Fractions
Fractions are revised in full in this NSW Year 9 Core maths subtopic: all four operations, mixed numbers included, from the Stage 4 work under NESA outcome MA4‑FRC‑C‑01. Several questions are worded problems, so choosing the operation matters as much as carrying it out.
You will add and subtract fractions with unrelated denominators, such as \(\frac{7}{24} + \frac{11}{40}\), and mixed numbers, including borrowing when the fraction part is too small. You will multiply by cancelling first, divide by multiplying by the reciprocal, convert mixed numbers to improper fractions, find a fraction of a quantity in different units, and scale up a result such as \(5\) out of every \(30\) cars. Real problems include packing oranges into bags and timing a slide show. It is part of Year 9 Core Working with Numbers.
Theory
Fractions follow two different habits: adding and subtracting need a common denominator; multiplying and dividing do not. This Year 9 maths page revises all four operations, mixed numbers, dividing with the reciprocal, and finding a fraction of a quantity.
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 \(\dfrac{4}{3}\).
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 \(10\); \(3\) goes into \(9\) and \(12\).
| \(\dfrac{9}{10} \times \dfrac{5}{12}\) | \(=\) | \(\dfrac{3}{2} \times \dfrac{1}{4}\) |
| \(\) | \(=\) | \(\dfrac{3 \times 1}{2 \times 4}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{3}{8}}\) |
The answer is smaller than \(\dfrac{9}{10}\), 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) Find \(\dfrac{3}{8}\) of \(2\) hours, in minutes.
(ii) One batch of muesli bars needs \(\dfrac{3}{4}\) of a cup of oats. How many batches can be made from \(6\) cups?
(i) Change the units first: \(2\) hours \(= 120\) minutes. “Of” means multiply:
| \(\dfrac{3}{8} \times 120\) | \(=\) | \(\dfrac{360}{8}\) |
| \(\) | \(=\) | \(\mathbf{45}\ \textbf{minutes}\) |
Check: \(\dfrac{1}{8}\) of \(120\) is \(15\), and three of those is \(45\).
(ii) “How many \(\dfrac{3}{4}\)s fit into \(6\)?” is a division:
| \(6 \div \dfrac{3}{4}\) | \(=\) | \(\dfrac{6}{1} \times \dfrac{4}{3}\) |
| \(\) | \(=\) | \(\dfrac{24}{3}\) |
| \(\) | \(=\) | \(\mathbf{8}\ \textbf{batches}\) |
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 \(45\).