Subtracting fractions

To subtract fractions, they need to have the same denominator (the bottom part of the fraction). If they don’t have the same denominator, we need to find the lowest common multiple (LCM) first.

Making the denominators the same

See lowest common multiple for more details on how to find the lowest common multiple.

Once you have the LCM of the denominators, you can convert each fraction to an equivalent fraction with the LCM as the new denominator. You do this by thinking ‘what does the denominator need to be multiplied by to get the LCM?’, and then multiplying both the numerator and denominator by that number. Do that for each fraction.

Example: make the denominators the same for \frac{2}{3} and \frac{5}{4}

• The LCM of 3 and 4 is 12.
• For \frac{2}{3}:
  • 3 \times 4 = 12, so we need to multiply both the numerator and denominator by 4:
  • \frac{2 \times 4}{3 \imes 4} = \frac{8}{12}
• For \frac{5}{4}:
  • 4 \times 3 = 12, so we need to multiply both the numerator and denominator by 3:
  • \frac{5 \times 3}{4 \times 3} = \frac{15}{12}
• So, \frac{2}{3} becomes \frac{8}{12} and \frac{5}{4} becomes \frac{15}{12}.

Subtracting fractions

Once the fractions have the same denominator, you can subtract them by simply subtracting the numerators and keeping the denominator the same.

This can be written as:

\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}

Example: subtract \frac{5}{4} from \frac{2}{3}

• First, we make the denominators the same (as shown above):
  • \frac{2}{3} = \frac{8}{12}
  • \frac{5}{4} = \frac{15}{12}
• Now we can subtract them:
  • = \frac{8 - 15}{12}
  • = \frac{-7}{12}
• Answer: \frac{-7}{12} or -\frac{7}{12}

Example: evaluate \frac{7}{10} - \frac{3}{5}

• The LCM of 10 and 5 is 10.
• For \frac{7}{10}:
  • The denominator is already 10, so we leave it as is: \frac{7}{10}
• For \frac{3}{5}:
  • 5 \times 2 = 10, so we multiply both the numerator and denominator by 2:
  • \frac{3 \times 2}{5 \times 2} = \frac{6}{10}
• Now we can subtract them:
  • = \frac{7 - 6}{10}
  • = \frac{1}{10}
• Answer: \frac{1}{10}

Example: evaluate \frac{9}{8} - \frac{5}{12}

• The LCM of 8 and 12 is 24.
• For \frac{9}{8}:
  • 8 \times 3 = 24, so we multiply both the numerator and denominator by 3:
  • \frac{9 \times 3}{8 \times 3} = \frac{27}{24}
• For \frac{5}{12}:
  • 12 \times 2 = 24, so we multiply both the numerator and denominator by 2:
  • \frac{5 \times 2}{12 \times 2} = \frac{10}{24}
• Now we can subtract them:
  • = \frac{27 - 10}{24}
  • = \frac{17}{24}
• Answer: \frac{17}{24}

Example: evaluate \frac{3x}{4} - \frac{5x}{6}

• The LCM of 4 and 6 is 12.
• For \frac{3x}{4}:
  • 4 \times 3 = 12, so we multiply both the numerator and denominator by 3:
  • \frac{3x \times 3}{4 \times 3} = \frac{9x}{12}
• For \frac{5x}{6}:
  • 6 \times 2 = 12, so we multiply both the numerator and denominator by 2:
  • \frac{5x \times 2}{6 \times 2} = \frac{10x}{12}
• Now we can subtract them:
  • = \frac{9x - 10x}{12}
  • = \frac{-1x}{12} or \frac{-x}{12}
• Answer: \frac{-x}{12} or -\frac{x}{12}