical to academic performance and requires an understanding of the situation, determining the logical sequence of events, and expressing it clearly.

• Subtest C: Negative Questions

The student is asked to explain why something would not occur or why one shouldn't take a given action in a situation.  Responses reveal how well your student notices, attends to, understands, and expresses an appropriate response on this subtest.

• Subtest D: Problem Solving

The student must recognize the problem, think of alternative solutions, evaluate the options, and state an appropriate solution that will work well.  It also includes how to avoid specific problems.

• Subtest E: Predicting

This subtest requires the student to anticipate what will happen in the future.  This requires him to draw from past experiences to reflect on the future.  This skill is an academic as well as a life skill.

• Subtest F: Determining Causes

The student must give a logical reason for a given aspect of the situation in the paragraph.  To be successful, the student must see the relationship between the action and the outcome.

• Examiner Qualifications

The test should only be administered by a trained professional familiar with language disorders (e.g., speech-language pathologist, psychologist).

Test Procedures

• All items are presented in a conversational style with normal intonation and speaking rate.
• The student looks at a picture in the Picture Stimuli Book and answers questions (presented verbally by the examiner) about the picture.
• Each task is presented in its entirety to every student. Basals and ceilings are not used in the TOPS 3 Elementary.  Prompts on the test form are allowed only if the student's response is unclear to the examiner.  It is not used to give the student a "second chance" after a clear, complete but incorrect response.
• Acceptable responses for each test item are indicated on the test form.

Discussion of Performance

The Discussion of Performance section in the Examiner's Manual was developed to guide the examiner to make appropriate and educationally-relevant recommendations for remediation based on a clear understanding of each subtest.

It includes a research-based rationale for the importance of teaching thinking skills, clinically sound information about each task, what is required for the student to be successful, how the task relates to academic and classroom behavior, the specific steps a student goes through to complete each thinking task, and the breakdown of what the student's responses reflect about his thinking skills

Standardization/Statistics

Two studies were conducted on the TOPS 3 Elementary – the item pool and standardization studies.  The item pool study consisted of 690 subjects and the standardization study consisted of 1,406 subjects.  The subjects in both studies represented the latest National Census for race, gender, age, and educational placement.  This included subjects with IEPs for special services but who attend regular education classes.

• Inter-Rater Reliability
• Test-Retest
• Reliability Based on Item Homogeneity (KR20)

The test-retest coefficient is .84 for the total test, the SEM is 9.88 for the total test.  Based on these tests, the TOPS 3 Elementary has satisfactory levels of reliability for all tasks and the total test at all age levels.

• Contrast Groups (t-values): Test discriminates between subjects with normal language development and subjects with language disorders.
• Point Biserial Correlations
• Subtest Intercorrelations
• Correlations Between Subtests and Total Test

The t-Values for differences between normal and language-disordered subjects were significant at the .01 level for five age levels and at the .05 level for two age levels.  The TOPS 3 Elementary clearly discriminates between these groups.  Inspection of all the biserial correlations reveals acceptable levels of item consistency with 85% of the individual items showing statistically significant pass/fail correlations with the task scores.

• Z-tests Chi Square analysis at the subtest level
• Analysis of Variance (ANOVA) F-tests

Of the more than 2,000 z-tests, only a small percentage showed any racial differences.  Percentages ranged from below 1% to 6%.  These low percentages indicate that neither race or SES are strong factors on the TOPS 3 Elementary.

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Tell Us About Your Problem-Solving Administrative Professional

For Administrative Professional Day, share your stories of a staff assistant, administrative assistant or executive assistant who makes a difference

Across Duke, roughly 1,000 administrative professionals in schools and departments help their units function smoothly.

And when things get busy, plans change quickly, or problems arise, these colleagues often find themselves in the middle of the solution.

In advance of National Administrative Professionals Day on April 24, we want to hear how Duke’s staff assistants, administrative assistants or executive assistants use their creativity, resourcefulness and problem-solving skills to help teams navigate the day to day.

Send a note and photo (if you have a picture) by 12 p.m. on April 12, to hr.duke.edu/sendnews . We’ll be in touch to chat more with you.

