Adding and Subtracting Fractions
Theory
Fractions can be added or subtracted only when the parts are the same size. Rename them with the lowest common denominator (LCD), then add or subtract the numerators: \(\dfrac{3}{7} - \dfrac{1}{3} = \dfrac{9}{21} - \dfrac{7}{21} = \dfrac{2}{21}\). This Year 8 maths page shows how.
Same denominators. Add or subtract the numerators and keep the denominator: \(\dfrac{3}{7} + \dfrac{2}{7} = \dfrac{5}{7}\).
The lowest common denominator (LCD) is the smallest number both denominators divide into. Rename each fraction over the LCD, combine the numerators, then simplify: \(\dfrac{3}{10} + \dfrac{9}{20} = \dfrac{6}{20} + \dfrac{9}{20} = \dfrac{15}{20} = \dfrac{3}{4}\).
Mixed numerals, adding. Add the whole numbers, add the fractions, then carry any extra whole: \(3\dfrac{5}{8} + 2\dfrac{5}{6} = 5 + \dfrac{15}{24} + \dfrac{20}{24} = 5 + 1\dfrac{11}{24} = 6\dfrac{11}{24}\).
Mixed numerals, subtracting. Subtract the whole numbers and the fractions. If the first fraction is too small, borrow one whole, or change both to improper fractions, which never needs borrowing.
From a whole number. Take one whole and write it over the denominator: \(7 - 2\dfrac{4}{9} = 6\dfrac{9}{9} - 2\dfrac{4}{9} = 4\dfrac{5}{9}\).
Negative fractions. Rename, then combine the numerators with their signs: \(-\dfrac{1}{3} - \dfrac{1}{4} = \dfrac{-4 - 3}{12} = -\dfrac{7}{12}\). Adding moves right on the number line and subtracting moves left. A negative mixed numeral is negative all through: \(-2\dfrac{1}{4} = -\dfrac{9}{4}\).
Brackets first, then add and subtract from left to right.
A missing fraction. In \(a + \square = b\), the box is \(b - a\).
Pronumerals on top. The denominators are still numbers, so use the LCD as usual: \(\dfrac{x}{2} + \dfrac{x}{5} = \dfrac{5x}{10} + \dfrac{2x}{10} = \dfrac{7x}{10}\).
Rename over the LCD, then combine the numerators:
| Sum | LCD | Working | Answer |
|---|---|---|---|
| \(\dfrac{1}{4} + \dfrac{1}{3}\) | \(12\) | \(\dfrac{3}{12} + \dfrac{4}{12}\) | \(\dfrac{7}{12}\) |
| \(-\dfrac{1}{3} - \dfrac{1}{4}\) | \(12\) | \(\dfrac{-4 - 3}{12}\) | \(-\dfrac{7}{12}\) |
| \(6 - 2\dfrac{1}{5}\) | \(5\) | \(5\dfrac{5}{5} - 2\dfrac{1}{5}\) | \(3\dfrac{4}{5}\) |
| \(4\dfrac{1}{6} - 2\dfrac{3}{4}\) | \(12\) | \(3\dfrac{14}{12} - 2\dfrac{9}{12}\) | \(1\dfrac{5}{12}\) |
Adding or subtracting fractions
- Find the LCD of the denominators.
- Rename each fraction over the LCD (multiply top and bottom by the same number).
- Combine the numerators, with their signs; keep the denominator.
- Simplify and write an improper answer as a mixed numeral.
Mixed numerals
- Add or subtract the whole numbers.
- Rename the fractions over the LCD and combine them.
- Borrow one whole if the first fraction is too small, or carry an extra whole after adding. With negatives, use improper fractions.
(i) \(\dfrac{3}{7} + \dfrac{2}{7}\)
(ii) \(\dfrac{3}{7} - \dfrac{1}{3}\)
(iii) \(\dfrac{3}{7} + \dfrac{5}{14}\)
| \(\dfrac{3}{7} + \dfrac{2}{7}\) | \(=\) | \(\dfrac{3 + 2}{7} = \dfrac{5}{7}\) |
| \(\dfrac{3}{7} - \dfrac{1}{3}\) | \(=\) | \(\dfrac{9}{21} - \dfrac{7}{21}\) |
| \(\) | \(=\) | \(\dfrac{2}{21}\) |
| \(\dfrac{3}{7} + \dfrac{5}{14}\) | \(=\) | \(\dfrac{6}{14} + \dfrac{5}{14}\) |
| \(\) | \(=\) | \(\dfrac{11}{14}\) |
(i) \(\dfrac{5}{7}\)
(ii) \(\dfrac{2}{21}\)
(iii) \(\dfrac{11}{14}\)
(i) \(2\dfrac{1}{4} + 3\dfrac{1}{3}\)
(ii) \(3\dfrac{1}{5} - 2\dfrac{1}{7}\)
(iii) \(3\dfrac{1}{7} - 2\dfrac{1}{5}\)
Work with the whole numbers and the fractions separately.
