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## Fractions - Adding and Subtracting Fractions

Fractions -, adding and subtracting fractions, fractions adding and subtracting fractions.

## Fractions: Adding and Subtracting Fractions

Lesson 3: adding and subtracting fractions.

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## Adding and subtracting fractions

In the previous lessons, you learned that a fraction is part of a whole. Fractions show how much you have of something, like 1/2 of a tank of gas or 1/3 of a cup of water.

In real life, you might need to add or subtract fractions. For example, have you ever walked 1/2 of a mile to work and then walked another 1/2 mile back? Or drained 1/4 of a quart of gas from a gas tank that had 3/4 of a quart in it? You probably didn't think about it at the time, but these are examples of adding and subtracting fractions.

Click through the slideshow to learn how to set up addition and subtraction problems with fractions.

Let's imagine that a cake recipe tells you to add 3/5 of a cup of oil to the batter.

You also need 1/5 of a cup of oil to grease the pan. To see how much oil you'll need total, you can add these fractions together.

When you add fractions, you just add the top numbers, or numerators .

That's because the bottom numbers, or denominators , show how many parts would make a whole.

We don't want to change how many parts make a whole cup ( 5 ). We just want to find out how many parts we need total.

So we only need to add the numerators of our fractions.

We can stack the fractions so the numerators are lined up. This will make it easier to add them.

And that's all we have to do to set up an addition example with fractions. Our fractions are now ready to be added.

We'll do the same thing to set up a subtraction example. Let's say you had 3/4 of a tank of gas when you got to work.

If you use 1/4 of a tank to drive home, how much will you have left? We can subtract these fractions to find out.

Just like when we added, we'll stack our fractions to keep the numerators lined up.

This is because we want to subtract 1 part from 3 parts.

Now that our example is set up, we're ready to subtract!

Try setting up these addition and subtraction problems with fractions. Don't try solving them yet!

You run 4/10 of a mile in the morning. Later, you run for 3/10 of a mile.

You had 7/8 of a stick of butter and used 2/8 of the stick while cooking dinner.

Your gas tank is 2/5 full, and you put in another 2/5 of a tank.

## Solving addition problems with fractions

Now that we know how to write addition problems with fractions, let's practice solving a few. If you can add whole numbers , you're ready to add fractions.

Click through the slideshow to learn how to add fractions.

Let's continue with our previous example and add these fractions: 3/5 of cup of oil and 1/5 of a cup of oil.

Remember, when we add fractions, we don't add the denominators.

This is because we're finding how many parts we need total. The numerators show the parts we need, so we'll add 3 and 1 .

3 plus 1 equals 4 . Make sure to line up the 4 with the numbers you just added.

The denominators will stay the same, so we'll write 5 on the bottom of our new fraction.

3/5 plus 1/5 equals 4/5 . So you'll need 4/5 of a cup of oil total to make your cake.

Let's try another example: 7/10 plus 2/10 .

Just like before, we're only going to add the numerators. In this example, the numerators are 7 and 2 .

7 plus 2 equals 9 , so we'll write that to the right of the numerators.

Just like in our earlier example, the denominator stays the same.

So 7/10 plus 2/10 equals 9/10 .

Try solving some of the addition problems below.

## Solving subtraction problems with fractions

Subtracting fractions is a lot like regular subtraction. If you can subtract whole numbers , you can subtract fractions too!

Click through the slideshow to learn how to subtract fractions.

Let's use our earlier example and subtract 1/4 of a tank of gas from 3/4 of a tank.

Just like in addition, we're not going to change the denominators.

We don't want to change how many parts make a whole tank of gas. We just want to know how many parts we'll have left.

We'll start by subtracting the numerators. 3 minus 1 equals 2 , so we'll write 2 to the right of the numerators.

Just like when we added, the denominator of our answer will be the same as the other denominators.

So 3/4 minus 1/4 equals 2/4 . You'll have 2/4 of a tank of gas left when you get home.

Let's try solving another problem: 5/6 minus 3/6 .

We'll start by subtracting the numerators.

5 minus 3 equals 2 . So we'll put a 2 to the right of the numerators.

As usual, the denominator stays the same.

So 5/6 minus 3/6 equals 2/6 .

Try solving some of the subtraction problems below.

After you add or subtract fractions, you may sometimes have a fraction that can be reduced to a simpler fraction. As you learned in Comparing and Reducing Fractions , it's always best to reduce a fraction to its simplest form when you can. For example, 1/4 plus 1/4 equals 2/4 . Because 2 and 4 can both be divided 2 , we can reduce 2/4 to 1/2 .

## Adding fractions with different denominators

On the last page, we learned how to add fractions that have the same denominator, like 1/4 and 3/4 . But what if you needed to add fractions with different denominators? For example, our cake recipe might say to blend 1/4 cup of milk in slowly and then dump in another 1/3 of a cup.

