problem solving converting metric units

Metric Math Conversion Problems

Related Pages Introduction to the Metric System Metric Worksheets Measurement Games

In these lessons, we will look into some methods that can be used for metric conversion problems. The first method uses the metric table and can also be regarded as a shortcut method. The second method uses the metric staircase and the third method uses the unit fraction method.

The following diagrams show some metric conversions for lengths and weights. Scroll down the page for examples and solutions.

Convert using a Metric Table or Shortcut Method

The metric system uses prefixes to denote multiple of 10.

The following table shows some common prefixes of metric units and how to convert between them.

Each prefix differs by a multiple of 10 from the next prefix. When converting between the different units of measure, we look at the number of “jumps” between the prefixes of the two units and then multiply or divide by the powers of 10 accordingly (moving to the right would mean to multiply and moving to the left would mean to divide).

Example: Convert 3 km to m

Solution: There are 3 “jumps” to the right from kilometer to meter.

So, we multiply by 10 3 = 1000

3 km = 3 × 1,000 = 3,000 m

Example: Convert 20 mm to cm

Solution: There is 1 “jump” to the left from millimeter to centimeter

So, we divide by 10.

20 mm = 20 ÷ 10 = 2 cm

How to convert to different metric units of measure for length, capacity, and mass? The metric system is widely used because it is based on the powers of 10. This allows for easy conversion between smaller and larger units. The basic unit of length is the meter (m). It is just over 1 yard. The basic unit of weight is the gram (g). One gram is the mass of 1 cubic centimeter of water. The basic unit of capacity or volume is liter (l). A liter equals 1000 cubic centimeters. Examples: Complete each conversion

• 6.3km = ____ m
• 62,300cm = ___ hm
• 37,200g = ___ kg
• 1.73hg = ___ dg
• 93.7L = ___ mL
• 62,300cL = ___ kL

How to perform metric conversion using the table or shortcut method?

Use the mnemonic “King Henry Drinks Ucky Dark Chocolate Milk”

• 0.521 mm to m
• 9501 g to kg
• 40.6 L to mL
• 1457 dm to hm
• 53.45 dag to mg .4 kL to mL

How to convert metric units of weight without using a calculator? The presentation focuses on conversion between kg, g, and mg. Imagine a 1 kilogram(kg) tablet which is made up of a 1000 smaller gram(g) tablets which is in turn made up of 1000 even smaller milligram (mg) tablets. Big to small: multiply Small to big: divide

How to multiply and divide by 1000 in conversion of metric units? This tutorial explains answers to exercises in converting metric units of weight. The exercises involve multiplying and dividing without a calculator.

Metric Staircase or Ladder Method

How to use the metric staircase to convert units within the metric system? Another method similar to the table method is to use the metric staircase. It is also called the ladder method.

Use the mnemonic “Kangaroo Helps Dingo Because Dingo Can’t Multiply”

How to use the metric staircase to convert between metric measurements? When you go up the stairs to the left, move the decimal to the left. When you go down the stairs to the right, move the decimal to the right.

Convert using the Unit Fraction Method

In this method, we multiply by a unit fraction in order to perform the metric conversion.

Common Metric Length Conversions Using Unit Fractions

Common Metric Weight Conversions Using Unit Fractions

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Converting M U

Converting metric units

Here you will learn about converting metric units, including metric units of length, metric units of mass and metric units of capacity (volume).

Students will first learn about converting metric units as part of measurement and data in 4 th grade and 5 th grade.

What is converting metric units?

Converting metric units is being able to convert between different metric units of measurement (including length, mass and volume).

To do this, you need to know what the metric units are and their conversion factors.

Certain prefixes are used before the base unit to show bigger and smaller metric units. They can help us remember the metric conversions.

• Metric units of length

The SI unit (international system of units) of length is the meter (m).

For example,

Covert 7 meters (m) to centimeters (cm).

Looking at the diagram above, meters can be converted to centimeters by multiplying by 100.

7 \mathrm{~m} \times 100=700 \mathrm{~cm}

• Metric units of mass

The metric system for mass is based around grams (g).

Covert 4 kilograms (kg) to grams (g).

Looking at the diagram above, kilograms can be converted to grams by multiplying by 1000.

4 \mathrm{~kg} \times 1000=4000 \mathrm{~g}

• Metric units of capacity (volume)

The metric system for capacity is based on liters (l).

Covert 6,000 milliliters (ml) to liters (l).

Looking at the diagram above, milliliters can be converted to liters by dividing by 10 then dividing by 100, which is equal to dividing by 1,000.

6,000 \mathrm{~ml} \div 1,000=6 \mathrm{~l}

Common Core State Standards

How does this relate to 4 th grade math and 5 th grade math?

• Grade 4 – Measurement and Data (4.MD.A.1) Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two- column table. For example, know that 1 \, ft is 12 times as long as 1 \, in. Express the length of a 4 \, ft snake as 48 \, in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …
• Grade 5 – Measurement and Data (5.MD.A.1) Convert among different-sized standard measurement units within a given measurement system (for example, convert 5 \, cm to 0.05 \, m ), and use these conversions in solving multi-step, real world problems.

How to convert metric units

In order to convert metric units:

Find the unit conversion.

Multiply or divide.

Write the answer.

[FREE] Units of Measurement Check for Understanding (Grade 4 to 5)

Use this quiz to check your grade 4 to 5 students’ understanding of units of measurement. 10+ questions with answers covering a range of 4th and 5th grade units of measurement topics to identify areas of strength and support!

Converting metric units examples

Example 1: converting length.

Convert 3.2 \mathrm{~m} to cm.

1 \mathrm{~m}=100 \mathrm{~cm}

2 Multiply or divide.

Look at the relationship in the original conversion.

From meters to centimeters, the relationship is multiplying by 100.

Multiply 3.2 meters by 100\text{:}

3.2 \text { meters } \times 100=320 \text { centimeters }

3 Write the answer.

3.2 \mathrm{~m}=320 \mathrm{~cm}

Example 2: converting length

Convert 780 \mathrm{~mm} to cm.

10 \mathrm{~mm}=1 \mathrm{~cm}

From millimeters to centimeters, the relationship is dividing by 10.

Divide 780 millimeters by 10\text{:}

780 \text { millimeters } \div 10=78 \text { centimeters }

780 \mathrm{~mm}=78 \mathrm{~cm}

Example 3: converting mass

Convert 12.5 \mathrm{~kg} to g.