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Baltimore Investigation Turns to Ship’s Deadly Mechanical Failure

The Dali reported a power blackout and steering problems before hitting the Francis Scott Key Bridge in Baltimore. But what went wrong so far has not been explained.

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By Mike Baker and Peter Eavis

Just minutes before the cargo ship Dali was set to glide under Baltimore’s Francis Scott Key Bridge, the ship’s alarms began to blare. The lights went out. The engine halted. Even the rudder, which the crew uses to maneuver the vessel, was frozen.

As a frantic effort to restore the ship was underway, the pilot soon recognized that the aimless vessel was drifting toward disaster, and called for help.

The cascading collapse of the vessel’s most crucial operating systems left the Dali adrift until it ultimately collided with the Key bridge, knocking the span into the river and killing six people. But as crews this week were still sorting out how to disentangle the ship and recover the bodies of those who died, investigators were also turning to the most central question: What could have caused such a catastrophic failure at the worst possible moment?

Engineers, captains and shipping officials around the world are waiting for that answer in an era when the industry’s largest ships can carry four times as much cargo as those just a few decades ago, navigating through congested urban ports under bridges that may carry tens of thousands of people a day,

Already, a few key questions are emerging, according to engineers and shipping experts monitoring the investigation, and most of them point to the electrical generators that power nearly every system on the 984-foot vessel, not only the lights, navigation and steering, but the pumps that provide fuel, oil and water to the massive diesel engine.

The “ complete blackout ” reported by the pilot is hard to explain in today’s shipping world, in which large commercial vessels now operate with a range of automation, computerized monitoring, and built-in redundancies and backup systems designed to avert just such a calamity.

“In the last 30 to 40 years, the level of that redundancy has been increasing quite considerably,” said John Carlton, a professor of marine engineering at City, University of London. “The ship of today is so very different to the one of 30 years ago.”

Yet there is a wide range of possible factors contributing to the failure that investigators will have to sift through as they interview crew members, examine fuel supplies and scrutinize the ship systems that broke down that night.

If there was faulty maintenance, it could have caused a delay in starting the emergency backup generator, or an electrical fault could have prevented it from remaining engaged. Contaminated fuel or an inadvertently closed valve could have fouled or starved the main generators. Human error could have set off problems or failed to overcome them. The ship’s own automation could have led to equipment glitches. Or a fire could have broken out and damaged crucial equipment.

The answers will have implications not only for international shipping but also for who is liable for damages that S&P Global Ratings estimated at more than $2 billion.

Grace Ocean Private, the Singapore-based company that owns the Dali, said it was “fully cooperating with federal and state government agencies.” Grace Ocean’s owner is Yoshimasa Abe, a Japanese citizen who owns at least two shipping lines and more than 50 vessels, including some of the world’s biggest container ships. While the Dali was insured, Mr. Abe’s company potentially faces large claims against it, depending on the findings of the accident investigators.

Given the scope of the failure, it is possible that there were multiple problems. Timothy McCoy, a professor specializing in marine engineering at the University of Michigan, said that much like a plane crash, an extensive breakdown of a ship’s systems typically involves a sequence of events.

A close look at the potential factors would include many of the most essential elements in the operation of a modern cargo vessel — including the fuel that feeds the ship’s 55,000-horsepower diesel engine that in turn powers the ship’s propeller.

Fuel also powers the huge generators that provide electricity to container ships. And a ship like the Dali needs electrical power to run its main engine — its fuel injectors are electrically powered, for instance — and steer its rudder. Without electricity, the ship can go adrift.

An outbreak of contaminated fuel led to reported problems with 32 vessels from Texas to Singapore last August, maritime industry officials reported , with some of them reporting loss of power and propulsion at sea.

In Washington State last year, a large passenger ferry ran aground after losing power as a result of bacterial and fungal growth in the vessel’s fuel tanks that fouled the ship’s filtration systems.

At the time it was built, 2015, the Dali had four generators, according to S&P Maritime Portal, a shipping data service. Not all of them run at once, usually, but container vessels leaving port will typically have an extra generator running, to provide reserve power if needed. “At least two should be online at the same time,” said Mark Bulaclac, an academic on maritime issues who has also served as an engineer on container ships.

