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# Subtraction of Fractions

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

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Subtraction of Fractions

Julian and Suz ordered a pizza that was cut into 10 slices. Suz ate 3 slices and Julian ate 4 slices. What fraction of the pizza did each person eat? What fraction of the pizza is left?

### Guidance

\begin{align*}\frac{5}{7}-\frac{2}{7}=?\end{align*}

The problem above can be represented with fraction strips:

To subtract fractions, the fractions must have the same bottom numbers (denominators). In this case, both fractions have a denominator of 7. The answer is the result of subtracting the top numbers (numerators).

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

#### Example A

\begin{align*}\frac{8}{11}-\frac{6}{11}=?\end{align*}

Solution:

#### Example B

Bessie is measuring the amount of soda in the two coolers in the cafeteria. She estimates that the first cooler is \begin{align*}\frac{2}{3}\end{align*} full and the second cooler is \begin{align*}\frac{1}{4}\end{align*} full. What single fraction could Bessie use to represent how much more soda is in the first cooler than in the second cooler?

Solution: 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 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 denominators 3 and 4.

Therefore, there is \begin{align*}\frac{5}{12}\end{align*} more soda in the first cooler than in the second.

#### Example C

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

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

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

#### Concept Problem Revisited

Julian and Suz ordered a pizza that was cut into 10 slices. Suz ate 3 slices and Julian ate 4 slices. What fraction of the pizza did each person eat? What fraction of the pizza is left?

Suz ate \begin{align*}\frac{3}{10}\end{align*} of the pizza because she ate 3 out of the 10 slices. Julian ate \begin{align*}\frac{4}{10}\end{align*} of the pizza. Together they ate \begin{align*}\frac{7}{10}\end{align*} of the pizza. \begin{align*}\frac{10}{10}-\frac{7}{10}=\frac{3}{10}\end{align*}. Therefore, \begin{align*}\frac{3}{10}\end{align*} of the pizza is left.

### 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. \begin{align*}\frac{5}{8}\end{align*} has denominator 8.
Fraction
A fraction is any rational number that is not an integer.
LCD
The least common denominator is the lowest common multiple of the denominators of two or more fractions. The least common denominator of \begin{align*}\frac{3}{4}\end{align*} and \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 5 is 30.
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. \begin{align*}\frac{5}{8}\end{align*} has 'numerator 5.

### Guided Practice

1. \begin{align*}\frac{7}{10}-\frac{2}{5}=?\end{align*}

2. \begin{align*}\frac{7}{8}-\frac{1}{2}\end{align*}.

3. \begin{align*}\frac{5}{8}-\frac{1}{3}=?\end{align*}

4. \begin{align*}\frac{4}{5}-\frac{1}{4}=?\end{align*}

1.

2. \begin{align*}\frac{7}{8}-\frac{4}{8}=\frac{7-4}{8}=\frac{3}{8}\end{align*}.

3. \begin{align*}\frac{5}{8}-\frac{1}{3}=\frac{7}{24}\end{align*}

4. \begin{align*}\frac{4}{5}-\frac{1}{4}=\frac{11}{20}\end{align*}

### Practice

Complete the following subtraction problems using any method.

1. \begin{align*}\frac{3}{4}-\frac{5}{8}\end{align*}
2. \begin{align*}\frac{4}{5}-\frac{2}{3}\end{align*}
3. \begin{align*}\frac{5}{9}-\frac{2}{3}\end{align*}
4. \begin{align*}\frac{6}{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{3}{5}-\frac{3}{10}\end{align*}
8. \begin{align*}\frac{7}{9}-\frac{1}{3}\end{align*}
9. \begin{align*}\frac{5}{8}-\frac{1}{4}\end{align*}
10. \begin{align*}\frac{2}{5}-\frac{2}{10}\end{align*}
11. \begin{align*}\frac{7}{11}-\frac{1}{2}\end{align*}
12. \begin{align*}\frac{5}{8}-\frac{5}{12}\end{align*}
13. \begin{align*}\frac{5}{6}-\frac{3}{4}\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 a subtraction statement and find the result.

1. Sally used \begin{align*}\frac{2}{3} \ cups\end{align*} of flour to make cookies. Terri used \begin{align*}\frac{1}{2} \ cups\end{align*} of flour to make a cake. Who used more flour? How much more flour did she use?
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. Alyssa used \begin{align*}\frac{3}{8} \ cup\end{align*} of milk, \begin{align*}2 \frac{1}{4} \ cups\end{align*} of flour and \begin{align*}\frac{1}{3} \ cup\end{align*} of melted butter to make waffles. Who used more cups of ingredients? How many more cups of ingredients did she use?
3. Write two fractions with different denominators whose difference is \begin{align*}\frac{3}{8}\end{align*}.
4. Jake’s dog ate \begin{align*}12 \frac{2}{3} \ cans\end{align*} of food in one week and \begin{align*}9 \frac{1}{4} \ cans\end{align*} the next week. How many more cans of dog food did Jake’s dog eat in week one?
5. Sierra and Clark each solved the same problem.
Sierra’s Solution
Clark’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 $8$.
fraction

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.
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.
Numerator

Numerator

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