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# 2.6: Add and Subtract Fractions and Mixed Numbers

Difficulty Level: Advanced Created by: CK-12
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“Look at this,” Trevor exclaimed rushing into student council meeting one afternoon. In his arms he held several boxes.

“Look at what?” Kelly asked.

Trevor stopped and put all of the boxes in the middle of the table. He was grinning from ear to ear.

“Look, Mr. Grimes the custodian found all of these boxes of pens in the store room. He said we can have them for free. We have \begin{align*}2 \frac{1}{2}\end{align*} boxes of pens here and another \begin{align*}\frac{3}{4}\end{align*} of a box,” Trevor said smiling.

“Great work! We also ordered a case of pens and there are 25 boxes in a case. Now we will have a lot of pens,” Kelly said.

“How many do we have altogether?” Mallory asked.

It is time to learn about fraction addition. Solving this problem will rely on your ability to add fractions. In this Concept, you will learn how to add and subtract fractions. When finished, you will see this problem again so be sure to pay close attention!

### Guidance

By this time in math class you have been working with fractions for a long time. Yet fractions are often a place of struggle for many students. We often think in terms of whole numbers and not in terms of fractions.

A fraction is a number that names a part of a whole or a part of a group.

If a rectangle is \begin{align*}\frac{1}{3}\end{align*} shaded, it means that if the rectangle were divided into three equal parts, one of those parts would be shaded. Most fractions will represents numbers less than 1, meaning that the numerator is less than the denominator.

To represent a number greater than 1, we use an improper fraction or a mixed number.

An improper fraction has a numerator that is larger than its denominator, such as \begin{align*}\frac{5}{3}\end{align*}.

This fraction can also be written as the mixed number \begin{align*}1 \frac{2}{3}\end{align*}.

A mixed number is a number that has both wholes and parts, so you will see a whole number and a fraction with mixed numbers.

Let's take a look at adding and subtracting fractions.

How do we add and subtract fractions?

Well, the first thing to look at is the bottom number of the fractions that you are adding or subtracting. The bottom number or denominator tells you how many parts the whole is divided into. If the denominator is a three, then we know that the whole is divided into three parts. The top number or numerator tells you how many parts you have out of the whole.

If the denominators of the fractions being added are the same, then the wholes are divided the same way so we can simply add or subtract the numerators.

\begin{align*}\frac{1}{8}+ \frac{2}{8} = \frac{3}{8}\end{align*}

Here you can see that both fractions have denominators of 8, so we can simply add the numerators.

Our answer is three-eighths.

We can also subtract fractions.

\begin{align*}\frac{10}{12}- \frac{3}{12} = \frac{7}{12}\end{align*}

Once again our denominators are common. So we can simply subtract the numerators.

Our answer is seven-twelfths.

What if the denominators are not the same?

In this case, we have to find a common denominator and rename the fractions in terms of this common denominator. A way to think about this is as equal fractions. Look at the fractions below.

\begin{align*}\frac{1}{2}=\frac{4}{8}\end{align*}

These two fractions are equal. This means that they both represent the same part of the whole. The fraction one-half has simply been renamed in terms of eighths. This is the same thing that we do when finding common denominators. We rename the fractions in terms of the common denominator.

Add: \begin{align*}\frac{1}{4}+\frac{2}{5}\end{align*}

First, find a common denominator by finding the least common multiple of the denominators, 4 and 5.

The first few multiples of 4 are 4, 8, 12, 16, 20. The first few multiples of 5 are 5, 10, 15, 20. So the least common multiple is 20.

Now, find how to rename each fraction to a fraction with 20 as the denominator.

For the first fraction, you need to multiply the denominator by 5 to get a denominator of 20. So, multiply the first fraction by the equivalent of 1, \begin{align*}\frac{5}{5}\end{align*}.

\begin{align*}\frac{1}{4} \times \frac{5}{5} = \frac{5}{20}\end{align*}

Now do the same for the other fraction.

You need to multiply the denominator by 4 to get a denominator of 20. So, multiply the second fraction by the equivalent of 1, \begin{align*}\frac{4}{4}\end{align*}.

\begin{align*}\frac{2}{5} \times \frac{4}{4}=\frac{8}{20}\end{align*}

Next, add the fractions.

\begin{align*}\frac{5}{20} \times \frac{8}{20}=\frac{13}{20}\end{align*}

Our answer is \begin{align*}\frac{13}{20}\end{align*}.

We can also work with mixed numbers and fractions. There is an added step when working with this combination.

Subtract: \begin{align*}2\frac{7}{8}-\frac{2}{3}\end{align*}

First change the mixed number to an improper fraction. To do this, multiply the denominator by the whole number, then add the numerator.

\begin{align*}8 \times 2+7 & =16+7=23\\ 2\frac{7}{8} & = \frac{23}{8}\end{align*}

Then find a common denominator.

The first few multiples of 8 are 8, 16, 24. Since 24 is also divisible by 3, it is the least common multiple.

