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Add fractions and mixed numbers with like and unlike denominators.

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Add and Subtract Fractions and Mixed Numbers

Donna and her friend Andrew work at The Balloon Factory. They are in charge of taking inventory before the big sale tomorrow. Donna opens the first two boxes and one 12\begin{align*}\frac{1}{2}\end{align*} is full while the other is 13\begin{align*}\frac{1}{3}\end{align*} full. The boxes are the same size so she empties one box into the other. What does she write on the top for the sum of the contents? Andrew is working in the stockroom with the helium balloons. He finds 314\begin{align*}3 \frac{1}{4}\end{align*} boxes of red balloons and 223\begin{align*}2 \frac{2}{3}\end{align*} boxes of green balloons. What does he record as the sum of these two types?

In this concept, you will learn to add and subtract fractions and mixed numbers.

A fraction is a number that names a part of a whole or a part of a group. If a rectangle is 13\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, you use an improper fraction or a mixed number. An improper fraction has a numerator that is larger than its denominator, such as 53\begin{align*}\frac{5}{3}\end{align*}, or can be written as the mixed, such as 123\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. 123\begin{align*}1 \frac{2}{3}\end{align*} is a mixed number.

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

The first thing you need to look at when adding and subtracting fractions is the denominator of the fractions to see if they are the same. Remember that the denominator tells you how many parts the whole is divided into. The numerator tells you how many parts you have out of the whole. In our rectangle example above, the fraction of shaded area 13\begin{align*}\frac{1}{3}\end{align*} is because there is 1 part of the 3 parts that is shaded.

When adding or subtracting fractions, the denominators of the fractions must be the same, then you can simply add or subtract the numerators. For example, notice in the fractions below, both denominators are 8 so you just add the numerators (1+2=3)\begin{align*}(1 + 2 = 3)\end{align*} .

Subtracting fractions that have the same denominator is the same. In the example below, both denominators are the same so you just need to subtract the numerators (103=7)\begin{align*}(10 – 3 = 7)\end{align*}.

1012312=712\begin{align*}\frac{10}{12} - \frac{3}{12} = \frac{7}{12}\end{align*}

If the denominators are not the same, you need to find a common denominator. You do this by creating an equal fraction that has the denominator you need. Look at the fractions below.

12=48\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.

Let’s look at an example.

Add: 14+25\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, and 20. The first few multiples of 5 are 5, 10, 15, and 20. So the least common multiple is 20.

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

You need to multiply the denominator by 5 to get a denominator of 20. So, multiply the first fraction by the equivalent of 1or 55\begin{align*}\frac{5}{5}\end{align*}.

14×55=520\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 1or 44\begin{align*}\frac{4}{4}\end{align*}.

25×44=820\begin{align*}\frac{2}{5} \times \frac{4}{4} = \frac{8}{20}\end{align*}

520+820=1320\begin{align*}\frac{5}{20} + \frac{8}{20} = \frac{13}{20}\end{align*}

The answer is 1320\begin{align*}\frac{13}{20}\end{align*}.

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

Subtract: 27823\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.

8×2+7278=23=238\begin{align*}8 \times 2 + 7& = 23\\ 2 \frac{7}{8} &= \frac{23}{8}\end{align*}

Next, find a common denominator.

The first few multiples of 8 are 8, 16, and 24. Since 24 is also divisible by 3, it is the least common multiple. Rename each fraction as a fraction with the common denominator.

238×3323×88=6924=1624\begin{align*}\frac{23}{8} \times \frac{3}{3} &= \frac{69}{24}\\ \frac{2}{3} \times \frac{8}{8} &= \frac{16}{24}\end{align*}

Then, find the difference.

69241624=5324\begin{align*}\frac{69}{24} - \frac{16}{24} = \frac{53}{24}\end{align*}

Notice that the answer is an improper fraction. You need to put your answer into a mixed fraction to finish this question.

Finally, simplify the difference to a mixed number.

5324+2524\begin{align*}\frac{53}{24} + 2 \frac{5}{24}\end{align*}

The answer is 2524\begin{align*}2 \frac{5}{24}\end{align*}.

Examples

Example 1

Earlier, you were given a problem about  Donna and Andrew with their inventory.

Donna combined two boxes of regular balloons into one box. She wanted to know what to record for her final total. She had 12\begin{align*}\frac{1}{2}\end{align*} box that she combined with 13\begin{align*}\frac{1}{3}\end{align*} of a box.

To find the total, Donna must first find a common denominator. Since 2 and 3 are both multiples of 6, it is the least common multiple.

Next, rename each fraction as a fraction with the common denominator.

12×3313×22=36=26\begin{align*}\frac{1}{2} \times \frac{3}{3} &= \frac{3}{6}\\ \frac{1}{3} \times \frac{2}{2} &= \frac{2}{6}\end{align*}

36+26=56\begin{align*}\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\end{align*}

So Donna would write 56\begin{align*}\frac{5}{6}\end{align*} on the top of the box of balloons.

