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# 1.5: Subtraction of Fractions

Difficulty Level: At Grade Created by: CK-12

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

5727=?\begin{align*}\frac{5}{7}-\frac{2}{7}=?\end{align*}

The problem can be represented by using fraction strips.

5727=527=37

To subtract fractions, the fractions must have the same bottom numbers (denominators). Both fractions have a denominator of 7. The answer is the result of subtracting the top numbers (numerators). The numbers in the numerator are 5 and 2. The difference of 5 and 2 is 3. This difference is written in the numerator over the denominator of 7. Therefore 5727=37\begin{align*}\frac{5}{7}-\frac{2}{7}=\frac{3}{7}\end{align*}.

When you subtract fractions that have different denominators, you must express the fractions as equivalent fractions with a LCD. Now, subtract the numerators and write the difference over the common denominator.

#### Example A

811611=?\begin{align*}\frac{8}{11}-\frac{6}{11}=?\end{align*}

811611=8611=211

#### Example B

Bessie is measuring the amount of soda in the two coolers in the cafeteria. She estimates that the first cooler is 23\begin{align*}\frac{2}{3}\end{align*} full and the second cooler is 14\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?

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 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 denominators 3 and 4.

#### Example C

A number line can also be used to subtract fractions. In the following example, a mixed number which is a whole number and a fraction will be added to a fraction by using a 14\begin{align*}\frac{1}{4}\end{align*} number line.

13412\begin{align*}1 \frac{3}{4}-\frac{1}{2}\end{align*}

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 1 plus 3 more intervals to the right. Mark the location. This represents 134\begin{align*}1 \frac{3}{4}\end{align*}.

From here, move to the left 12\begin{align*}\frac{1}{2}\end{align*} or 12\begin{align*}\frac{1}{2}\end{align*} of 4, which is 2 intervals. An equivalent fraction for 12\begin{align*}\frac{1}{2}\end{align*} is 24\begin{align*}\frac{2}{4}\end{align*}.

The difference of 134\begin{align*}1 \frac{3}{4}\end{align*} and 12\begin{align*}\frac{1}{2}\end{align*} is 114\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 at 310\begin{align*}\frac{3}{10}\end{align*} of the pizza because she ate 3 out of the 10 slices. Julian ate 410\begin{align*}\frac{4}{10}\end{align*} of the pizza. Together they ate 710\begin{align*}\frac{7}{10}\end{align*} of the pizza. 1010710=310\begin{align*}\frac{10}{10}-\frac{7}{10}=\frac{3}{10}\end{align*}. Therefore, 310\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. 58\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 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 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. 58\begin{align*}\frac{5}{8}\end{align*} has numerator 5.

### Guided Practice

1. Use a model to answer the problem 71025=?\begin{align*}\frac{7}{10}-\frac{2}{5}=?\end{align*}

2. Use a number line to determine the answer to the problem 7812\begin{align*}\frac{7}{8}-\frac{1}{2}\end{align*}.

3. Determine the answer to 5813=?\begin{align*}\frac{5}{8}-\frac{1}{3}=?\end{align*} and 4514=?\begin{align*}\frac{4}{5}-\frac{1}{4}=?\end{align*} by using the rules for subtracting fractions.

1.

Use fraction strips to represent each fraction.

The fractions do not have a common denominator. An equivalent fraction for 25\begin{align*}\frac{2}{5}\end{align*} is (22)(25)=410\begin{align*}\left(\frac{2}{2}\right) \left(\frac{2}{5}\right)=\frac{4}{10}\end{align*}

When four strips have been removed from 710\begin{align*}\frac{7}{10}\end{align*}, there are 310\begin{align*}\frac{3}{10}\end{align*} remaining.

710410=7410=310

2. 7812\begin{align*}\frac{7}{8}-\frac{1}{2}\end{align*}

Use a 18\begin{align*}\frac{1}{8}\end{align*} number line. The number line is labeled in intervals of 8. Place the starting point at 78\begin{align*}\frac{7}{8}\end{align*}.

