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# Addition of Fractions

## 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?

### Guidance

25+15=?\begin{align*}\frac{2}{5}+\frac{1}{5}=?\end{align*}

The problem above can be represented using fraction strips.

25+15=2+15=35

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 25+15=35\begin{align*}\frac{2}{5}+\frac{1}{5}=\frac{3}{5}\end{align*}.

A number line can also be used to show the addition of fractions, as you will explore in Example C.

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.

#### Example A

37+27=?\begin{align*}\frac{3}{7}+\frac{2}{7}=?\end{align*}

Solution:

37+27=3+27=57

#### Example B

Louise is taking inventory of the amount of water in the water coolers located in the school. She estimates that one cooler is 23\begin{align*}\frac{2}{3}\end{align*} full and the other is 14\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?

Solution: Use fraction strips to represent each fraction.

23\begin{align*}\frac{2}{3}\end{align*} and 812\begin{align*}\frac{8}{12}\end{align*} are equivalent fractions. 23(44)=812\begin{align*}\frac{2}{3} \left(\frac{4}{4}\right)=\frac{8}{12}\end{align*}.

14\begin{align*}\frac{1}{4}\end{align*} and 312\begin{align*}\frac{3}{12}\end{align*} are equivalent fractions. 14(33)=312\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 1112\begin{align*}\frac{11}{12}\end{align*}.

23+14=812+312=1112

The denominator of 12 is the LCD (least common denominator) of 23\begin{align*}\frac{2}{3}\end{align*} and 14\begin{align*}\frac{1}{4}\end{align*} because it is the LCM (least common multiple) of the numbers 3 and 4.

#### Example C

What is 234+12\begin{align*}2\frac{3}{4}+\frac{1}{2}\end{align*}?

Solution: The number line is labeled in intervals of 4 which indicates that each interval represents 14\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 234\begin{align*}2 \frac{3}{4}\end{align*}.

From here, move to the right 12\begin{align*}\frac{1}{2}\end{align*} or 12\begin{align*}\frac{1}{2}\end{align*} of 4, which is 2 intervals.

The sum of 234\begin{align*}2 \frac{3}{4}\end{align*} and 12\begin{align*}\frac{1}{2}\end{align*} is 314\begin{align*}3\frac{1}{4}\end{align*}.

#### Concept Problem Revisited

Lily ate 38\begin{align*}\frac{3}{8}\end{align*} of the pizza because she ate 3 out of the 8 slices. Howard ate 48\begin{align*}\frac{4}{8}\end{align*} (or 12\begin{align*}\frac{1}{2}\end{align*}) of the pizza. Together they ate 7 slices which is 78\begin{align*}\frac{7}{8}\end{align*} of the pizza.

### Vocabulary

Denominator
The denominator of a fraction is the number on the bottom that indicates the total number of equal parts in the whole or the group. 58\begin{align*}\frac{5}{8}\end{align*} has denominator 8.
Fraction
A fraction is any rational number that is not an integer.
Improper Fraction
An improper fraction is a fraction in which the numerator is larger than the denominator.
83\begin{align*}\frac{8}{3}\end{align*} is an improper fraction.
LCD
The least common denominator is the lowest common multiple of the denominators of two or more fractions. The least common denominator of 34\begin{align*}\frac{3}{4}\end{align*} and 15\begin{align*}\frac{1}{5}\end{align*} is 20.
LCM
The least common multiple is the lowest common multiple that two or more numbers share. The least common multiple of 6 and 4 is 12.
Mixed Number
A mixed number is a number made up of a whole number and a fraction such as 435\begin{align*}4\frac{3}{5}\end{align*}.
Numerator
The numerator of a fraction is the number on top that is the number of equal parts being considered in the whole or the group. 58\begin{align*}\frac{5}{8}\end{align*} has numerator 5.

### Guided Practice

1. 12+16=?\begin{align*}\frac{1}{2}+\frac{1}{6}=?\end{align*}

2. 16+34=?\begin{align*}\frac{1}{6}+\frac{3}{4}=?\end{align*}

3. 25+23=?\begin{align*}\frac{2}{5}+\frac{2}{3}=?\end{align*}

1.

12+16=36+16=46=23

2.

16+34=212+912=1112

3.

25+23=615+1015=1615=1115

1615\begin{align*}\frac{16}{15}\end{align*} is an improper fraction. An improper fraction is one with a larger numerator than denominator. 1515=1\begin{align*}\frac{15}{15}=1\end{align*} plus there is 115\begin{align*}\frac{1}{15}\end{align*} left over. This can be written as a whole number and a fraction 1115\begin{align*}1 \frac{1}{15}\end{align*}. This representation is called a mixed number.

### Practice

Complete the following addition problems using any method.

1. 14+58\begin{align*}\frac{1}{4}+\frac{5}{8}\end{align*}
2. 25+13\begin{align*}\frac{2}{5}+\frac{1}{3}\end{align*}
3. 29+23\begin{align*}\frac{2}{9}+\frac{2}{3}\end{align*}
4. 37+23\begin{align*}\frac{3}{7}+\frac{2}{3}\end{align*}
5. 710+15\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:
Justin’s Solution:
Who is correct? What would you tell the person who has the wrong answer?

### Vocabulary Language: English

Denominator

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

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

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

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

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

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

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

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

Mixed Number

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