If all generators were running on a common source of bad fuel, that might have caused them all to fail.

Henry Lipian, a forensic crash investigator who previously worked in the Coast Guard, said the sudden loss of the ship’s generators led him to think of fuel problems as a potential culprit.

He said investigators would need to look at the fuel on board, how it was delivered, whether it had been tested beforehand and what filtration systems were on the ship. But he said that a problem with the fuel valves could be another explanation.

“I’d want to start tracing all of those fuel lines,” he said.

In Baltimore, investigators were in the process of collecting a fuel sample from the Dali in order to examine the quality, viscosity and signs of any contaminants, said Jennifer Homendy, the chair of the National Transportation Safety Board.

Yet other experts said there were also reasons to doubt the contaminated fuel scenario. New fuels typically undergo testing, and duplicate filtration systems can help clean out problematic components that were not flagged in testing. No reports have emerged of other ships having a problem from the same batch of fuel.

Maritime engineers say an electrical chain reaction could also have caused all the generators to go down. When one generator fails, it can create a situation in which there is too much demand for too little supply of electricity. Other generators are then at risk of being damaged, so the system will shut them down, too, said Richard Burke, a professor of naval architecture and marine engineering at SUNY Maritime College in New York.

“It’s as if you and I are both holding up a heavy weight and I let go,” he said, “You can’t hold it by yourself, so you drop the weight.”

A haywire generator could also zap the electrical distribution system on the ship, said Capt. Morgan McManus, an instructor at SUNY Maritime College.

When all the main generators fail, ships rely on a backup generator that is typically situated above the water line in another area of the ship, with its own fuel source.

Marine engineers say backup generators provide electricity to run some lights, the navigation system — and, crucially, the ship’s steering system. Without at least backup power, the rudder cannot be moved.

Because some lights came back on after the Dali experienced its initial blackout, it appears the backup generator did activate, but only after a roughly one-minute delay. Even then, the lights appeared to go back off, then on again, raising the possibility of a problem with the backup generator.

Ms. Homendy of the N.T.S.B. said this week that investigators had collected data “consistent with a power outage” but were still trying to determine the extent.

Clay Diamond, the head of the American Pilots’ Association, a trade group that has been in close contact with the harbor pilots in Maryland, said that steering was restored after the emergency generator came online. But even with a hard turn to the left and the dropping of an anchor, there was not enough time to turn or stop the ship.

Mr. Bulaclac, the shipping engineer, said backup generators are meant to be regularly tested by turning them on for two hours once a month. “What I would like to know is when that emergency diesel generator was last tested,” he said.

The Coast Guard inspected the Dali when it docked in the Port of New York in September but found no deficiencies on the ship. The Coast Guard did not provide details of what it inspected.

The modernization of ships may have introduced other ways vessels can fail. They have increasingly depended on computers to monitor for troubles and take action when a problem is identified. In some ways, this is a built-in layer of automatic protection: If one component gets overloaded, it can be automatically shut down to prevent further damage. But those shutdowns can cause problems on their own.

“I could not rule out that some computer failure shut all the valves off or shut off pumps that provide the fuel,” Mr. Lipian said.

Michael Forsythe and Jenny Gross contributed reporting.

Mike Baker is a national reporter for The Times, based in Seattle. More about Mike Baker

Peter Eavis reports on business, financial markets, the economy and companies across different sectors. More about Peter Eavis

North Korea aims to adopt solid-fuel missiles for faster launches

• Tuesday's launch condemned by Japan, S.Korea and U.S.
• Missile shows superiority of North's defence tech, leader says
• Implications for stocks of liquid-fuel weapons seen unclear

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Reporting by Joyce Lee and Josh Smith; Additional reporting by Hyonhee Shin; Editing by Clarence Fernandez

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Israel approves reopening of Erez crossing into Gaza, use of Ashdod port for aid

Israel said it approved the reopening of the Erez crossing into northern Gaza and the temporary use of Ashdod port in southern Israel, following U.S. demands to increase humanitarian aid supplies into Gaza.