Andrew was doing inventory on helium balloons. He wanted to know the total of his two boxes to record on his inventory. He had 314\begin{align*}3 \frac{1}{4}\end{align*}  of one box and 223\begin{align*}2 \frac{2}{3}\end{align*} of another box.

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

4×3+1314=133×2+2=8=134223=83\begin{align*}4 \times 3 + 1 &= 13 \qquad \qquad \qquad \qquad 3 \times 2 + 2 = 8\\ 3 \frac{1}{4} &= \frac{13}{4} \qquad \qquad \qquad \qquad \qquad 2 \frac{2}{3} = \frac{8}{3}\end{align*}

Next, find a common denominator.

The first few multiples of 4 are 4, 8, and 12. Since 12 is also divisible by 3, it is the least common multiple. Rename each fraction as a fraction with the common denominator.

3912+3212=7112\begin{align*}\frac{39}{12} + \frac{32}{12} = \frac{71}{12}\end{align*}

Notice that the answer is an improper fraction. You need to put your answer into a mixed fraction to finish this problem.

Finally, simplify the solution to a mixed number.

7112=51112\begin{align*}\frac{71}{12} = 5 \frac{11}{12}\end{align*}

So, Andrew should write 51112\begin{align*}5 \frac{11}{12}\end{align*} as the total number of red and green helium balloons.

Example 2

46+58=\begin{align*}\frac{4}{6} + \frac{5}{8} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

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

Next, rename each fraction with a denominator of 24.

46×4458×33=1624=1524\begin{align*}\frac{4}{6} \times \frac{4}{4} &= \frac{16}{24}\\ \frac{5}{8} \times \frac{3}{3} &= \frac{15}{24}\end{align*}

1624+1524=3124\begin{align*}\frac{16}{24} + \frac{15}{24} = \frac{31}{24}\end{align*}

Since the numerator is greater than the denominator you can put you final answer into an improper fraction.

3124=1724\begin{align*}\frac{31}{24} = 1 \frac{7}{24}\end{align*}

The answer is 1724\begin{align*}1 \frac{7}{24}\end{align*}.

Example 3

Subtract: 4916\begin{align*}\frac{4}{9} - \frac{1}{6}\end{align*}

First, find a common denominator. The first few multiples of 9 are 9 and 18. Since 18 is also divisible by 6, it is the least common multiple.

Next, rename each fraction as a fraction with the common denominator.

49×2216×33=818=318\begin{align*}\frac{4}{9} \times \frac{2}{2} &= \frac{8}{18}\\ \frac{1}{6} \times \frac{3}{3} &= \frac{3}{18}\end{align*}

Then, find the difference.

818318=518\begin{align*}\frac{8}{18} - \frac{3}{18} = \frac{5}{18}\end{align*}

The answer is 518\begin{align*}\frac{5}{18}\end{align*}.

Example 4

Add: 1012+26\begin{align*}\frac{10}{12} + \frac{2}{6}\end{align*}

First, find a common denominator. Since 12 is also divisible by 6, it is the least common multiple.

Next, rename each fraction as a fraction with the common denominator.

26×22=412\begin{align*}\frac{2}{6} \times \frac{2}{2} = \frac{4}{12}\end{align*}

1012+412=1412\begin{align*}\frac{10}{12} + \frac{4}{12} = \frac{14}{12}\end{align*}

Since the numerator is greater than the denominator you can put you final answer into an improper fraction. R,emember you need to put your answer into simplest form.

1421212=1212=116\begin{align*}\frac{14}{2} &= 1 \frac{2}{12}\\ 1 \frac{2}{12} &= 1 \frac{1}{6}\end{align*}

The answer is \begin{align*}1 \frac{1}{6}\end{align*}.

Example 5

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

First, find a common denominator. Since 8 is also divisible by 4, it is the least common multiple.

Next, rename each fraction as a fraction with the common denominator.

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

Then, find the difference.

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

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

The answer is \begin{align*}\frac{1}{4}\end{align*}.

Review

1.  \begin{align*}\frac{3}{6}+\frac{1}{6}\end{align*}
2.  \begin{align*}\frac{2}{5}+\frac{1}{5}\end{align*}
3.  \begin{align*}\frac{6}{10}+\frac{1}{10}\end{align*}
4.  \begin{align*}\frac{8}{12}+\frac{2}{12}\end{align*}
5.  \begin{align*}\frac{9}{16}+\frac{1}{16}\end{align*}
6.  \begin{align*}\frac{10}{20}+\frac{3}{20}\end{align*}
7.  \begin{align*}\frac{18}{20}-\frac{3}{20}\end{align*}
8.  \begin{align*}\frac{20}{21}-\frac{13}{21}\end{align*}
9.  \begin{align*}\frac{16}{18}-\frac{10}{18}\end{align*}
10.  \begin{align*}\frac{24}{25}-\frac{9}{25}\end{align*}
11.  \begin{align*}\frac{18}{38}-\frac{2}{36}\end{align*}
12.  \begin{align*}\frac{28}{30}-\frac{10}{30}\end{align*}

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

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

To see the Review answers, open this PDF file and look for section 2.6.

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Color Highlighted Text Notes

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.