1 \mathrm{~kg}=1,000 \mathrm{~g}

From kilograms to grams, the relationship is multiplying by 1,000.

Multiply 12.5 kilograms by 1,000\text{:}

12.5 \text { kilograms } \times 1,000=12,500 \text { grams }

12.5 \mathrm{~kg}=12,500 \mathrm{~g}

Example 4: converting mass

Convert 3,800 \mathrm{~g} to kg.

From grams to kilograms, the relationship is dividing by 1,000.

Divide 3,800 grams by 1,000\text{:}

3,800 \text { grams } \div 1,000=3.8 \text { kilograms }

3,800 \mathrm{~g}=3.8 \mathrm{~kg}

Example 5: converting capacity

Convert 7.1 \mathrm{~l} to ml.

1,000 \mathrm{~ml}=1 \, l

From liters to milliliters, the relationship is multiplying by 1,000.

Multiply 7.1 liters by 1,000\text{:}

7.1 \text { liters } \times 1,000=7,100 \text { milliliters }

7.1 \mathrm{~l}=7,100 \mathrm{~ml}

Example 6: converting capacity

Convert 750 \mathrm{~ml} to l.

1 \mathrm{~l}=1,000 \mathrm{~ml}

From milliliters to liters, the relationship is dividing by 1,000.

Divide 750 milliliters by 1,000\text{:}

750 \text { milliliters } \div 1,000=0.75 \text { liters }

750 \mathrm{~ml}=0.75 \, l

Teaching tips for converting metric units

• Begin by providing students with many real world measurement experiences. This can include measuring the same object in multiple units, like the weight of a stack of books in grams, centigrams, and kilograms, or activities that require students to discuss the most appropriate units to use. Once students have a good foundation in how to measure metric units and their relative sizes, they are ready to start converting.
• Instead of directly telling students about the patterns in metric conversions, wait to see when they notice them as they work with them. Given enough practice converting, the patterns in the placement of the decimal point or the number of zeroes is something many students will pick up on. Once they begin discussing this, you can highlight this pattern as a class and keep track of what students are noticing on the white board or an anchor chart. With time, you can formalize this idea with students and connect it to the powers of 10 and/or our base 10 place value system.
• Worksheets can be a useful tool to help students practice, but ensure that the worksheet includes many different units, not just the most common ones. This is a good way for students to practice converting all units in the metric system and make use of less common prefixes, like deci.
• For students that are struggling, give them time to work with a conversion calculator that completes the conversion for them. Have them explain each conversion, including why the units changed and what equation was used. Once they become more comfortable, they can begin solving conversions on their own. However, it may be a good idea to give them access to a conversion chart until they have memorized all the conversion factors.

Easy mistakes to make

• from larger units to smaller units – multiply
• from smaller units to larger units – divide
• Forgetting about uncommon units The metric system has a new unit for each power of 10. However, within each base unit there are common units and uncommon units. For example, In units of weight, the most commonly used units are grams, milligrams, and kilograms. However, there are also centigrams, decigrams, and many other units. In units of capacity, the most commonly used units are liters and milliliters. However, there are also kiloliters, centiliters, and many other units.
• Adding or subtracting to convert The relationship between units is multiplicative, which is why multiplication and division are used to convert. Using addition or subtraction will not correctly convert between units.

Related units of measurement lessons

• Units of measurement
• Converting units of time
• Conversion of measurement units
• Metric units of measurement

Practice converting metric units questions

1. Convert: 390 \mathrm{~cm} to m

From centimeters to meters, the relationship is dividing by 100.

Divide 390 centimeters by 100\text{:}

390 \text { centimeters } \div 100=3.9 \text { meters }

2. Convert: 57 \mathrm{~cm} to mm

From centimeters to millimeters, the relationship is multiplying by 10.

Multiply 57 centimeters by 10\text{:}

57 \text { centimeters } \times 10=570 \text { millimeters }

3. Convert: 81,000 \mathrm{~g} to kg

Divide 81,000 grams by 1,000\text{:}

81,000 \text { grams } \div 1,000=81 \text { kilograms }

4. Convert: 630 \mathrm{~kg} to tonnes

From kilograms to tonnes, the relationship is dividing by 1,000.

Divide 630 kilograms by 1,000\text{:}

630 \text { kilograms } \div 1,000=0.63 \text { tonnes }

5. Convert: 4.8 \mathrm{~l} to ml

Multiply 4.8 liters by 1,000\text{:}

4.8 \text { liters } \times 1,000=4,800 \text { milliliters }

6. Convert: 560 \mathrm{~ml} to l

Divide 560 milliliters by 1,000\text{:}

560 \text { milliliters } \div 1,000=0.56 \text { liters }

Converting metric units FAQs

The customary system is a different measurement system. Some examples of customary units are inches, feet, ounces, pounds, cups and gallons. This system of measurement is commonly used in the United States.

Yes, any measurement that is not a complete unit can be shown with a fraction or a decimal.

The next lessons are

• Represent and interpret data
• Converting fractions, decimals, and percentages

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6.2.2: Converting within the Metric System

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• Page ID 62186

• The NROC Project

Learning Objectives

• Perform arithmetic calculations on metric units of length, mass, and volume.

Introduction

While knowing the different units used in the metric system is important, the real purpose behind learning the metric system is for you to be able to use these measurement units to calculate the size, mass, or volume of different objects. In practice, it is often necessary to convert one metric measurement to another unit. This happens frequently in the medical, scientific, and technical fields, where the metric system is commonly used.

If you have a prescription for 5,000 milligrams of medicine, and upon getting it filled, the dosage reads 5 grams of medicine, did the pharmacist make a mistake?

For a moment, imagine that you are a pharmacist. You receive three prescriptions for liquid amoxicillin: one calls for 2.5 centiliters, one calls for 0.3 deciliters, and one calls for 450 milliliters. Amoxicillin is stored in the refrigerator in 1 liter, 1 deciliter, and 1 centiliter containers. Which container should you use to ensure you are not wasting any of the unused drug?

To solve this problem, you need to know how to convert from one measurement to another as well as how to add different quantities together. Let’s take a look at how to do this.

Converting from Larger to Smaller Units

Converting between measurements in the metric system is simply a matter of identifying the unit that you have, the unit that you want to convert to, and then counting the number of units between them. A basic example of this is shown below.

Convert 1 kilometer to decimeters.