(iii) \(\dfrac{5}{35}\) is smaller than \(\dfrac{7}{35}\), so borrow one whole.
| \(2\dfrac{1}{4} + 3\dfrac{1}{3}\) | \(=\) | \(5 + \dfrac{3}{12} + \dfrac{4}{12}\) |
| \(\) | \(=\) | \(5\dfrac{7}{12}\) |
| \(3\dfrac{1}{5} - 2\dfrac{1}{7}\) | \(=\) | \(1 + \dfrac{7}{35} - \dfrac{5}{35}\) |
| \(\) | \(=\) | \(1\dfrac{2}{35}\) |
| \(3\dfrac{1}{7} - 2\dfrac{1}{5}\) | \(=\) | \(3\dfrac{5}{35} - 2\dfrac{7}{35}\) |
| \(\) | \(=\) | \(2\dfrac{40}{35} - 2\dfrac{7}{35}\) |
| \(\) | \(=\) | \(\dfrac{33}{35}\) |
(i) \(5\dfrac{7}{12}\)
(ii) \(1\dfrac{2}{35}\)
(iii) \(\dfrac{33}{35}\)
(ii) Evaluate \(-2\dfrac{1}{4} + 3\dfrac{1}{2}\).
(iii) Find the fraction in the box: \(\dfrac{2}{5} + \square = -\dfrac{1}{10}\).
(ii) The sign belongs to the whole mixed numeral: \(-2\dfrac{1}{4} = -\dfrac{9}{4}\).
(iii) The box is \(-\dfrac{1}{10} - \dfrac{2}{5}\).
| \(-\dfrac{20}{24} + \dfrac{9}{24}\) | \(=\) | \(\dfrac{-20 + 9}{24} = -\dfrac{11}{24}\) |
| \(-\dfrac{9}{4} + \dfrac{14}{4}\) | \(=\) | \(\dfrac{5}{4} = 1\dfrac{1}{4}\) |
| \(-\dfrac{1}{10} - \dfrac{4}{10}\) | \(=\) | \(-\dfrac{5}{10} = -\dfrac{1}{2}\) |
(i) \(-\dfrac{11}{24}\)
(ii) \(1\dfrac{1}{4}\)
(iii) \(-\dfrac{1}{2}\). Check: \(\dfrac{4}{10} - \dfrac{5}{10} = -\dfrac{1}{10}\).
(i) \(4\dfrac{2}{9} - 1\dfrac{5}{6}\)
(ii) \(7 - 2\dfrac{4}{9}\)
(i) The LCD is \(18\), and \(\dfrac{4}{18}\) is smaller than \(\dfrac{15}{18}\), so borrow one whole, \(\dfrac{18}{18}\).
| \(4\dfrac{4}{18} - 1\dfrac{15}{18}\) | \(=\) | \(3\dfrac{22}{18} - 1\dfrac{15}{18}\) |
| \(\) | \(=\) | \(2\dfrac{7}{18}\) |
| \(6\dfrac{9}{9} - 2\dfrac{4}{9}\) | \(=\) | \(4\dfrac{5}{9}\) |
(i) \(2\dfrac{7}{18}\). Check with improper fractions: \(\dfrac{38}{9} - \dfrac{11}{6} = \dfrac{76}{18} - \dfrac{33}{18} = \dfrac{43}{18} = 2\dfrac{7}{18}\).
(ii) \(4\dfrac{5}{9}\)
(ii) A pond is \(3\dfrac{1}{2}\) m below its full mark, written \(-3\dfrac{1}{2}\) m. It rises \(1\dfrac{3}{4}\) m. What is the new level?
| \(\text{Left}\) | \(=\) | \(1\dfrac{1}{2} - \left(\dfrac{2}{5} + \dfrac{1}{3}\right)\) |
| \(\) | \(=\) | \(\dfrac{3}{2} - \dfrac{11}{15}\) |
| \(\) | \(=\) | \(\dfrac{45}{30} - \dfrac{22}{30} = \dfrac{23}{30}\) |
| \(-\dfrac{7}{2} + \dfrac{7}{4}\) | \(=\) | \(-\dfrac{14}{4} + \dfrac{7}{4}\) |
| \(\) | \(=\) | \(-\dfrac{7}{4} = -1\dfrac{3}{4}\) |
(i) \(\dfrac{23}{30}\) L is left.
(ii) \(-1\dfrac{3}{4}\) m, still below the full mark.
(i) \(\dfrac{3n}{5} - \dfrac{n}{4}\)
(ii) \(\dfrac{5p}{8} + \dfrac{p}{6}\)
| \(\dfrac{3n}{5} - \dfrac{n}{4}\) | \(=\) | \(\dfrac{12n}{20} - \dfrac{5n}{20}\) |
| \(\) | \(=\) | \(\dfrac{7n}{20}\) |
| \(\dfrac{5p}{8} + \dfrac{p}{6}\) | \(=\) | \(\dfrac{15p}{24} + \dfrac{4p}{24}\) |
| \(\) | \(=\) | \(\dfrac{19p}{24}\) |
(i) \(\dfrac{7n}{20}\)
(ii) \(\dfrac{19p}{24}\). Check (i) with \(n = 20\): \(12 - 5 = 7\) and \(\dfrac{7 \times 20}{20} = 7\).
Common pitfalls
Frequently asked questions
How do you add fractions with different denominators?
Rename both fractions over the lowest common denominator, then add the numerators: \(\dfrac{1}{4} + \dfrac{1}{3} = \dfrac{3}{12} + \dfrac{4}{12} = \dfrac{7}{12}\).
What is the lowest common denominator?
The smallest number that all the denominators divide into exactly. For 6 and 8 it is 24.
How do I subtract mixed numerals when the fraction is too small?
Borrow one whole from the whole number and add it to the fraction, or change both mixed numerals to improper fractions.
How do I subtract a fraction from a whole number?
Write one of the wholes as a fraction: \(5 - \dfrac{2}{3} = 4\dfrac{3}{3} - \dfrac{2}{3} = 4\dfrac{1}{3}\).
How do I add negative fractions?
Rename over the LCD, then add the numerators with their signs, just like adding integers.