Operations with Fractions
Every question with a fully worked solution.
- Operations with Fractions - Video - Adding Fractions Word Problems | Fraction Word Problems Watch
- Operations with Fractions - Video - Subtracting Fractions Word Problems | Fraction Word Problems Watch
- Operations with Fractions - Video - Multiplying Fractions Word Problems | Fraction Word Problems Watch
- Operations with Fractions - Video - Dividing Fractions Word Problems | Fraction Word Problems Watch
Theory
Fractions follow two different habits: adding and subtracting need a common denominator; multiplying and dividing do not. This Year 9 maths page revises the Year 8 work on all four operations, mixed numbers, dividing with the reciprocal, and negative fractions.
A fraction \(\dfrac{a}{b}\) has a numerator \(a\) on top and a denominator \(b\) underneath. A mixed number such as \(2\dfrac{1}{3}\) has a whole part and a fraction part; as an improper fraction it is \(\dfrac{7}{3}\).
To add or subtract, the pieces must be the same size. Rewrite both fractions with a common denominator (the lowest one both denominators divide into), then add or subtract the numerators only.
To multiply, multiply the numerators and multiply the denominators. Cancel any common factor between a top and a bottom first.
To divide, multiply by the reciprocal of the second fraction: turn it upside down. The reciprocal of \(\dfrac{3}{4}\) is \(\dfrac{4}{3}\).
Negative fractions follow the integer rules: adding a negative is subtracting, and for \(\times\) and \(\div\) the same signs give a positive answer and different signs give a negative answer. A reciprocal keeps its sign: the reciprocal of \(-\dfrac{2}{5}\) is \(-\dfrac{5}{2}\).
A fraction of a quantity means multiply: \(\dfrac{2}{3}\) of \(60\) is \(\dfrac{2}{3} \times 60 = 40\). Give answers in simplest form.
The four operations:
| Operation | Common denominator? | What to do |
|---|---|---|
| \(+\) and \(-\) | yes | rewrite both, then add or subtract the numerators |
| \(\times\) | no | cancel, then multiply across |
| \(\div\) | no | flip the second fraction, then multiply |
How to work with fractions
- Change mixed numbers to improper fractions when multiplying or dividing (or when subtracting needs it).
- Add or subtract: find the lowest common denominator, rewrite both fractions, then work with the numerators.
- Multiply: cancel a top with a bottom, then multiply across. Divide: flip the second fraction, then multiply.
- Simplify the answer and write an improper fraction as a mixed number if that suits the question.
- Check the size: multiplying by a fraction less than \(1\) makes a number smaller; dividing by one makes it bigger.
(i) \(\dfrac{3}{4} + \dfrac{5}{6}\)
(ii) \(4\dfrac{1}{6} - 1\dfrac{3}{4}\)
(i) Both \(4\) and \(6\) divide into \(12\). Rewrite both fractions as twelfths, then add the numerators only:
| \(\dfrac{3}{4} + \dfrac{5}{6}\) | \(=\) | \(\dfrac{9}{12} + \dfrac{10}{12}\) |
| \(\) | \(=\) | \(\dfrac{19}{12}\) |
| \(\) | \(=\) | \(\mathbf{1\dfrac{7}{12}}\) |
(ii) \(\dfrac{3}{4}\) is bigger than \(\dfrac{1}{6}\), so change both to improper fractions first:
| \(4\dfrac{1}{6} - 1\dfrac{3}{4}\) | \(=\) | \(\dfrac{25}{6} - \dfrac{7}{4}\) |
| \(\) | \(=\) | \(\dfrac{50}{12} - \dfrac{21}{12}\) |
| \(\) | \(=\) | \(\dfrac{29}{12}\) |
| \(\) | \(=\) | \(\mathbf{2\dfrac{5}{12}}\) |
No common denominator is needed. Cancel first: \(5\) goes into \(5\) and \(15\); \(4\) goes into \(8\) and \(12\).
| \(\dfrac{8}{15} \times \dfrac{5}{12}\) | \(=\) | \(\dfrac{2}{3} \times \dfrac{1}{3}\) |
| \(\) | \(=\) | \(\dfrac{2 \times 1}{3 \times 3}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{2}{9}}\) |
The answer is smaller than \(\dfrac{8}{15}\), as it must be when you multiply by a fraction less than \(1\).
Change both mixed numbers to improper fractions first:
| \(3\dfrac{1}{2}\) | \(=\) | \(\dfrac{7}{2}\) |
| \(1\dfrac{1}{4}\) | \(=\) | \(\dfrac{5}{4}\) |
Keep the first fraction, flip the second, and multiply:
| \(\dfrac{7}{2} \div \dfrac{5}{4}\) | \(=\) | \(\dfrac{7}{2} \times \dfrac{4}{5}\) |
| \(\) | \(=\) | \(\dfrac{28}{10}\) |
| \(\) | \(=\) | \(\dfrac{14}{5}\) |
| \(\) | \(=\) | \(\mathbf{2\dfrac{4}{5}}\) |
(i) \(-\dfrac{1}{2} + \dfrac{3}{10}\)
(ii) the reciprocal of \(-1\dfrac{2}{3}\)
(iii) \(-\dfrac{5}{6} \div \left(-\dfrac{10}{9}\right)\)
(i) Use tenths. Start at \(-\dfrac{5}{10}\) and move \(\dfrac{3}{10}\) right; you are still below zero:
| \(-\dfrac{1}{2} + \dfrac{3}{10}\) | \(=\) | \(-\dfrac{5}{10} + \dfrac{3}{10}\) |
| \(\) | \(=\) | \(-\dfrac{2}{10} = \mathbf{-\dfrac{1}{5}}\) |
(ii) Change to an improper fraction, then turn it upside down. The sign stays:
| \(-1\dfrac{2}{3}\) | \(=\) | \(-\dfrac{5}{3}\) |
| \(\text{reciprocal}\) | \(=\) | \(\mathbf{-\dfrac{3}{5}}\) |
Check: \(-\dfrac{5}{3} \times \left(-\dfrac{3}{5}\right) = 1\).
(iii) Flip the second fraction and multiply. Two negatives give a positive:
| \(-\dfrac{5}{6} \div \left(-\dfrac{10}{9}\right)\) | \(=\) | \(-\dfrac{5}{6} \times \left(-\dfrac{9}{10}\right)\) |
| \(\) | \(=\) | \(\dfrac{1}{2} \times \dfrac{3}{2}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{3}{4}}\) |
Common pitfalls
Frequently asked questions
How do you add fractions with different denominators?
Find a number both denominators divide into, rewrite each fraction with that denominator, then add the numerators. For example, \(\dfrac{3}{4} + \dfrac{5}{6} = \dfrac{9}{12} + \dfrac{10}{12} = \dfrac{19}{12} = 1\dfrac{7}{12}\).
How do you divide fractions?
Keep the first fraction, flip the second fraction upside down, and multiply. For example, \(\dfrac{7}{2} \div \dfrac{5}{4} = \dfrac{7}{2} \times \dfrac{4}{5} = \dfrac{14}{5}\).
Do you need a common denominator to multiply fractions?
No. Multiply the numerators together and the denominators together. Cancelling common factors first keeps the numbers small.
How do you multiply mixed numbers?
Change each mixed number to an improper fraction first, then multiply across and simplify. Multiplying the whole numbers and the fractions separately gives the wrong answer.
How do you find a fraction of an amount?
Multiply the fraction by the amount. To find \(\dfrac{3}{8}\) of \(120\), divide \(120\) by \(8\) to get \(15\), then multiply by \(3\) to get \(45\).