\(\ 1 \text { kilometer }=10,000 \text { decimeters }\)

This problem is straightforward because you are converting 1 kilometer to another unit. The example below shows how you would solve this problem if you were asked to convert 8.2 kilometers to decimeters . Notice that most steps are the same; the critical difference is that you multiply by 8.2 in the final step.

Convert 8.2 kilometers to decimeters.

\(\ 8.2 \text { kilometers }=82,000 \text { decimeters }\)

You can also apply the rules of base 10 to use the “move the decimal” shortcut method in this example. Notice how decimeters (dm) is four places to the right of kilometers (km); similarly, you move the decimal point four places to the right when converting 8.2 kilometers to decimeters.

Convert 0.55 liters to centiliters.

\(\ 0.55 \text { liters }=55 \text { centiliters }\)

How many dekaliters are in 0.5 deciliters?

• Incorrect. A dekaliter is larger than a deciliter, so you would expect the number of dekaliters in 0.5 deciliters to be smaller than 0.5. The correct answer is 0.005.
• Incorrect. Deciliters and dekaliters are different units of measurement, so you would not expect 0.5 deciliters to equal 0.5 dekaliters. The correct answer is 0.005.
• Correct. One deciliter is 100 times smaller than a dekaliter, so you move the decimal point two places to the left to convert 0.5 deciliters to 0.005 dekaliters.

Converting from Smaller to Larger Units

You can use similar processes when converting from smaller to larger units. When converting a larger unit to a smaller one, you multiply; when you convert a smaller unit to a larger one, you divide. Here is an example.

Convert 739 centigrams to grams.

\(\ 739 \text { centigrams }=7.39 \text { grams }\)

Notice that the shortcut method of counting prefixes and moving the decimal the same number of places also works here. Just make sure you are moving the decimal point in the correct direction for the conversion.

Convert 205.5 milliliters to kiloliters.

\(\ 205.5 \text { milliliters }=0.0002055 \text { kiloliters }\)

Convert 3,085 milligrams to grams.

• \(\ 3,085,000 \text { grams }\)
• \(\ 308.5 \text { grams }\)
• \(\ 3.085 \text { grams }\)
• \(\ 0.3085 \text { grams }\)
• Incorrect. Grams are larger than milligrams, so you would expect the number of grams in 3,085 milligrams to be less than 3,085. The correct answer is 3,085 grams.
• Incorrect. One gram is more than 10 times larger than a milligram, so you would expect the number of grams to be less than 308.5. The correct answer is 3,085 grams.
• Correct. One gram is 1,000 times larger than a milligram, so you can move the decimal point in 3,085 three places to the left.
• Incorrect. This is too small; one gram is 1,000, not 10,000, times larger than a milligram. The correct answer is 3,085 grams.

Factor Label Method

There is yet another method that you can use to convert metric measurements: the factor label method . You used this method when you were converting measurement units within the U.S. customary system.

The factor label method works the same in the metric system; it relies on the use of unit fractions and the cancelling of intermediate units. The table below shows some of the unit equivalents and unit fractions for length in the metric system. (You should notice that all of the unit fractions contain a factor of 10. Remember that the metric system is based on the notion that each unit is 10 times larger than the one that came before it.)

Also, notice that two new prefixes have been added here: mega- (which is very big) and micro- (which is very small).

When applying the factor label method in the metric system, be sure to check that you are not skipping over any intermediate units of measurement!

Convert 7,225 centimeters to meters.

\(\ \text { 7, } 225 \text { centimeters }=72.25 \text { meters }\)

Using whichever method you prefer, convert 32.5 kilometers to meters.

• \(\ 32,500 \text { meters }\)
• \(\ 325 \text { meters }\)
• \(\ 0.325 \text { meters }\)
• \(\ 0.00325 \text { meters }\)
• Correct. To find the number of meters in 32.5 kilometers, you can set up the following equation: \(\ \frac{32.5 \text { kilometers }}{1} \cdot \frac{1,000 \text { meters }}{1 \text { kilometer }}=\frac{32,500 \text { meters }}{1}\). The kilometers units cancel, leaving the answer in meters.
• Incorrect. A kilometer is more than 10 times the size of a meter; look at the unit fractions and try your calculations again. The correct answer is 32,500 meters.
• Incorrect. A kilometer is larger than a meter, so you would expect the number of meters in 32.5 kilometers to be more than 32.5. Look at the unit fractions and try your calculations again. The correct answer is 32,500 meters.

Now that you have seen how to convert among metric measurements in multiple ways, let’s revisit the problem posed earlier.

\(\ 5 \text { grams }=5,000 \text { milligrams }\), so the pharmacist did not make a mistake.

To convert among units in the metric system, identify the unit that you have, the unit that you want to convert to, and then count the number of units between them. If you are going from a larger unit to a smaller unit, you multiply by 10 successively. If you are going from a smaller unit to a larger unit, you divide by 10 successively. The factor label method can also be applied to conversions within the metric system. To use the factor label method, you multiply the original measurement by unit fractions; this allows you to represent the original measurement in a different measurement unit.

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Resources tagged with: Metric and imperial units

There are 9 NRICH Mathematical resources connected to Metric and imperial units , you may find related items under Measuring and calculating with units .

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These Olympic quantities have been jumbled up! Can you put them back together again?

Triathlon and Fitness

The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?

All in a Jumble

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Robot Camera

Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

Weighing Fruit

Can you use this information to estimate how much the different fruit selections weigh in kilos and pounds?

At the Pumps

How will you find out how much a tank of petrol costs?

A Day with Grandpa

Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?

Watching the Wheels Go 'round and 'round

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

A security camera, taking pictures each half a second, films a cyclist going by. In the film, the cyclist appears to go forward while the wheels appear to go backwards. Why?

Using Metric Conversions to Solve Problems

Learning Objective(s)

·          Solve application problems involving metric units of length, mass, and volume.

Introduction

Learning how to solve real-world problems using metric conversions is as important as learning how to do the conversions themselves. Mathematicians, scientists, nurses, and even athletes are often confronted with situations where they are presented with information using metric measurements, and must then make informed decisions based on that data.

To solve these problems effectively, you need to understand the context of a problem, perform conversions, and then check the reasonableness of your answer. Do all three of these steps and you will succeed in whatever measurement system you find yourself using.

Understanding Context and Performing Conversions

The first step in solving any real-world problem is to understand its context. This will help you figure out what kinds of solutions are reasonable (and the problem itself may give you clues about what types of conversions are necessary). Here is an example.

