Adding and Subtracting Fractions
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Theory
Fractions can be added or subtracted only when they have the same denominator. Then add or subtract the numerators and keep the denominator. This Year 7 maths page shows how to find the lowest common denominator, add and subtract fractions, take fractions from a whole and simplify the answer.
Same denominator: add or subtract the numerators and keep the denominator: \(\dfrac{1}{5} + \dfrac{3}{5} + \dfrac{4}{5} = \dfrac{8}{5} = 1\dfrac{3}{5}\). Simplify if you can.
Different denominators: change the fractions to equivalent fractions with the same denominator, then add or subtract: \(\dfrac{1}{5} + \dfrac{3}{4} = \dfrac{4}{20} + \dfrac{15}{20} = \dfrac{19}{20}\).
The lowest common denominator (LCD) is the smallest number both denominators divide into. If one denominator is a multiple of the other, it is the LCD (\(12\) for \(4\) and \(12\)). If their HCF is \(1\), multiply them (\(15\) for \(3\) and \(5\)). Otherwise list multiples (\(18\) for \(6\) and \(9\)).
Taking from a whole: write the whole with the same denominator: \(1 - \dfrac{2}{7} = \dfrac{7}{7} - \dfrac{2}{7} = \dfrac{5}{7}\).
Make the parts the same size first:
How to add or subtract fractions
- Find the LCD of the denominators.
- Rename each fraction with the LCD (multiply top and bottom by the same number).
- Add or subtract the numerators; keep the denominator.
- Simplify, and change an improper answer to a mixed numeral.
| \(\dfrac{2}{3} + \dfrac{3}{4}\) | \(=\) | \(\dfrac{8}{12} + \dfrac{9}{12}\) |
| \(\) | \(=\) | \(\dfrac{17}{12}\) |
| \(\) | \(=\) | \(1\dfrac{5}{12}\) |
\(1\dfrac{5}{12}\)
| \(\dfrac{2}{3} - \dfrac{5}{9}\) | \(=\) | \(\dfrac{6}{9} - \dfrac{5}{9}\) |
| \(\) | \(=\) | \(\dfrac{1}{9}\) |
\(9\) is a multiple of \(3\), so the LCD is \(9\).
| \(\dfrac{11}{12} - \dfrac{5}{12}\) | \(=\) | \(\dfrac{6}{12}\) |
| \(\) | \(=\) | \(\dfrac{1}{2}\) |
\(\dfrac{1}{2}\)
| \(\dfrac{7}{10} + \dfrac{1}{2}\) | \(=\) | \(\dfrac{7}{10} + \dfrac{5}{10}\) |
| \(\) | \(=\) | \(\dfrac{12}{10} = \dfrac{6}{5}\) |
| \(\) | \(=\) | \(1\dfrac{1}{5}\) |
\(1\dfrac{1}{5}\)
Multiples of \(6\): \(6, 12, 18, 24\). Multiples of \(8\): \(8, 16, 24\). The LCD is \(24\).
| \(\dfrac{5}{6} - \dfrac{3}{8}\) | \(=\) | \(\dfrac{20}{24} - \dfrac{9}{24}\) |
| \(\) | \(=\) | \(\dfrac{11}{24}\) |
\(\dfrac{11}{24}\)
| \(\dfrac{3}{4} + \dfrac{2}{3}\) | \(=\) | \(\dfrac{9}{12} + \dfrac{8}{12} = \dfrac{17}{12}\) |
| \(2 - \dfrac{17}{12}\) | \(=\) | \(\dfrac{24}{12} - \dfrac{17}{12}\) |
| \(\) | \(=\) | \(\dfrac{7}{12}\) |
\(\dfrac{7}{12}\) L is left.
Common pitfalls
Frequently asked questions
Can you add fractions with different denominators?
Only after renaming them with a common denominator.
How do you find the lowest common denominator?
Find the smallest number both denominators divide into. For \(6\) and \(8\) it is \(24\).
Do you add the denominators?
No. Keep the common denominator and add only the numerators.
How do you subtract a fraction from 1?
Write \(1\) with the same denominator. \(1 - \dfrac{2}{7} = \dfrac{5}{7}\).
What if the answer is an improper fraction?
Write it as a mixed numeral. \(\dfrac{17}{12} = 1\dfrac{5}{12}\).