Rename each fraction as a fraction with the common denominator.

\begin{align*}\frac{23}{8} \times \frac{3}{3} &= \frac{69}{24}\\ \frac{2}{3} \times \frac{8}{8} &= \frac{16}{24}\end{align*}

Now find the difference.

\begin{align*}\frac{69}{24}-\frac{16}{24}=\frac{53}{24}\end{align*}

Notice that the answer is an improper fraction. We can’t leave it this way.

Finally, simplify the difference to a mixed number.

\begin{align*}\frac{53}{24}=2\frac{5}{24}\end{align*}

This is our answer.

Take a few minutes to write some notes about adding and subtracting fractions. Be sure to write down the steps to renaming fractions with common denominators.

Try these. Be sure your answers are in simplest form.

#### Example A

Subtract: \begin{align*}\frac{4}{9}-\frac{1}{6}\end{align*}

Solution:\begin{align*}\frac{5}{18}\end{align*}

#### Example B

Add: \begin{align*}\frac{10}{12}+\frac{2}{6}\end{align*}

Solution: \begin{align*}1 \frac{1}{6}\end{align*}

#### Example C

Subtract:\begin{align*}\frac{4}{8}-\frac{1}{4}\end{align*}

Solution: \begin{align*}\frac{1}{4}\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

To solve this problem, we need to find a sum. A sum means addition, so we are going to figure out how many boxes of pens the student council has in all.

First, let’s find the sum of the extras.

The students were given \begin{align*}2 \frac{1}{2}\end{align*} boxes of pens and another \begin{align*}\frac{3}{4}\end{align*} of a box.

\begin{align*}2 \frac{1}{2} + \frac{3}{4}\end{align*}

You might be able to complete this math in your head or you might need to do it out the long way.

For the long way, we rename each fraction. First, we make the mixed number into an improper fraction.

\begin{align*}\frac{5}{2}+\frac{3}{4}\end{align*}

Next, we rename these in terms of fourths since four is the least common denominator.

\begin{align*}\frac{10}{4}+\frac{3}{4} = \frac{13}{4} =3 \frac{1}{4}\end{align*}

Now we can take the sum of the extras and add it to the number of boxes in a case. Be careful that you don’t add it to the number of cases. We are talking about boxes here.

There are 25 boxes in 1 case.

The students will have a sum of \begin{align*}28 \frac{1}{4}\end{align*} boxes of pens.

### Vocabulary

Fraction
represents a part of a whole.
Improper fraction
a fraction where the numerator is greater than the denominator. It means that we have more than one whole represented.
Mixed Number
a whole number and a fraction written together.
Denominator
the bottom number in a fraction tells you how many parts the whole has been divided into.
Numerator
the top number in a fraction. It tells you how many parts you have out of the whole.

### Guided Practice

Here is one for you to try on your own.

\begin{align*}\frac{4}{6}+\frac{5}{8}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Solution

First, notice that these two fractions have uncommon denominators. The lowest common denominator for 6 and 8 is 24.

Next, we rename each fraction in terms of 24ths.

\begin{align*}\frac{16}{24} + \frac{15}{24}\end{align*}

Now we add the numerators.

\begin{align*}\frac{31}{24}\end{align*}

This is an improper fraction, so we have to change it to a mixed number.

\begin{align*}\frac{31}{24} = 1 \frac{7}{24}\end{align*}

This is our answer.

### Practice

Directions: Add or subtract the following fractions. Be sure that your answer is in simplest form.

1. \begin{align*}\frac{3}{6}+\frac{1}{6}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{2}{5}+\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{6}{10}+\frac{1}{10}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\frac{8}{12}+\frac{2}{12}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}\frac{9}{16}+\frac{1}{16}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}\frac{10}{20}+\frac{3}{20}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}\frac{18}{20}-\frac{3}{20}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}\frac{20}{21}-\frac{13}{21}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}\frac{16}{18}-\frac{10}{18} =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}\frac{24}{25}-\frac{9}{25}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}\frac{18}{36}-\frac{2}{36}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}\frac{28}{30}-\frac{10}{30}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Directions: Add or subtract the following mixed numbers and fractions. Be sure that your answer is in simplest form.

1. \begin{align*}2\frac{1}{2}+3=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}6\frac{4}{5}-\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}8\frac{1}{2}+\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}9\frac{4}{5}-2\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}6\frac{4}{9}-4\frac{1}{9}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}5\frac{1}{2}+2\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}8\frac{4}{6}+2\frac{2}{6}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}12\frac{5}{8}-2\frac{1}{8}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Directions: Add or subtract the following fractions with unlike denominators. Be sure that your answer is in simplest form.

1. \begin{align*}\frac{4}{5}+\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{6}{8}+\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{1}{3}+\frac{4}{9}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\frac{8}{9}-\frac{2}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}\frac{10}{12}-\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

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### Vocabulary Language: English

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.

LCD

The least common denominator or lowest common denominator (LCD) of two fractions is the smallest number that is a multiple of both of the original denominators.

LCM

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both of the original numbers.

Numerator

The numerator is the number above the fraction bar in a fraction.

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