This may not be likely to happen (a runner would have to be quite an athlete to compete in all of these races) but it is an interesting question to consider. The problem required you to find the total distance that the runner would run (in kilometers). The example showed how to add the distances, in meters, and then convert that number to kilometers.

An example with a different context, but still requiring conversions, is shown below.

This problem asked for the difference between two quantities. The easiest way to find this is to convert one quantity so that both quantities are measured in the same unit, and then subtract one from the other.

Checking your Conversions

Sometimes it is a good idea to check your conversions using a second method. This usually helps you catch any errors that you may make, such as using the wrong unit fractions or moving the decimal point the wrong way.

Having come up with the answer, you could also check your conversions using the quicker “move the decimal” method, shown below.

The initial answer checks out—0.72 liter of liquid is needed to fill the bottle. Checking one conversion with another method is a good practice for catching any errors in scale.

Understanding the context of real-life application problems is important. Look for words within the problem that help you identify what operations are needed, and then apply the correct unit conversions. Checking your final answer by using another conversion method (such as the “move the decimal” method, if you have used the factor label method to solve the problem) can cut down on errors in your calculations.

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Chapter 1: Algebra Review

1.6 Unit Conversion Word Problems

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.

Example 1.6.1

 If 1 pound = 16 ounces, how many pounds are in 435 ounces?

[latex]\begin{array}{rrll} 435\text{ oz}&=&435\cancel{\text{oz}}\times \dfrac{1\text{ lb}}{16\cancel{ \text{oz}}} \hspace{0.2in}& \text{This operation cancels the oz and leaves the lbs} \\ \\ &=&\dfrac{435\text{ lb}}{16} \hspace{0.2in}& \text{Which reduces to } \\ \\ &=&27\dfrac{3}{16}\text{ lb} \hspace{0.2in}& \text{Solution} \end{array}[/latex]

The same process can be used to convert problems with several units in them. Consider the following example.

Example 1.6.2

A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second?

[latex]\begin{array}{rrll} 45 \text{ mi/h}&=&\dfrac{45\cancel{\text{mi}}}{\cancel{\text{hr}}}\times \dfrac{5280 \text{ ft}}{1\cancel{ \text{mi}}}\times \dfrac{1\cancel{\text{hr}}}{3600\text{ s}}\hspace{0.2in}&\text{This will cancel the miles and hours} \\ \\ &=&45\times \dfrac{5280}{1}\times \dfrac{1}{3600} \text{ ft/s}\hspace{0.2in}&\text{This reduces to} \\ \\ &=&66\text{ ft/s}\hspace{0.2in}&\text{Solution} \end{array}[/latex]

Example 1.6.3

Convert 8 ft 3 to yd 3 .

[latex]\begin{array}{rrll} 8\text{ ft}^3&=&8\text{ ft}^3 \times \dfrac{(1\text{ yd})^3}{(3\text{ ft})^3}&\text{Cube the parentheses} \\ \\ &=&8\text{ }\cancel{\text{ft}^3}\times \dfrac{1\text{ yd}^3}{27\text{ }\cancel{\text{ft}^3}}&\text{This will cancel the ft}^3\text{ and replace them with yd}^3 \\ \\ &=&8\times \dfrac{1\text{ yd}^3}{27}&\text{Which reduces to} \\ \\ &=&\dfrac{8}{27}\text{ yd}^3\text{ or }0.296\text{ yd}^3&\text{Solution} \end{array}[/latex]

Example 1.6.4

A room is 10 ft by 12 ft. How many square yards are in the room? The area of the room is 120 ft 2 (area = length × width).

Converting the area yields:

[latex]\begin{array}{rrll} 120\text{ ft}^2&=&120\text{ }\cancel{\text{ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ }\cancel{\text{ft}})^2}&\text{Cancel ft}^2\text{ and replace with yd}^2 \\ \\ &=&\dfrac{120\text{ yd}^2}{9}&\text{This reduces to} \\ \\ &=&13\dfrac{1}{3}\text{ yd}^2&\text{Solution} \\ \\ \end{array}[/latex]

The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.

Example 1.6.5

A child is prescribed a dosage of 12 mg of a certain drug per day and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 refills)?

[latex]\begin{array}{rrll} 3\text{ prescriptions}&=&3\cancel{\text{pres.}}\times \dfrac{60\cancel{\text{tablets}}}{1\cancel{\text{pres.}}}\times \dfrac{4\cancel{\text{mg}}}{1\cancel{\text{tablet}}}\times \dfrac{1\text{ dosage}}{12\cancel{\text{mg}}}&\text{This cancels all unwanted units} \\ \\ &=&\dfrac{3\times 60\times 4\times 1}{1\times 1\times 12}\text{ or }\dfrac{720}{12}\text{ dosages}&\text{Which reduces to} \\ \\ &=&60\text{ daily dosages}&\text{Solution} \\ \\ \end{array}[/latex]

Metric and Imperial (U.S.) Conversions

[latex]\begin{array}{rrlrrl} 12\text{ in}&=&1\text{ ft}\hspace{1in}&10\text{ mm}&=&1\text{ cm} \\ 3\text{ ft}&=&1\text{ yd}&100\text{ cm}&=&1\text{ m} \\ 1760\text{ yds}&=&1\text{ mi}&1000\text{ m}&=&1\text{ km} \\ 5280\text{ ft}&=&1\text{ mi}&&& \end{array}[/latex]

Imperial to metric conversions:

[latex]\begin{array}{rrl} 1\text{ inch}&=&2.54\text{ cm} \\ 1\text{ ft}&=&0.3048\text{ m} \\ 1\text{ mile}&=&1.61\text{ km} \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 144\text{ in}^2&=&1\text{ ft}^2\hspace{1in}&10,000\text{ cm}^2&=&1\text{ m}^2 \\ 43,560\text{ ft}^2&=&1\text{ acre}&10,000\text{ m}^2&=&1\text{ hectare} \\ 640\text{ acres}&=&1\text{ mi}^2&100\text{ hectares}&=&1\text{ km}^2 \end{array}[/latex]

[latex]\begin{array}{rrl} 1\text{ in}^2&=&6.45\text{ cm}^2 \\ 1\text{ ft}^2&=&0.092903\text{ m}^2 \\ 1\text{ mi}^2&=&2.59\text{ km}^2 \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 57.75\text{ in}^3&=&1\text{ qt}\hspace{1in}&1\text{ cm}^3&=&1\text{ ml} \\ 4\text{ qt}&=&1\text{ gal}&1000\text{ ml}&=&1\text{ litre} \\ 42\text{ gal (petroleum)}&=&1\text{ barrel}&1000\text{ litres}&=&1\text{ m}^3 \end{array}[/latex]