In Comparing and Reducing Fractions , we compared fractions with a different bottom number, or denominator. We had to change the fractions so their denominators were the same. To do that, we found the lowest common denominator , or LCD .

We can only add or subtract fractions if they have the same denominators. So we'll need to find the lowest common denominator before we add or subtract these fractions. Once the fractions have the same denominator, we can add or subtract as usual.

Click through the slideshow to learn how to add fractions with different denominators.

Let's add 1/4 and 1/3 .

Before we can add these fractions, we'll need to change them so they have the same denominator .

To do that, we'll have to find the LCD , or lowest common denominator, of 4 and 3 .

It looks like 12 is the smallest number that can be divided by both 3 and 4, so 12 is our LCD .

Since 12 is the LCD, it will be the new denominator for our fractions.

Now we'll change the numerators of the fractions, just like we changed the denominators.

First, let's look at the fraction on the left: 1/4 .

To change 4 into 12 , we multiplied it by 3 .

Since the denominator was multiplied by 3 , we'll also multiply the numerator by 3 .

1 times 3 equals 3 .

1/4 is equal to 3/12 .

Now let's look at the fraction on the right: 1/3 . We changed its denominator to 12 as well.

Our old denominator was 3 . We multiplied it by 4 to get 12.

We'll also multiply the numerator by 4 . 1 times 4 equals 4 .

So 1/3 is equal to 4/12 .

Now that our fractions have the same denominator, we can add them like we normally do.

3 plus 4 equals 7 . As usual, the denominator stays the same. So 3/12 plus 4/12 equals 7/12 .

Try solving the addition problems below.

## Subtracting fractions with different denominators

We just saw that fractions can only be added when they have the same denominator. The same thing is true when we're subtracting fractions. Before we can subtract, we'll have to change our fractions so they have the same denominator.

Click through the slideshow to learn how to subtract fractions with different denominators.

Let's try subtracting 1/3 from 3/5 .

First, we'll change the denominators of both fractions to be the same by finding the lowest common denominator .

It looks like 15 is the smallest number that can be divided evenly by 3 and 5 , so 15 is our LCD.

Now we'll change our first fraction. To change the denominator to 15 , we'll multiply the denominator and the numerator by 3 .

5 times 3 equals 15 . So our fraction is now 9/15 .

Now let's change the second fraction. To change the denominator to 15 , we'll multiply both numbers by 5 to get 5/15 .

Now that our fractions have the same denominator, we can subtract like we normally do.

9 minus 5 equals 4 . As always, the denominator stays the same. So 9/15 minus 5/15 equals 4/15 .

Try solving the subtraction problems below.

## Adding and subtracting mixed numbers

Over the last few pages, you've practiced adding and subtracting different kinds of fractions. But some problems will need one extra step. For example, can you add the fractions below?

In Introduction to Fractions , you learned about mixed numbers . A mixed number has both a fraction and a whole number . An example is 2 1/2 , or two-and-a-half . Another way to write this would be 5/2 , or five-halves . These two numbers look different, but they're actually the same.

5/2 is an improper fraction . This just means the top number is larger than the bottom number. Even though improper fractions look strange, you can add and subtract them just like normal fractions. Mixed numbers aren't easy to add, so you'll have to convert them into improper fractions first.

Let's add these two mixed numbers: 2 3/5 and 1 3/5 .

We'll need to convert these mixed numbers to improper fractions. Let's start with 2 3/5 .

As you learned in Lesson 2 , we'll multiply the whole number, 2 , by the bottom number, 5 .

2 times 5 equals 10 .

Now, let's add 10 to the numerator, 3 .

10 + 3 equals 13 .

Just like when you add fractions, the denominator stays the same. Our improper fraction is 13/5 .

Now we'll need to convert our second mixed number: 1 3/5 .

First, we'll multiply the whole number by the denominator. 1 x 5 = 5 .

Next, we'll add 5 to the numerators. 5 + 3 = 8 .

Just like last time, the denominator remains the same. So we've changed 1 3/5 to 8/5 .

Now that we've changed our mixed numbers to improper fractions, we can add like we normally do.

13 plus 8 equals 21 . As usual, the denominator will stay the same. So 13/5 + 8/5 = 21/5 .

Because we started with a mixed number, let's convert this improper fraction back into a mixed number.

As you learned in the previous lesson , divide the top number by the bottom number. 21 divided by 5 equals 4, with a remainder of 1 .

The answer, 4, will become our whole number.

And the remainder , 1, will become the numerator of the fraction.

So 2 3/5 + 1 3/5 = 4 1/5 .

/en/fractions/multiplying-and-dividing-fractions/content/

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## Unit 7: Add and subtract fractions

Decomposing fractions.

- Decomposing a fraction visually (Opens a modal)
- Decomposing a mixed number (Opens a modal)
- Decomposing fractions review (Opens a modal)
- Decompose fractions visually Get 3 of 4 questions to level up!
- Decompose fractions Get 5 of 7 questions to level up!

