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## Add fractions and mixed numbers with like and unlike denominators.

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Luke is making cookies to share with his teachers for the holidays. He has decided to make three different types of cookies. For the first type of cookie he needs \begin{align*}\frac{3}{4}\end{align*} cups of sugar. For the second type of cookie he needs \begin{align*}1 \frac{1}{3}\end{align*} cups of sugar. For the third type of cookie he needs \begin{align*}2 \frac{1}{2}\end{align*} cups of sugar.  His mom just told him that they only have about 5 cups of sugar in the house. How can Luke figure out if he will have enough sugar to make all of his cookies?

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

### Adding Fractions and Mixed Numbers

A fraction describes a part of a whole number. The number written below the bar in a fraction is the denominator, and it tells you how many parts the whole is divided into. The number above the bar in a fraction is the numerator, and it tells you how many parts of the whole you have. Equivalent fractions are fractions that describe the same part of a whole.

1. Rewrite each fraction as an equivalent fraction so that the denominators are the same.
2. Add the numerators of the equivalent fractions. Keep the denominator the same.

Here is an example.

Find the sum of \begin{align*}\frac{1}{2}\end{align*} and \begin{align*}\frac{1}{3}\end{align*}.

First, find a common denominator. You are looking for a number that is a multiple of both 2 and 3. The product of 2 and 3 is 6 and in this case that is the least common multiple of 2 and 3.

Now, rewrite each fraction as an equivalent fraction with a denominator of 6. Remember to always multiply the numerator and denominator of the fraction by the same number.

\begin{align*}\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \\ \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\end{align*}

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same.

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

Finally, check to make sure your answer is in simplest form. \begin{align*}\frac{5}{6}\end{align*} is in simplest form because 5 and 6 do not have any common factors besides 1.

The answer is \begin{align*}\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\end{align*}.

Numbers that consist of a whole number and a fraction are called mixed numbers.

1. Add the fractional parts of the mixed numbers.
2. Add the whole parts of the mixed numbers.
3. Add the sum of the fractional parts to the sum of the whole numbers.

Here is an example.

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

First, add the fractional parts: \begin{align*}\frac{3}{4}+ \frac{1}{3}\end{align*}. Find a common denominator. You are looking for a number that is a multiple of both 4 and 3. 12 is the least common multiple of 4 and 3.

Now, rewrite each fraction as an equivalent fraction with a denominator of 12.

\begin{align*}\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \\ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \end{align*}

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same. Rewrite your result as a mixed number.

\begin{align*}\frac{9}{12}+\frac{4}{12} = \frac{13}{12} = 1 \frac{1}{12}\end{align*}

Now, add the whole part of the original mixed number to your sum so far.

\begin{align*}2+1 \frac{1}{12}=3 \frac{1}{12}\end{align*}

Finally, check to make sure your answer is in simplest form. \begin{align*}3 \frac{1}{12}\end{align*} is in simplest form because 1 and 12 do not have any common factors besides 1.

The answer is \begin{align*}\frac{3}{4}+2 \frac{1}{3}=3 \frac{1}{12}\end{align*}.

### Examples

#### Example 1

He is making three different types of cookies. For the first type of cookie he needs \begin{align*}\frac{3}{4}\end{align*} cups of sugar, for the second type of cookie he needs \begin{align*}1\frac{1}{3}\end{align*} cups of sugar, and for the third type he needs \begin{align*}2\frac{1}{2}\end{align*} cups of sugar. His mom told him that they only have about 5 cups of sugar in the house. Luke needs to make sure he has enough sugar to make all of his cookies.

To figure this out, Luke will need to add the three amounts of sugar and see if he needs less than 5 cups.

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of 4, 3, and 2. 12 is the least common multiple of 4, 3, and 2.

Now, rewrite each fraction as an equivalent fraction with a denominator of 12.

\begin{align*}\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \\ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \\ \frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} \end{align*}

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same. Rewrite your result as a mixed number.

\begin{align*}\frac{9}{12}+\frac{4}{12}+\frac{6}{12}=\frac{19}{12}=1 \frac{7}{12} \end{align*}

Now, add the whole parts of the original mixed numbers.

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

Next, add the sum of the fractional parts to the sum of the whole numbers.

\begin{align*}3+1 \frac{7}{12}=4 \frac{7}{12}\end{align*}

Luke needs \begin{align*}4 \frac{7}{12}\end{align*} cups of sugar.

The answer is that, because Luke needs \begin{align*}4 \frac{7}{12}\end{align*} cups of sugar and he has 5 cups of sugar, he will have enough sugar to make all three types of cookies.

#### Example 2

Donte is making a costume with blue, red, and black fabric. He has \begin{align*}6 \frac{1}{2}\end{align*} yards of blue fabric, \begin{align*}3 \frac{2}{3}\end{align*} yards of red fabric, and \begin{align*}5 \frac{4}{5}\end{align*} yards of black fabric. How many yards of fabric does Donte have all together?

In order to solve this problem, you need to add the three given mixed numbers.

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of 2, 3, and 5. 30 is the least common multiple of 2, 3, and 5.