[latex]\begin{array}{rrl} 16.39\text{ cm}^3&=&1\text{ in}^3 \\ 1\text{ ft}^3&=&0.0283168\text{ m}^3 \\ 3.79\text{ litres}&=&1\text{ gal} \end{array}[/latex]

[latex]\begin{array}{rrlrrl} 437.5\text{ grains}&=&1\text{ oz}\hspace{1in}&1000\text{ mg}&=&1\text{ g} \\ 16\text{ oz}&=&1\text{ lb}&1000\text{ g}&=&1\text{ kg} \\ 2000\text{ lb}&=&1\text{ short ton}&1000\text{ kg}&=&1\text{ metric ton} \end{array}[/latex]

[latex]\begin{array}{rrl} 453\text{ g}&=&1\text{ lb} \\ 2.2\text{ lb}&=&1\text{ kg} \end{array}[/latex]

Temperature

Fahrenheit to Celsius conversions:

[latex]\begin{array}{rrl} ^{\circ}\text{C} &= &\dfrac{5}{9} (^{\circ}\text{F} - 32) \\ \\ ^{\circ}\text{F}& =& \dfrac{9}{5}(^{\circ}\text{C} + 32) \end{array}[/latex]

For questions 1 to 18, use dimensional analysis to perform the indicated conversions.

• 7 miles to yards
• 234 oz to tons
• 11.2 mg to grams
• 1.35 km to centimetres
• 9,800,000 mm to miles
• 4.5 ft 2 to square yards
• 435,000 m 2 to square kilometres
• 8 km 2 to square feet
• 0.0065 km 3 to cubic metres
• 14.62 in. 2 to square centimetres
• 5500 cm 3 to cubic yards
• 3.5 mph (miles per hour) to feet per second
• 185 yd per min. to miles per hour
• 153 ft/s (feet per second) to miles per hour
• 248 mph to metres per second
• 186,000 mph to kilometres per year
• 7.50 tons/yd 2 to pounds per square inch
• 16 ft/s 2 to kilometres per hour squared

For questions 19 to 27, solve each conversion word problem.

• On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?
• A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day.
• An average human heart beats 60 times per minute. If the average person lives to the age of 86, how many times does the average heart beat in a lifetime?
• Blood sugar levels are measured in milligrams of glucose per decilitre of blood volume. If a person’s blood sugar level measured 128 mg/dL, what is this in grams per litre?
• You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?
• A cargo container is 50 ft long, 10 ft wide, and 8 ft tall. Find its volume in cubic yards and cubic metres.
• A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy more than ¼ of the lot it is built on. Suppose you own a [latex]\frac{1}{3}[/latex]-acre lot (1 acre = 43,560 ft 2 ). What is the maximum allowed footprint for your house in square feet? In square metres?
• A car travels 23 km in 15 minutes. How fast is it going in kilometres per hour? In metres per second?
• The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams. What is the mass of this diamond in milligrams? Weight in pounds?

Answer Key 1.6

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Solving Problems Involving Conversion of Units of Measurement

Measuring is one of the human activities that we perform daily. A tailor measures the length of the dress; a butcher measures the weight of the meat; a surveyor measures the area of large land masses, and so on. 

The ability to measure objects is always connected to the history of our civilization. During ancient times, our ancestors used their fingers, hands, and feet to determine the length of an object. In this era, humans had varying ways of measuring things. 

Eventually, after the French Revolution in the late 18th century, a standardized way of measurement was developed (i.e., the metric system). Today, we have a convention of measurement units and advanced technological tools to measure objects. 

In this module, you’ll learn what measurement is, the units of measurement (for length, weight, volume, time, and temperature), and how to convert units in the metric system. 

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Table of Contents

What is measurement.

Measurement is the provision of a numerical value to present and describe the magnitude or amount of a particular object.  

We use measurement units to provide a more accurate description of the object’s measurement. Some examples of measurement units are meters, liters, grams, inches, Fahrenheit, and so on. 

For instance, if we want to determine how long a piece of wood is, we measure its length. To do this, we use a particular SI unit of measurement (e.g., meters) and provide a number that describes the length of this wood (e.g., this wood is 3 meters long). 

Since there are a lot of measurement units being used around the world, a standardized set of measurement units have been adopted by several countries. This is called the SI Units of Measurement , more commonly known as the Metric System (e.g., meter, gram, liter). On the other hand, there’s also the Imperial System or US Standard Units , which is also commonly used in the Philippines (e.g., feet, yards, pounds, etc.).

What Are the Different Ways To Measure Objects?

There are different ways to measure objects depending on the particular trait we want to describe or show. 

This refers to the distance from one point to another. In other words, this describes how long or short an object is. Standard measurement units are meters, centimeters, inches, feet, etc.

The SI base unit for length is a meter.

This refers to the amount of space occupied by a two-dimensional figure. In other words, it tells us how much surface a plane figure covers . Commonly used measurement units are square meters, square kilometers, square yards, etc. 

Metric units of the area have an exponent of 2 to indicate that we are measuring the amount of two-dimensional space occupied (e.g., the square meter is written as m 2 ).

The SI base unit for the area is square meters (m 2 ).

3. Volume or Capacity

The volume is the space occupied or enclosed by a three-dimensional figure . Commonly used measurement units are cubic meters, cubic kilometers, cubic yards, etc. 

Metric volume units have an exponent of 3 to indicate that we are measuring the amount of three-dimensional space occupied (e.g., a cubic meter is written as m 3 ).

In Physics, mass and weight mean differently . Mass refers to the amount of matter an object has, while weight refers to the force that gravity exerts on an object. 

The SI base units for mass and weight are different. The kilogram is the SI base unit for mass, while Newton is the SI base unit for weight.

However, outside a Physics classroom, these terms are often used interchangeably. Many people perceive mass and weight as the same thing, which refers to how heavy an object is. To avoid ambiguity and confusion in our discussion, we will strictly use in this article the word “mass” to refer to the heaviness of an object 

This refers to the duration of the sequence of events. For instance, we measure time to determine how long you read this reviewer. 

The SI base unit for time is seconds.

