Fraction Problems
Theory
Every fraction in a problem is a fraction of something, so first ask: a fraction of what? If a hall with \(960\) seats is \(\dfrac{3}{5}\) full, then \(\dfrac{2}{5}\) is empty, and \(\dfrac{2}{5} \times 960 = 384\) seats are left. This Year 8 maths page shows how.
A fraction of a quantity. Divide by the denominator, then multiply by the numerator: \(\dfrac{3}{5}\) of \(45\) is \(45 \div 5 \times 3 = 27\).
Same whole. Fractions can be added, subtracted or compared only when they are parts of the same whole. The rest is \(1\) minus the parts used.
A fraction of what is left is a fraction of a fraction, and ‘of’ means times: \(\dfrac{3}{5}\) of \(\dfrac{2}{3}\) of a tank is \(\dfrac{3}{5} \times \dfrac{2}{3} = \dfrac{2}{5}\) of the tank.
How many fit. Divide, and keep only the whole pieces. The fraction left over is part of a piece, so multiply it by the size of one piece to find what is left.
Working backwards. When a known amount is \(\dfrac{a}{b}\) of the whole, divide it by \(a\) to get \(\dfrac{1}{b}\) of the whole, then multiply by \(b\).
A difference. If one group is \(\dfrac{3}{8}\) of the class and another is \(\dfrac{1}{6}\), the extra students are \(\dfrac{3}{8} - \dfrac{1}{6}\) of the class; then work backwards.
Rates that combine. A job that takes \(n\) minutes is \(\dfrac{1}{n}\) of the job each minute. Add the rates, then divide \(1\) by the total.
Match the units first (for example change minutes to seconds), then answer in a sentence with units and check the size.
Choose the step from the words:
| The question says | Do this |
|---|---|
| \(\dfrac{a}{b}\) of an amount | amount \(\div b \times a\) |
| what fraction is left | \(1 -\) the parts used |
| a fraction of what is left | multiply the two fractions |
| how many pieces fit | divide; keep the whole pieces |
| \(\dfrac{a}{b}\) of the whole is known | \(\div a\), then \(\times b\) |
| working together | add \(\dfrac{1}{n}\) rates, then \(1 \div\) total |
Solving a fraction problem
- Name the whole each fraction belongs to.
- Choose the step: of (multiply), left (\(1 -\)), fit (divide), backwards (\(\div a \times b\)).
- Work with fractions, matching units first.
- Answer in a sentence with units, and check the size.
Working backwards
- Find the fraction that the known amount stands for.
- Divide the amount by the numerator to get one part.
- Multiply by the denominator to get the whole, then check.
\(\dfrac{3}{5}\) full means \(\dfrac{2}{5}\) of the seats are left.
| \(960 \times \dfrac{2}{5}\) | \(=\) | \(\dfrac{960}{5} \times 2\) |
| \(\) | \(=\) | \(192 \times 2\) |
| \(\) | \(=\) | \(384\) |
\(384\) seats are left.
(ii) A roll of shade cloth is \(9\) m long and each strip uses \(1\dfrac{1}{4}\) m. How many whole strips can be cut, and how much cloth is left over?
(i) \(1\) minute \(= 60\) seconds.
(ii) Divide, keep the whole strips, then turn the leftover part of a strip into metres.
| \(60 \div 2\dfrac{1}{2}\) | \(=\) | \(60 \times \dfrac{2}{5}\) |
| \(\) | \(=\) | \(24\) |
| \(9 \div 1\dfrac{1}{4}\) | \(=\) | \(9 \times \dfrac{4}{5} = 7\dfrac{1}{5}\) |
| \(\dfrac{1}{5} \times \dfrac{5}{4}\) | \(=\) | \(\dfrac{1}{4}\) |
(i) \(24\) pictures
(ii) \(7\) strips with \(\dfrac{1}{4}\) m left. Check: \(9 - 7 \times \dfrac{5}{4} = \dfrac{1}{4}\).
\(\dfrac{3}{8}\) of the fuel is used, so \(\dfrac{5}{8}\) of the fuel stays.
| \(\dfrac{5}{8} \times \dfrac{4}{5}\) | \(=\) | \(\dfrac{1}{2} \times \dfrac{1}{1} = \dfrac{1}{2}\) |
\(\dfrac{1}{2}\) of the tank is still full. Check: used \(= \dfrac{3}{8} \times \dfrac{4}{5} = \dfrac{3}{10}\), and \(\dfrac{4}{5} - \dfrac{3}{10} = \dfrac{1}{2}\).
(ii) In a class, \(\dfrac{3}{8}\) of the students walk to school and \(\dfrac{1}{6}\) ride bikes. \(30\) more students walk than ride. How many students are there, and how many walk?
| \(1 - \left(\dfrac{3}{8} + \dfrac{2}{8}\right)\) | \(=\) | \(\dfrac{3}{8}\) |
| \(117 \div 3 \times 8\) | \(=\) | \(312\) |
| \(\dfrac{3}{8} - \dfrac{1}{6}\) | \(=\) | \(\dfrac{9}{24} - \dfrac{4}{24} = \dfrac{5}{24}\) |
| \(30 \div 5 \times 24\) | \(=\) | \(144\) |
| \(\dfrac{3}{8} \times 144\) | \(=\) | \(54\) |
(i) Her pay is \(\$312\). Check: \(\dfrac{3}{8} \times 312 = 117\).
(ii) \(144\) students; \(54\) walk and \(24\) ride, a difference of \(30\).
Each minute, A does \(\dfrac{1}{10}\) of the job and B does \(\dfrac{1}{15}\).
| \(\dfrac{1}{10} + \dfrac{1}{15}\) | \(=\) | \(\dfrac{3}{30} + \dfrac{2}{30} = \dfrac{1}{6}\) |
| \(1 \div \dfrac{1}{6}\) | \(=\) | \(6\) |
Together they take \(6\) minutes, less than the faster printer alone, as it should be.
(ii) Is \(\dfrac{7}{10}\) of \(40\) more than \(\dfrac{2}{3}\) of \(60\)?
| \(\dfrac{1}{4} \times \dfrac{2}{3} + \dfrac{1}{2} \times \dfrac{1}{3}\) | \(=\) | \(\dfrac{1}{6} + \dfrac{1}{6} = \dfrac{1}{3}\) |
| \(\dfrac{1}{3} \times 600\) | \(=\) | \(200\) |
| \(\dfrac{7}{10} \times 40\) | \(=\) | \(28\) |
| \(\dfrac{2}{3} \times 60\) | \(=\) | \(40\) |
(i) \(\dfrac{1}{3}\) of the club, so \(200\) members swim.
(ii) No. \(\dfrac{7}{10}\) is the bigger fraction, but \(28 < 40\) because the wholes are different.
Common pitfalls
Frequently asked questions
How do I find a fraction of an amount?
Divide the amount by the denominator, then multiply by the numerator: \(\dfrac{3}{4}\) of \(20\) is \(20 \div 4 \times 3 = 15\).
What does a fraction of what is left mean?
Multiply the fraction by the fraction that was left. The answer is a fraction of the whole.
How do I find the whole from a part?
Divide the known amount by the numerator of its fraction to get one part, then multiply by the denominator.
How do I work out how long two people take together?
Write what each does in one unit of time as a fraction of the job, add these, then divide 1 by the total.
Why keep only whole pieces?
A part of a piece is not a full piece, so it cannot be counted as one; it is the material left over.