## Adding and subtracting fractions with like denominators

- Adding fractions with like denominators (Opens a modal)
- Subtracting fractions with like denominators (Opens a modal)
- Add fractions with common denominators Get 5 of 7 questions to level up!
- Subtract fractions with common denominators Get 5 of 7 questions to level up!

## Adding and subtracting fractions: word problems

- Fraction word problem: pizza (Opens a modal)
- Fraction word problem: spider eyes (Opens a modal)
- Fraction word problem: piano (Opens a modal)
- Add and subtract fractions word problems (same denominator) Get 5 of 7 questions to level up!

## Mixed numbers

- Writing mixed numbers as improper fractions (Opens a modal)
- Writing improper fractions as mixed numbers (Opens a modal)
- Comparing improper fractions and mixed numbers (Opens a modal)
- Mixed numbers and improper fractions review (Opens a modal)
- Write mixed numbers and improper fractions Get 5 of 7 questions to level up!

## Adding and subtracting mixed numbers

- Intro to adding mixed numbers (Opens a modal)
- Intro to subtracting mixed numbers (Opens a modal)
- Add and subtract mixed numbers (no regrouping) Get 5 of 7 questions to level up!
- Add and subtract mixed numbers (with regrouping) Get 3 of 4 questions to level up!

## Adding and subtracting mixed numbers word problems

- Fraction word problem: lizard (Opens a modal)
- Subtracting mixed numbers with like denominators word problem (Opens a modal)
- Add and subtract mixed numbers word problems (like denominators) Get 3 of 4 questions to level up!

## Fractions with denominators of 10 and 100

- Visually converting tenths and hundredths (Opens a modal)
- Decomposing hundredths (Opens a modal)
- Decomposing hundredths on number line (Opens a modal)
- Adding fractions (denominators 10 & 100) (Opens a modal)
- Adding fractions: 7/10+13/100 (Opens a modal)
- Equivalent fractions with fraction models (denominators 10 & 100) Get 5 of 7 questions to level up!
- Equivalent fractions (denominators 10 & 100) Get 5 of 7 questions to level up!
- Decompose fractions with denominators of 100 Get 3 of 4 questions to level up!
- Equivalent expressions with common denominators (denominators 10 & 100) Get 5 of 7 questions to level up!
- Add fractions (denominators 10 & 100) Get 3 of 4 questions to level up!

## Line plots with fractions

- Making line plots with fractional data (Opens a modal)
- Interpreting line plots with fractions (Opens a modal)
- Reading a line plot with fractions (Opens a modal)
- Graph data on line plots (through 1/8 of a unit) Get 3 of 4 questions to level up!
- Interpret line plots with fraction addition and subtraction Get 3 of 4 questions to level up!

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

CHAPTER 2 Operations with Rational Numbers and Introduction to Real Numbers

## 2.2 Add and Subtract Fractions

Learning Objectives

By the end of this section, you will be able to:

- Add or subtract fractions with a common denominator
- Add or subtract fractions with different denominators
- Use the order of operations to simplify complex fractions
- Evaluate variable expressions with fractions

## Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Fraction Addition and Subtraction

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

Now we will do an example that has both addition and subtraction.

## Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

HOW TO: Add or Subtract Fractions

- Yes—go to step 2.
- No—rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
- Add or subtract the fractions.
- Simplify, if possible.

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2

We will apply this method as we subtract the fractions in (Example 6) .

Do the fractions have a common denominator? No, so we need to find the LCD.

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

The fractions have different denominators.

We now have all four operations for fractions. The table below summarizes fraction operations .

To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.

First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

## Use the Order of Operations to Simplify Complex Fractions

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

HOW TO: Simplify Complex Fractions

- Simplify the numerator.
- Simplify the denominator.
- Divide the numerator by the denominator. Simplify if possible.

It may help to put parentheses around the numerator and the denominator.

TRY IT 10.1

TRY IT 10.2

## Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

TRY IT 11.1

TRY IT 11.2

TRY IT 12.1

TRY IT 12.2

Substitute the values into the expression.

TRY IT 13.1

TRY IT 13.2

The next example will have only variables, no constants.

TRY IT 14.1

TRY IT 14.2

## Key Concepts

- Do they have a common denominator? Yes—go to step 2. No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
- Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.

## Practice Makes Perfect

Add and subtract fractions with a common denominator.

In the following exercises, add.

## Mixed Practice

In the following exercises, simplify.

In the following exercises, add or subtract.

In the following exercises, evaluate.

## Everyday Math

Writing exercises, attributions.

This chapter has been adapted from “Add and Subtract Fractions” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence . Adapted by Izabela Mazur. See the Copyright page for more information.

Introductory Algebra by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

## IMAGES

## VIDEO

## COMMENTS

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