6. Temperature

Temperature tells us how hot or cold an object is. In Physics, the temperature is the average kinetic energy of the particles of an object . The primary way to measure temperature is through a thermometer .

SI base unit for temperature is Kelvin (K). However, Celsius (°C) and Fahrenheit (°F) are the more popular units.

This review will discuss the conversion of measurement units for length, area, volume, mass, time, and temperature. However, remember that these are not just the ways to measure objects. For instance, we can also measure their luminous intensity, electric current, amount of substance, and so on. 

Metric System Units of Length

We learned earlier that meter (m) is the SI base unit for length. This means that other metric units for length are derived from the meter. For instance, a kilometer (km) means 1000 meters. 

In the metric system, we use prefixes to indicate that a particular metric unit is a multiple of the base unit. For example, the prefix “kilo” means 1000 times the base unit. So, 1 km = 1000 meters.

Six prefixes are used in the metric system, and we list them below together with their equivalent value in the base unit. 

An easier way to visualize these prefixes is by using a table:

Whether you want to memorize the conversion table above is up to you. However, it is advisable to remember the equivalent value of each prefix in terms of the base unit. These prefixes also apply to metric units for area and mass. 

Converting Metric Units of Length

The easiest way to convert metric units is by simply moving decimal places. 

For instance, let us convert 375 meters to kilometers by looking at the table of prefixes below. 

Note that the table above has three steps to the left from meter to kilometer. This means we must move three decimal places to the left in 375 meters to get its equivalent in kilometers. 

This means that 375 meters are equal to 0.375 kilometers.

Sample Problem 1: Convert 98.35 decameters to centimeters

Solution: Looking at the table of metric units of length, there are three steps to the right, from decameters to centimeters. 

This implies that we must move three decimal places to the right to convert 98.35 decameters to centimeters.

Hence, 98.35 decameters = 98,350 centimeters

Sample Problem 2: A ribbon was divided into two strips. The first strip measures 176.50 centimeters, while the second measures 89.56 centimeters. What is the length of the original ribbon in meters?

Solution: There are two steps to the left from centimeters to meters. Hence, we move two decimal places to the left to convert centimeters to meters. 

• Moving two decimal places in 176.50 centimeters results in 1.7650 meters.
• Moving two decimal places in 89.56 centimeters results in 0.8956 meters.

Adding the length of the strips converted to meters: 1.7650 m + 0.8956 m = 2.6606 m.

Hence, the length of the original ribbon is 2.6606 or 2.66 meters. 

Metric System Units of Area

The SI base unit for the area is a square meter (m 2 ). Like the metric units of length, the metric units for the area are derived from the base unit (i.e., square meters). 

The prefixes you have learned in the metric units for length also apply to metric units for the area. Again, these prefixes indicate that a particular metric unit is a multiple of the base unit. 

Six prefixes are used in the metric system, and we list them below together with their equivalent value in the base unit. Note that all metric units for the area have a superscript of 2 to indicate that we are dealing with square units. 

Just like for length, it is easier to visualize these prefixes by using a table:

Converting Metric Units of Area

The method of converting metric units of the area is similar to the one we use to convert metric units of length. That is, by moving decimal places.

Let us convert 520 m 2 to km 2 . By looking at the table, there are three steps to the left, from square meters to square kilometers. 

We move three decimal places to the left in 520 m 2 to obtain its equivalent in km 2 . 

Thus, 520 m 2 is equal to 0.52 km 2 .

Since we always use the table of prefixes of metric units, I highly recommend memorizing the horizontal arrangement of these prefixes. They are not that hard to remember since there are only six prefixes. This is much easier than memorizing the conversion units. 

Sample Problem: Every square meter of land in a province costs ₱4,000. Jennie plans to buy a 15-square decameter of land in this province. How much will Jennie have to pay to purchase the land? 

Solution: Since the pricing of the land is expressed as ₱4,000 per square meter (m 2 ), we have to convert 15 square decameters (dam 2 ) to square meters (m 2 ) to calculate the land price accurately. 

From square decameter to square meter, there’s one step to the right.

Hence, we move one decimal to the right in 15 dam 2 to convert it into m 2 :

Thus, 15 dam 2 is equal to 150 m 2 .

Now, since the price of the land is ₱4000 per square meter (m 2 ), then a 150 m 2 of land will cost:

150 x 4000 = 600,000

The answer is ₱600,000.

Metric System Units of Volume

The SI base unit for volume is cubic meters (m 3 ). Like length, the metric units for volume or capacity are derived from cubic meters (m 3 ). 

Six prefixes are used in the metric system, and we list them below together with their equivalent value in the base unit. Note that all metric units for volume have a superscript of 3 to indicate that we are dealing with cubic units. 

Just like for length and area, it is easier to visualize these prefixes by using a table:

The liter is another metric unit used for volume/capacity. The liter is a special name for cubic decimeter (dm 3 ). Thus, 1 liter equals 1 cubic decimeter (1 L = 1 dm 3 ). 

Like any metric unit, prefixes are also used to derive other metric units for volume. For instance, the prefix “milli” in milliliter indicates that this unit is equal to thousandths (0.001) of a liter. 

Although a liter is a metric unit for volume, there’s no need to put a superscript of 3 (which also applies to other metric units based on it). 

Here are the other six prefixes associated with liter:

Again, it’s easier to visualize these prefixes by using a table:

Converting Metric Units of Volume

Converting metric units of volume is similar to converting metric units for length and area. 

Sample Problem 1 : A shoe box has a volume of 750 cm 3 . Determine its volume in m 3 .

Solution : There are two steps to the left from cm 3 to m 3 . Hence, we move two decimal places to the left in 750 cm 3 to transform it into m 3 .

Therefore, 750 cm 3 is equivalent to 7.5 m 3 .

Sample Problem 2 : A large tank can be filled with 250 L of water. A water pipe puts 200 cL of water into the tank per minute. How long can the pipe fill the large tank?

Solution : To determine how long the pipe can fill the large tank, we divide 250L by 200 cL. However, we cannot perform this immediately since the given measurements differ in units.  

First, let us convert 200 cL to L. 

Looking at the table of prefixes, notice two steps to the left from cL to L. Hence, we move two decimal places to the left in 200 cL to convert it into L.

Hence, 200 cL = 2 L

We can now divide 250 L by 2 L. Dividing 250 L by 2 L, we’ll obtain

250 ÷ 2 = 125

This means that the pipe can fill the tank after 125 minutes.

