Addition and Subtraction of Fractions

The same basic steps apply when subtracting a fraction from a fraction. Information from the previous section is applicable.

When subtracting like fractions, subtract the numerators.

For example, \(\frac{8}{9}-\frac{2}{9}=\frac{6}{9}=\frac{2}{3}\).

In this example, the answer was reduced to \(\frac{2}{3}\).

Addition and Subtraction of Fractions with Like Denominators Review

Adding Fractions Without Like Denominators

Fractions will not always have like denominators when adding them together. Rewrite the fractions to have a like denominator in order to add the fractions. To determine what the denominator should be, find the lowest common multiple between the numbers and convert the fractions to have the value as the denominator. This value is considered the least common denominator, or LCD.

Convert the fraction to have the LCD by multiplying the fraction by the necessary numerical value written as a fraction equal to 1.

For example, add \(\frac{2}{5}+\frac{7}{15}\). The fractions must be converted to have like denominators in order to add them together. The number 15 is a multiple of 5 by multiplying it by the value 3. Multiply the numerator and denominator by \(\frac{3}{3}\)  a value equivalent to 1. Multiply, \(\frac{2}{5}\times\frac{3}{3}=\frac{2\times3}{5\times3}=\frac{6}{15}\). Now that the fraction is converted to have the LCD, the fractions can be added together by adding the numerators, \( \frac{6}{15}+\frac{7}{15}=\frac{13}{15}\).

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KEEP IN MIND

Change the fractions to have a common denominator in order to complete the operation.

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Sometimes both fractions will have to be converted to have the LCD. The fractions can be converted by different forms of the value 1. For example, add \(\frac{1}{6}+\frac{4}{9}\). The LCD between the fractions is 18. The first fraction must be converted by multiplying it by \(\frac{3}{3}\) because \(\frac{1}{6}\times\frac{3}{3}=\frac{1\times3}{6\times3}=\frac{3}{18}\). The second fraction must be converted by multiplying it by \(\frac{2}{2}\) because \(\frac{4}{9}\times\frac{2}{2}=\frac{4\times2}{9\times2}=\frac{8}{18}\). The fractions can be added together with like denominators, \(\frac{3}{18}+\frac{8}{18}=\frac{11}{18}\).

When adding a fraction to an improper fraction or mixed number, the process is similar.

For example, add \(\frac{7}{11}+2\frac{1}{2}\). Change the mixed number to an improper fraction, \(\frac{7}{11}+\frac{5}{2}\). Convert each fraction to have the LCD, \(\frac{7}{11}\times\frac{2}{2}=\frac{14}{22}\) and \(\frac{5}{2}\times\frac{11}{11}=\frac{55}{22}\). Add the numerators together and reduce if necessary, \(\frac{14}{22}+\frac{55}{22}=\frac{69}{22}\). The fraction \(\frac{69}{22}\) cannot be reduced, but can be rewritten as a mixed number, \(3\frac{3}{22}\).

Subtracting Fractions Without Like Denominators

The same basic steps apply when subtracting fractions without like denominators. Similar steps can be used when subtracting a fraction from a mixed number or improper fraction, or subtracting a mixed number or improper fraction from a fraction.

STEP BY STEP

Step 1. Write any whole number as a fraction with a denominator of 1. Write any mixed numbers as improper fractions.

Step 2. Determine the least common multiple between denominators.

Step 3. Convert the fractions to have the least common multiple as the denominator.

Step 4. Subtract the numerators.

Step 5. Rewrite the fraction as a mixed number and reduce the fraction completely.

Subtract.

\(2\frac{1}{10}-\frac{6}{7}\)

Rewrite the expression as \(\frac{21}{10}-\frac{6}{7}\). Convert both fractions to have the lowest common denominator, \(\frac{21}{10}\times\frac{7}{7}=\frac{147}{70}\) and \(\frac{6}{7}\times\frac{10}{10}=\frac{60}{70}\). Subtract the numerators and reduce completely, \(\frac{147}{70}-\frac{60}{70}=\frac{87}{70}=1\frac{17}{70}\).

Subtract.

\(\frac{12}{5}-\frac{4}{9}\)

Convert the fractions to have the lowest common denominator, \(\frac{12}{5}\times\frac{9}{9}=\frac{108}{45}\) and \(\frac{4}{9}\times\frac{5}{5}=\frac{20}{45}\). Subtract the numerators and reduce as necessary, \(\frac{108}{45}-\frac{20}{45}=\frac{88}{45}=1\frac{43}{45}\).

Addition and Subtraction of Fractions Without Like Denominators Review

Let’s Review!

• The process of adding or subtracting like fractions requires the numerators to be added or subtracted. The answer may be reduced.
• Operations with fractions may produce an answer greater than the value of 1. The answer can be written as an improper fraction or a mixed number.
• The process to add or subtract fractions without like denominators requires the fractions to be converted in order to have the same denominator, also known as the least common denominator.