Now, rewrite each fraction as an equivalent fraction with a denominator of 30.

\begin{align*}\frac{1}{2} = \frac{1 \times 15}{2 \times 15} = \frac{15}{30} \\ \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \\ \frac{4}{5} = \frac{4 \times 6}{5 \times 6}\ = \frac{24}{30}\end{align*}

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same. Rewrite your result as a mixed number.

\begin{align*}\frac{15}{30}+\frac{20}{30}+\frac{24}{30}=\frac{59}{30}=1 \frac{29}{30}\end{align*}

Now, add the whole parts of the original mixed numbers.

\begin{align*}6+3+5=14\end{align*}

Next, add the sum of the fractional parts to the sum of the whole numbers.

\begin{align*}14+1 \frac{29}{30}=15 \frac{29}{30}\end{align*}

Finally, check to make sure your answer is in simplest form. \begin{align*}15 \frac{29}{30}\end{align*} is in simplest form because 29 and 30 do not have any common factors besides 1.

The answer is that Donte has \begin{align*}15 \frac{29}{30}\end{align*} yards of fabric all together.

#### Example 3

Add \begin{align*}9 \frac{1}{2}+22 \frac{1}{4}\end{align*}.

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of both 2 and 4. 4 is the least common multiple of 2 and 4.

Now, rewrite each fraction as an equivalent fraction with a denominator of 4.

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

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same.

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

Now, add the whole parts of the original mixed numbers.

\begin{align*}9+22=31\end{align*}

Next, add the sum of the fractional parts to the sum of the whole numbers.

\begin{align*}31+ \frac{3}{4} = 31 \frac{3}{4}\end{align*}

Finally, check to make sure your answer is in simplest form. \begin{align*}31 \frac{3}{4}\end{align*} is in simplest form because 3 and 4 do not have any common factors besides 1.

The answer is \begin{align*}9 \frac{1}{2}+22 \frac{1}{4} = 31 \frac{3}{4}\end{align*}.

#### Example 4

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

First, add the fractional parts of the mixed numbers. You will add the numerators and keep the denominator the same. Rewrite your result as a whole number.

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

Now, add the whole parts of the original mixed numbers.

\begin{align*}2+8=10\end{align*}

Next, add the sum of the fractional parts to the sum of the whole numbers.

\begin{align*}10+1=11\end{align*}

The answer is \begin{align*}2 \frac{1}{3}+8 \frac{2}{3}=11\end{align*}.

#### Example 5

Add \begin{align*}5 \frac{1}{3}+\frac{1}{7}\end{align*}.

First, add the fractional parts of the mixed numbers. Find a common denominator. You are looking for a number that is a multiple of both 3 and 7. 21 is the least common multiple of 3 and 7.

Now, rewrite each fraction as an equivalent fraction with a denominator of 21.

\begin{align*}\frac{1}{3} = \frac{1 \times 7}{3 \times 7}= \frac{7}{21} \\ \frac{1}{7}= \frac{1 \times 3}{7 \times 3}= \frac{3}{21} \end{align*}

Next, add the equivalent fractions. You will add the numerators and keep the denominator the same.

\begin{align*}\frac{7}{21}+ \frac{3}{21} = \frac{10}{21}\end{align*}

Now, add the sum of the fractional parts to the original whole number.

\begin{align*}5+\frac{10}{21}=5 \frac{10}{21}\end{align*}

Finally, check to make sure your answer is in simplest form. \begin{align*}5 \frac{10}{21}\end{align*} is in simplest form because 10 and 21 do not have any common factors besides 1.

The answer is \begin{align*}5 \frac{1}{3}+\frac{1}{7}=5 \frac{10}{21}\end{align*}.

### Review

1. \begin{align*}\frac{3}{7} + \frac{1}{14}\end{align*}
2. \begin{align*}\frac{3}{4} + \frac{1}{3}\end{align*}
3. \begin{align*}\frac{2}{5} + \frac{3}{10}\end{align*}
4. \begin{align*}\frac{1}{9} + \frac{1}{6}\end{align*}
5. \begin{align*}2\frac{3}{5} + \frac{17}{20}\end{align*}
6. \begin{align*}8\frac{5}{12} + 2\frac{1}{6}\end{align*}
7. \begin{align*}1\frac{2}{3} + 1\frac{3}{4}\end{align*}
8. \begin{align*}2\frac{1}{5} + 4\frac{14}{15}\end{align*}
9. \begin{align*}2\frac{1}{5} + 4\frac{2}{15}\end{align*}
10. \begin{align*}12\frac{1}{5} + 4\frac{6}{15}\end{align*}
11. \begin{align*}2\frac{3}{4} + 4\frac{1}{6}\end{align*}
12. \begin{align*}3\frac{4}{10} + 6\frac{1}{9}\end{align*}
13. \begin{align*}2\frac{1}{30} + \frac{12}{15}\end{align*}
14. \begin{align*}4\frac{1}{2} + 14\frac{4}{5}\end{align*}
15. \begin{align*}16\frac{8}{9} + 14\frac{9}{12}\end{align*}

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### Vocabulary Language: English

TermDefinition
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.
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.
Mixed Number A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.