Metric System Units of Mass

The SI base unit for mass is the kilogram (kg). However, note that “gram” is the primary basis for deriving other metric units for mass. “Gram” can be viewed as the “meter” in terms of mass.

Six prefixes are used in the metric system, and we list them here with their equivalent value in the base unit. 

It’s easier to visualize these prefixes by using a table:

Converting Metric Units of Mass

Sample Problem 1 : Myrna bought cough syrup with a mass of 50 grams. Determine its mass in milligrams.

Solution : Referring to the table of prefixes, there are three steps to the right from gram (g) to milligram (mg). Thus, we have to move three decimal places to the right to convert 50 g to mg:

Therefore, the cough syrup’s mass is equal to 50,000 mg.

Sample Problem 2: Rosie bought 1250.50 grams of mangoes. What is the mass of the mangoes that Rosie bought in kilograms?

Solution : Let us convert 1250.50 grams to kilograms. There are three steps to the left, from grams (g) to kilograms (kg). Hence, we must move three decimal places to the left in 1250.50 g to convert it into kilograms (kg). 

Hence, 1250.50 grams is equal to 1.25050 kilograms. 

Converting Units of Time

Let’s now discuss how to convert units of time. Moving decimal places is not applicable for converting time units, unlike the metric units for length, area, volume, or mass. Instead, we have to refer to the conversion values for each unit.

Shown below is the conversion of time units:

To convert one unit of time to another, follow these steps:

• Identify the given and to which unit we will convert it.
• Determine the relationship between the given units.
• Express the relationship between the given units as a conversion factor in a fractional form such that the denominator has a unit that is the same as the original unit.
• Multiply the given measurement by the conversion factor.

Let us apply the steps above to answer some examples.

Sample Problem 1 : How many hours are there in 5 days?

Step 1: Identify the given and to which unit we will convert it. 

The problem is asking us to convert 5 days into hours.

Step 2: Determine the relationship between the given units.

There are 24 hours in one day. In other words, 1 day = 24 hours.

Step 3: Express the relationship between the given units as a conversion factor in a fractional form such that the denominator has a unit that is the same as the original unit.

In the previous step, we’re able to determine that 1 day is equivalent to 24 hours. Express this as a fraction with the unit that matches the original unit as the denominator. Since the original unit is “days,” we must express the conversion factor as 24 hours/1 day.

Step 4: Multiply the given measurement by the conversion factor.

Now, let us multiply 5 days by 24 hours/1 day:

Hence, there are 120 hours in 5 days.

Sample Problem 2 : Rhodora plans to go on a vacation to Lemongate Beach for 8 weeks. How many days will Rhodora be on vacation?

The problem is asking us to convert 8 weeks to days

Step 2 : Determine the relationship between the given units.

There are 7 days in one week or 1 week = 7 days.

In the previous step, we’re able to determine that 1 week is equivalent to 7 days. Express this as a fraction with the unit that matches the original unit as the denominator. Since the original unit is “week,” we must express the conversion factor as 7 days/1 week. 

Now, let us multiply 8 weeks by 7 days/1 week:

Hence, Rhodora will be on vacation for 56 days.

Sample Problem 3 : A worker is paid ₱0.5 per minute for his job. How much will the worker earn if he works for a total amount of time equivalent to 25 days?

Solution: The worker’s wage is expressed as ₱0.5 per minute. Therefore, we must convert 25 days to minutes first before determining the worker’s earnings. 

The problem is asking us to convert 25 days to minutes.

Note that before converting days to minutes, we must first convert days to hours. Afterward, we will convert the result to minutes. This means that we will be dealing with two relationships in this problem. 

• Relationship #1 (days to hours): 24 hours in 1 day or 1 day = 24 hours.
• Relationship #2 (hours to minutes): 60 minutes in 1 hour or 1 hour = 60 minutes.

Express the relationships we derived from Step 2 as conversion factors:

• For relationship #1 (days to hours), the original unit is days, so we have 24 hours/1 day 
• For relationship #2 (hours to minutes), the original unit now is hours, so we have 60 minutes/1 hour 

Multiply the 25 days by the two conversion factors. 

Thus, there are 36,000 minutes in 25 days. 

Remember, the worker’s wage is expressed as ₱0.5 per minute. If the worker renders 36,000 minutes of work, he will earn 36,000 x 0.5 = ₱18,000.

Sample Problem 4 : 504 hours is equivalent to how many weeks?

The problem is asking us to convert 504 hours to weeks.

To convert hours to weeks, we first need to convert hours to days. Afterward, we convert days to weeks. Thus, we will be dealing with two relationships of the unit of time in this problem:

• Relationship #1 (hours to days): 24 hours in 1 day or 1 day = 24 hours
• Relationship #2 (days to weeks): 7 days in 1 week or 1 week = 7 days
• For relationship #1 (hours to days), the original unit is hours, so we have 1 day/24 hours 
• For relationship #2 (days to weeks), the original unit now is days, so we have 1 week/7 days 

Multiply the 504 hours by the two conversion factors. 

Thus, 504 hours is equal to 3 weeks.

Converting Units of Temperature

Kelvin, Celsius, and Fahrenheit are the units of measurement for temperature. This section will focus only on converting Celsius to Fahrenheit and vice versa.

To convert Celsius to Fahrenheit, use a conversion formula. We will discuss these formulas in this section.

How To Convert Celsius to Fahrenheit

Shown below is the formula to convert Celsius to Fahrenheit:

To use this formula, insert the given temperature expressed in Celsius into the formula and perform the calculation. The resulting value is the equivalent temperature in Fahrenheit.

Sample Problem 1: The freezing point of water is 0°C. What is the freezing point of water in Fahrenheit?

Solution : Applying the formula to convert Celsius to Fahrenheit:

Hence, the freezing point of water in Fahrenheit is 32°F.

Sample Problem 2: The average body temperature is 37°C. What is the average body temperature in Fahrenheit?

Hence, the average body temperature in Fahrenheit is 98.6°F.

How To Convert Fahrenheit to Celsius

Shown below is the formula to convert Fahrenheit to Celsius:

The formula above can be derived using the formula for converting Celsius to Fahrenheit. All you have to do is perform some basic algebra (in particular, solving a linear equation ).

To use the formula, insert the given temperature expressed in Fahrenheit into the formula and perform the calculation. The resulting value is the equivalent temperature in Celsius.

Sample Problem 1 : Convert 32 °F to °C

Hence, 32 °F is equal to 0°C.