From this point, move to the left a total of 4 intervals. 12\begin{align*}\frac{1}{2}\end{align*} of 8=4\begin{align*}8=4\end{align*}. An equivalent fraction for 12\begin{align*}\frac{1}{2}\end{align*} is 12(44)=48\begin{align*}\frac{1}{2} \left(\frac{4}{4} \right)=\frac{4}{8}\end{align*}. The point where you stop is the difference of 78\begin{align*}\frac{7}{8}\end{align*} and 12\begin{align*}\frac{1}{2}\end{align*}.

On the number line you stopped at the point 38\begin{align*}\frac{3}{8}\end{align*}. This is equal to 7848=748=38\begin{align*}\frac{7}{8}-\frac{4}{8}=\frac{7-4}{8}=\frac{3}{8}\end{align*}.

3. 5813=?\begin{align*}\frac{5}{8}-\frac{1}{3}=?\end{align*}

The least common multiple of 8 and 3 is 24. This means that both fractions must have a common denominator of 24 before they can be subtracted.

58(33)=1524\begin{align*}\frac{5}{8} \left(\frac{3}{3}\right)=\frac{15}{24}\end{align*}

13(88)=824\begin{align*}\frac{1}{3} \left(\frac{8}{8}\right)=\frac{8}{24}\end{align*}

58131524824=15824=724

4514=?\begin{align*}\frac{4}{5}-\frac{1}{4}=?\end{align*}

The least common multiple of 5 and 4 is 20. This means that both fractions must have a common denominator of 20 before they can be added.

45(44)=1620\begin{align*}\frac{4}{5} \left(\frac{4}{4}\right)=\frac{16}{20}\end{align*}

14(55)=520\begin{align*}\frac{1}{4} \left(\frac{5}{5}\right)=\frac{5}{20}\end{align*}

45141620520=16520=1120

### Practice

Use fraction strips to represent the following subtraction problems and use that model to determine the answer.

1. 3458\begin{align*}\frac{3}{4}-\frac{5}{8}\end{align*}
2. 4523\begin{align*}\frac{4}{5}-\frac{2}{3}\end{align*}
3. 5923\begin{align*}\frac{5}{9}-\frac{2}{3}\end{align*}
4. 6723\begin{align*}\frac{6}{7}-\frac{2}{3}\end{align*}
5. 71015\begin{align*}\frac{7}{10}-\frac{1}{5}\end{align*}

Use a number line to represent the following subtraction problems and use the number line to determine the answer.

1. 2312\begin{align*}\frac{2}{3}-\frac{1}{2}\end{align*}
2. 35310\begin{align*}\frac{3}{5}-\frac{3}{10}\end{align*}
3. 7913\begin{align*}\frac{7}{9}-\frac{1}{3}\end{align*}
4. 5814\begin{align*}\frac{5}{8}-\frac{1}{4}\end{align*}
5. 25210\begin{align*}\frac{2}{5}-\frac{2}{10}\end{align*}

Use the rules that you have learned for subtracting fractions to answer the following problems.

1. 71112\begin{align*}\frac{7}{11}-\frac{1}{2}\end{align*}
2. 58512\begin{align*}\frac{5}{8}-\frac{5}{12}\end{align*}
3. 5634\begin{align*}\frac{5}{6}-\frac{3}{4}\end{align*}
4. 5625\begin{align*}\frac{5}{6}-\frac{2}{5}\end{align*}
5. 4534\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 23 cups\begin{align*}\frac{2}{3} \ cups\end{align*} of flour to make cookies. Terri used 12 cups\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 34 cup\begin{align*}\frac{3}{4} \ cup\end{align*} of milk, 113 cups\begin{align*}1 \frac{1}{3} \ cups\end{align*} of flour and 38 cup\begin{align*}\frac{3}{8} \ cup\end{align*} of oil to make pancakes. Alyssa used 38 cup\begin{align*}\frac{3}{8} \ cup\end{align*} of milk, 214 cups\begin{align*}2 \frac{1}{4} \ cups\end{align*} of flour and 13 cup\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*}. Use fraction strips to model your answer.
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?

## Date Created:

Jan 16, 2013

Apr 29, 2014
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