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

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Lily and Howard ordered a pizza that was cut into 8 slices. Lily ate 3 slices and Howard ate 4 slices. What fraction of the pizza did each person eat? What fraction of the pizza did they eat all together?

What is the answer to \begin{align*}\frac{2}{5}+\frac{1}{5}=?\end{align*}

The problem above can be represented using fraction strips.

\begin{align*}\boxed{\frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5}}\end{align*}

To add fractions, the fractions must have the same bottom numbers (denominators). In this case, both fractions have a denominator of 5. The answer is the result of adding the top numbers (numerators). The numbers in the numerator are 1 and 2. The sum of 1 and 2 is 3. This sum is written in the numerator over the denominator of 5. Therefore \begin{align*}\frac{2}{5}+\frac{1}{5}=\frac{3}{5}\end{align*}.

The sum of two fractions will sometimes result in an answer that is an improper fraction. An improper fraction is a fraction that has a larger numerator than denominator. This answer can be written as a mixed number. A mixed number is a number made up of a whole number and a fraction.

In order to add fractions that have different denominators, the fractions must be expressed as equivalent fractions with a least common denominator (LCD). The sum of the numerators can be written over the common denominator.

Both fraction strips and number lines can be used to visualize the addition of fractions.

#### Let's practice adding fractions using fraction strips:

1. \begin{align*}\frac{3}{7}+\frac{2}{7}=?\end{align*}

\begin{align*}\boxed{\frac{3}{7} + \frac{2}{7} = \frac{3+2}{7} = \frac{5}{7}}\end{align*}

1. Louise is taking inventory of the amount of water in the water coolers located in the school. She estimates that one cooler is \begin{align*}\frac{2}{3}\end{align*} full and the other is \begin{align*}\frac{1}{4}\end{align*} full. What single fraction could Louise use to represent the amount of water of the two coolers together?

Use fraction strips to represent each fraction.

\begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{8}{12}\end{align*} are equivalent fractions. \begin{align*}\frac{2}{3} \left(\frac{4}{4}\right)=\frac{8}{12}\end{align*}.

\begin{align*}\frac{1}{4}\end{align*} and \begin{align*}\frac{3}{12}\end{align*} are equivalent fractions. \begin{align*}\frac{1}{4} \left(\frac{3}{3}\right)=\frac{3}{12}\end{align*}.

The two green pieces will be replaced with eight purple pieces and the one blue piece will be replaced with three purple pieces.

The amount of water in the two coolers can be represented by the single fraction \begin{align*}\frac{11}{12}\end{align*}.

\begin{align*} \frac{2}{3}+\frac{1}{4}=\frac{8}{12}+\frac{3}{12}=\frac{11}{12}\end{align*}

The denominator of 12 is the LCD (least common denominator) of \begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{1}{4}\end{align*} because it is the LCM (least common multiple) of the numbers 3 and 4.

#### Now, let's add the fractions using a number line:

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

The number line is labeled in intervals of 4 which indicates that each interval represents \begin{align*}\frac{1}{4}\end{align*}. From zero, move to the number 2 plus 3 more intervals to the right. Mark the location. This represents \begin{align*}2 \frac{3}{4}\end{align*}.

From here, move to the right \begin{align*}\frac{1}{2}\end{align*} or \begin{align*}\frac{1}{2}\end{align*} of 4, which is 2 intervals.

The sum of \begin{align*}2 \frac{3}{4}\end{align*} and \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}3\frac{1}{4}\end{align*}.

### Examples

#### Example 1

Earlier, you were told that Lily ate 3 slices and Howard ate 4 slices of an 8 slice pizza. What fraction of the pizza did each person eat? What fraction of the pizza did they eat all together?

Lily ate \begin{align*}\frac{3}{8}\end{align*} of the pizza because she ate 3 out of the 8 slices. Howard ate \begin{align*}\frac{4}{8}\end{align*} (or \begin{align*}\frac{1}{2}\end{align*}) of the pizza. Together they ate 7 slices, which is \begin{align*}\frac{7}{8}\end{align*} of the pizza.

#### Example 2

Add the fractions: \begin{align*}\frac{1}{2}+\frac{1}{6}=?\end{align*}

\begin{align*}& \frac{1}{2}+\frac{1}{6}=\frac{3}{6}+\frac{1}{6}= \frac{4}{6}=\frac{2}{3}\end{align*}

#### Example 3

Add the fractions: \begin{align*}\frac{1}{6}+\frac{3}{4}=?\end{align*}

\begin{align*}& \frac{1}{6}+\frac{3}{4}=\frac{2}{12}+\frac{9}{12}=\frac{11}{12}\end{align*}

#### Example 4

Add the fractions: \begin{align*}\frac{2}{5}+\frac{2}{3}=?\end{align*}

\begin{align*}& \frac{2}{5}+\frac{2}{3}=\frac{6}{15}+\frac{10}{15}=\frac{16}{15}=1 \frac{1}{15}\end{align*}

\begin{align*}\frac{16}{15}\end{align*} is an improper fraction. An improper fraction is one with a larger numerator than denominator. \begin{align*}\frac{15}{15}=1\end{align*} plus there is \begin{align*}\frac{1}{15}\end{align*} left over. This can be written as a whole number and a fraction \begin{align*}1 \frac{1}{15}\end{align*}. This representation is called a mixed number.

### Review

Complete the following addition problems using any method.

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

For each of the following questions, write an addition statement and find the result. Express all answers as either proper fraction or mixed numbers.

1. Karen used \begin{align*}\frac{5}{8} \ cups\end{align*} of flour to make cookies. Jenny used \begin{align*}\frac{15}{16} \ cups\end{align*} of flour to make a cake. How much flour did they use altogether?
2. Lauren used \begin{align*}\frac{3}{4} \ cup\end{align*} of milk, \begin{align*}1 \frac{1}{3} \ cups\end{align*} of flour and \begin{align*}\frac{3}{8} \ cup\end{align*} of oil to make pancakes. How many cups of ingredients did she use in total?
3. Write two fractions with different denominators whose sum is \begin{align*}\frac{5}{6}\end{align*}.
4. Allan’s cat ate \begin{align*}2 \frac{2}{3} \ cans\end{align*} of food in one week and \begin{align*}3 \frac{1}{4} \ cans\end{align*} the next week. How many cans of food did the cat eat in two weeks?
5. Amanda and Justin each solved the same problem.
Amanda’s Solution:
\begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{24}\end{align*}
Justin’s Solution:
\begin{align*}& \frac{1}{6}+\frac{3}{4}\\ & \frac{2}{12}+\frac{9}{12}\\ & =\frac{11}{12}\end{align*}
Who is correct? What would you tell the person who has the wrong answer?

To see the Review answers, open this PDF file and look for section 1.2.

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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.
Least Common Denominator The least common denominator or lowest common denominator of two fractions is the smallest number that is a multiple of both of the original denominators.
Least Common Multiple The least common multiple 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 $4\frac{3}{5}$.

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