Sample Problem 2 : The required storage temperature of a particular drug is 59 °F. What is its equivalent in °C?

Solution : 

Therefore, 59 °F is equal to 15 °C.

Next topic:  Perimeter and Area of Plane Figures

Previous topic:  Triangles: Classification and Theorems

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1.19: Problem Solving and Unit Conversions

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  ​​​​​​​   Learning Objectives

• To convert a value reported in one unit to a corresponding value in a different unit using conversion factors.

During your studies of chemistry (and physics also), you will note that mathematical equations are used in many different applications. Many of these equations have a number of different variables with which you will need to work. You should also note that these equations will often require you to use measurements with their units. Algebra skills become very important here!

Converting Between Units with Conversion Factors

A conversion factor is a factor used to convert one unit of measurement into another. A simple conversion factor can convert meters into centimeters, or a more complex one can convert miles per hour into meters per second. Since most calculations require measurements to be in certain units, you will find many uses for conversion factors. Always remember that a conversion factor has to represent a fact; this fact can either be simple or more complex. For instance, you already know that 12 eggs equal 1 dozen. A more complex fact is that the speed of light is \(1.86 \times 10^5\) miles/\(\text{sec}\). Either one of these can be used as a conversion factor depending on what type of calculation you are working with (Table \(\PageIndex{1}\)).

*Pounds and ounces are technically units of force, not mass, but this fact is often ignored by the non-scientific community.

Of course, there are other ratios which are not listed in Table \(\PageIndex{1}\). They may include:

• Ratios embedded in the text of the problem (using words such as per or in each , or using symbols such as / or %).
• Conversions in the metric system, as covered earlier in this chapter.
• Common knowledge ratios (such as 60 seconds \(=\) 1 minute).

If you learned the SI units and prefixes described, then you know that 1 cm is 1/100th of a meter.

\[ 1\; \rm{cm} = \dfrac{1}{100} \; \rm{m} = 10^{-2}\rm{m} \nonumber \]

\[100\; \rm{cm} = 1\; \rm{m} \nonumber \]

Suppose we divide both sides of the equation by \(1 \text{m}\) (both the number and the unit):

\[\mathrm{\dfrac{100\:cm}{1\:m}=\dfrac{1\:m}{1\:m}} \nonumber \]

As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the right side of the equation; it now has the same quantity in the numerator (the top) as it has in the denominator (the bottom). Any fraction that has the same quantity in the numerator and the denominator has a value of 1:

\[ \dfrac{ \text{100 cm}}{\text{1 m}} = \dfrac{ \text{1000 mm}}{\text{1 m}}= \dfrac{ 1\times 10^6 \mu \text{m}}{\text{1 m}}= 1 \nonumber \]

We know that 100 cm is 1 m, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units.

Performing Dimensional Analysis

Dimensional analysis is amongst the most valuable tools that physical scientists use. Simply put, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors. This is valuable because certain measurements are more accurate or easier to find than others. The use of units in a calculation to ensure that we obtain the final proper units is called dimensional analysis .

Here is a simple example. How many centimeters are there in 3.55 m? Perhaps you can determine the answer in your head. If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as \(\mathrm{\dfrac{100\:cm}{1\:m}}\) and multiply:

\[ 3.55 \; \rm{m} \times \dfrac{100 \; \rm{cm}}{1\; \rm{m}} \nonumber \]

The 3.55 m can be thought of as a fraction with a 1 in the denominator. Because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out:

\[\dfrac{3.55 \; \cancel{\rm{m}}}{ 1} \times \dfrac{100 \; \rm{cm}}{1 \; \cancel{\rm{m}}} \nonumber \]

The final step is to perform the calculation that remains once the units have been canceled:

\[ \dfrac{3.55}{1} \times \dfrac{100 \; \rm{cm}}{1} = 355 \; \rm{cm} \nonumber \]

In the final answer, we omit the 1 in the denominator. Thus, by a more formal procedure, we find that 3.55 m equals 355 cm. A generalized description of this process is as follows:

quantity (in old units) × conversion factor = quantity (in new units)

You may be wondering why we use a seemingly complicated procedure for a straightforward conversion. In later studies, the conversion problems you encounter will not always be so simple . If you master the technique of applying conversion factors, you will be able to solve a large variety of problems.

In the previous example, we used the fraction \(\dfrac{100 \; \rm{cm}}{1 \; \rm{m}}\) as a conversion factor. Does the conversion factor \(\dfrac{1 \; \rm m}{100 \; \rm{cm}}\) also equal 1? Yes, it does; it has the same quantity in the numerator as in the denominator (except that they are expressed in different units). Why did we not use that conversion factor? If we had used the second conversion factor, the original unit would not have canceled, and the result would have been meaningless. Here is what we would have gotten:

\[ 3.55 \; \rm{m} \times \dfrac{1\; \rm{m}}{100 \; \rm{cm}} = 0.0355 \dfrac{\rm{m}^2}{\rm{cm}} \nonumber \]

For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out . Figure \(\PageIndex{1}\) shows a concept map for constructing a proper conversion.

General Steps in Performing Dimensional Analysis

• Identify the " given " information in the problem. Look for a number with units to start this problem with.
• What is the problem asking you to " find "? In other words, what unit will your answer have?
• Use ratios and conversion factors to cancel out the units that aren't part of your answer, and leave you with units that are part of your answer.
• When your units cancel out correctly, you are ready to do the math . You are multiplying fractions, so you multiply the top numbers and divide by the bottom numbers in the fractions.

Significant Figures in Conversions

How do conversion factors affect the determination of significant figures?

• Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact.
• Exact numbers are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. (In other words, 1 kg is exactly 1,000 g, by the definition of kilo-.)
• Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact.
• In contrast, conversion factors that come from measurements (such as density, as we will see shortly) or that are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer.

Example \(\PageIndex{1}\)

Exercise \(\pageindex{1}\).

Perform each conversion.

• 101,000 ns to seconds
• 32.08 kg to grams
• 1.53 grams to cg
• Conversion factors are used to convert one unit of measurement into another.
• Dimensional analysis (unit conversions) involves the use of conversion factors that will cancel unwanted units and produce the appropriate units.

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Course: 5th grade   >   Unit 14

• Measurement word problem: tea party
• Time word problem: Susan's break

Convert units word problems (metrics)

• Convert units multi-step word problems (metric)
• Your answer should be
• an integer, like 6 ‍  
• an exact decimal, like 0.75 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