1.5: 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?
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Khan Academy Adding and Subtracting Fractions
Guidance
\begin{align*}\frac{5}{7}\frac{2}{7}=?\end{align*}
The problem can be represented by using fraction strips.
\begin{align*}\boxed{\frac{5}{7}\frac{2}{7}=\frac{52}{7}=\frac{3}{7}}\end{align*}
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 \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
\begin{align*}\frac{8}{11}\frac{6}{11}=?\end{align*}
\begin{align*}\boxed{\frac{8}{11}\frac{6}{11}=\frac{86}{11}=\frac{2}{11}}\end{align*}
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*}
Use fraction strips to represent each fraction.
\begin{align*}\frac{2}{3}\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.
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 \begin{align*}\frac{1}{4}\end{align*} number line.
\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 \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 here, 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 at \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. Use a model to answer the problem \begin{align*}\frac{7}{10}\frac{2}{5}=?\end{align*}
2. Use a number line to determine the answer to the problem \begin{align*}\frac{7}{8}\frac{1}{2}\end{align*}.
3. Determine the answer to \begin{align*}\frac{5}{8}\frac{1}{3}=?\end{align*} and \begin{align*}\frac{4}{5}\frac{1}{4}=?\end{align*} by using the rules for subtracting fractions.
Answers:
1.
Use fraction strips to represent each fraction.
The fractions do not have a common denominator. An equivalent fraction for \begin{align*}\frac{2}{5}\end{align*} is \begin{align*}\left(\frac{2}{2}\right) \left(\frac{2}{5}\right)=\frac{4}{10}\end{align*}
When four strips have been removed from \begin{align*}\frac{7}{10}\end{align*}, there are \begin{align*}\frac{3}{10}\end{align*} remaining.
\begin{align*}\boxed{\frac{7}{10}\frac{4}{10}=\frac{74}{10}=\frac{3}{10}}\end{align*}
2. \begin{align*}\frac{7}{8}\frac{1}{2}\end{align*}
Use a \begin{align*}\frac{1}{8}\end{align*} number line. The number line is labeled in intervals of 8. Place the starting point at \begin{align*}\frac{7}{8}\end{align*}.
From this point, move to the left a total of 4 intervals. \begin{align*}\frac{1}{2}\end{align*} of \begin{align*}8=4\end{align*}. An equivalent fraction for \begin{align*}\frac{1}{2}\end{align*} is \begin{align*}\frac{1}{2} \left(\frac{4}{4} \right)=\frac{4}{8}\end{align*}. The point where you stop is the difference of \begin{align*}\frac{7}{8}\end{align*} and \begin{align*}\frac{1}{2}\end{align*}.
On the number line you stopped at the point \begin{align*}\frac{3}{8}\end{align*}. This is equal to \begin{align*}\frac{7}{8}\frac{4}{8}=\frac{74}{8}=\frac{3}{8}\end{align*}.
3. \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.
\begin{align*}\frac{5}{8} \left(\frac{3}{3}\right)=\frac{15}{24}\end{align*}
\begin{align*}\frac{1}{3} \left(\frac{8}{8}\right)=\frac{8}{24}\end{align*}
\begin{align*}& \frac{5}{8}\frac{1}{3}\\ & \frac{15}{24}\frac{8}{24}\\ & =\frac{158}{24}=\frac{7}{24}\end{align*}
\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.
\begin{align*}\frac{4}{5} \left(\frac{4}{4}\right)=\frac{16}{20}\end{align*}
\begin{align*}\frac{1}{4} \left(\frac{5}{5}\right)=\frac{5}{20}\end{align*}
\begin{align*}& \frac{4}{5}\frac{1}{4}\\ & \frac{16}{20}\frac{5}{20}\\ & =\frac{165}{20}=\frac{11}{20}\end{align*}
Practice
Use fraction strips to represent the following subtraction problems and use that model to determine the answer.
 \begin{align*}\frac{3}{4}\frac{5}{8}\end{align*}
 \begin{align*}\frac{4}{5}\frac{2}{3}\end{align*}
 \begin{align*}\frac{5}{9}\frac{2}{3}\end{align*}
 \begin{align*}\frac{6}{7}\frac{2}{3}\end{align*}
 \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.
 \begin{align*}\frac{2}{3}\frac{1}{2}\end{align*}
 \begin{align*}\frac{3}{5}\frac{3}{10}\end{align*}
 \begin{align*}\frac{7}{9}\frac{1}{3}\end{align*}
 \begin{align*}\frac{5}{8}\frac{1}{4}\end{align*}
 \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.
 \begin{align*}\frac{7}{11}\frac{1}{2}\end{align*}
 \begin{align*}\frac{5}{8}\frac{5}{12}\end{align*}
 \begin{align*}\frac{5}{6}\frac{3}{4}\end{align*}
 \begin{align*}\frac{5}{6}\frac{2}{5}\end{align*}
 \begin{align*}\frac{4}{5}\frac{3}{4}\end{align*}
For each of the following questions, write a subtraction statement and find the result.
 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?
 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?
 Write two fractions with different denominators whose difference is \begin{align*}\frac{3}{8}\end{align*}. Use fraction strips to model your answer.
 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?
 Sierra and Clark each solved the same problem.

 Sierra’s Solution


 \begin{align*}& \frac{3}{4}\frac{1}{6}\\ & \frac{9}{12}\frac{2}{12}\\ & =\frac{7}{12}\end{align*}


 Clark’s Solution


 \begin{align*}& \frac{3}{4}\frac{1}{6}\\ & \frac{9}{12}\frac{2}{12}\\ & =\frac{7}{0}\end{align*}


 Who is correct? What would you tell the person who has the wrong answer?
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Image Attributions
In this concept you will learn to subtract real numbers using different representations. You will learn to subtract fractions by using appropriate models and by using the number line. These methods will lead to the formation of rules for subtracting fractions.
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