Fractions of a Quantity
Every question with a fully worked solution.
- Fractions of a Quantity - Video - Fractions of a Quantity 2 Watch
Theory
A fraction of a quantity is the fraction multiplied by the quantity. Divide by the denominator to find one part, then multiply by the numerator: \(\dfrac{4}{9}\) of \(927\) g is \(927 \div 9 \times 4 = 412\) g. This Year 8 maths page shows how.
The two steps. For \(\dfrac{5}{6}\) of \(42\), divide by the denominator to find one part, then multiply by the numerator: \(42 \div 6 \times 5 = 35\).
Change the unit first when the answer needs a different one: \(1\) h \(= 60\) min, \(1\) day \(= 24\) h, \(1\) year \(= 12\) months, \(1\) t \(= 1000\) kg, \(1\) kg \(= 1000\) g, \(1\) m \(= 100\) cm, \(1\) L \(= 1000\) mL.
Mixed numerals. Change to an improper fraction first: \(1\dfrac{1}{2}\) of \(20\) is \(\dfrac{3}{2} \times 20 = 30\).
The size of the fraction tells you the size of the answer. A fraction between \(0\) and \(1\) makes the quantity smaller; a fraction greater than \(1\) makes it larger. Dividing does the opposite: \(20 \div \dfrac{4}{5} = 25\).
The rest is the whole take away the part: \(1 - \dfrac{3}{7} = \dfrac{4}{7}\). A fraction of what is left is a fraction of a fraction: multiply the two fractions, then take that fraction of the quantity.
An increase by a fraction. Adding \(\dfrac{1}{3}\) of an amount gives \(1 + \dfrac{1}{3} = \dfrac{4}{3}\) of it. The gain itself is just \(\dfrac{1}{3}\) of the amount.
Working backwards. When \(\dfrac{a}{b}\) of the whole is known, divide by \(a\) to get \(\dfrac{1}{b}\) of the whole, then multiply by \(b\).
Estimate, then check. Compare the fraction with \(\dfrac{1}{2}\) or \(1\), and the answer with half the quantity or the quantity itself.
Units to change first:
| Quantity | In the smaller unit | Example |
|---|---|---|
| 1 hour | 60 minutes | \(\dfrac{3}{5}\) h \(= 36\) min |
| 1 day | 24 hours | \(\dfrac{3}{4}\) day \(= 18\) h |
| 1 tonne | 1000 kg | \(\dfrac{3}{8}\) t \(= 375\) kg |
| 1 kilogram | 1000 g | \(\dfrac{2}{9}\) of \(4.5\) kg \(= 1000\) g |
| 1 litre | 1000 mL | \(\dfrac{1}{4}\) L \(= 250\) mL |
Finding a fraction of a quantity
- Change the quantity to the unit the answer needs.
- Change any mixed numeral to an improper fraction.
- Divide by the denominator to find one part.
- Multiply by the numerator, then compare with your estimate.
Shares and the whole
- What is left: \(1 -\) the fraction used.
- A fraction of what is left: multiply the fractions, then use the quantity.
- From a part to the whole: divide by the numerator, multiply by the denominator.
(ii) Find \(\dfrac{3}{5}\) of an hour, in minutes.
| \(\dfrac{4}{9} \times 927\) | \(=\) | \(4 \times \dfrac{927}{9}\) |
| \(\) | \(=\) | \(4 \times 103 = 412\) |
| \(\dfrac{3}{5} \times 60\) | \(=\) | \(3 \times \dfrac{60}{5}\) |
| \(\) | \(=\) | \(3 \times 12 = 36\) |
(i) \(412\) g
(ii) \(36\) minutes
(ii) Find \(\dfrac{3}{8}\) of a tonne, in kilograms.
| \(4.5 \times 1000\) | \(=\) | \(4500~\text{g}\) |
| \(4500 \div 9 \times 2\) | \(=\) | \(1000\) |
| \(1000 \div 8 \times 3\) | \(=\) | \(375\) |
(i) \(1000\) g. Estimate: \(\dfrac{2}{9}\) is a little under \(\dfrac{1}{4}\), and \(\dfrac{1}{4}\) of \(4500\) g is \(1125\) g.
(ii) \(375\) kg
\(2\dfrac{1}{4} = \dfrac{9}{4}\) is more than \(1\), so the answer is more than \(6.4\) L.
| \(6.4 \div 4\) | \(=\) | \(1.6\) |
| \(9 \times 1.6\) | \(=\) | \(14.4\) |
\(14.4\) L. Check: \(2 \times 6.4 + 1.6 = 14.4\).
| \(1 - \dfrac{5}{8}\) | \(=\) | \(\dfrac{3}{8}\) |
| \(\dfrac{2}{3} \times \dfrac{3}{8}\) | \(=\) | \(\dfrac{1}{4}\) |
| \(840 \div 4\) | \(=\) | \(210\) |
The hospital gets \(\$210\). Check: the club gets \(\$525\), leaving \(\$315\), and \(\dfrac{2}{3} \times 315 = 210\).
(i) What does the bike cost?
(ii) How much more must he save?
| \(132 \div 4\) | \(=\) | \(33\) |
| \(7 \times 33\) | \(=\) | \(231\) |
| \(231 - 132\) | \(=\) | \(99\) |
(i) \(\$231\). Check: \(\dfrac{4}{7} \times 231 = 132\).
(ii) \(\$99\)
(ii) Find \(\dfrac{3}{8}\) of \(\$25.60\).
(iii) Without working them out, which is greater than \(36\): \(\dfrac{2}{3}\) of \(36\), \(\dfrac{5}{4}\) of \(36\) or \(\dfrac{1}{2}\) of \(36\)?
| \(\dfrac{5}{4} \times 4.80\) | \(=\) | \(1.20 \times 5 = 6.00\) |
| \(25.60 \div 8 \times 3\) | \(=\) | \(3.20 \times 3 = 9.60\) |
(i) \(\$6.00\)
(ii) \(\$9.60\)
(iii) \(\dfrac{5}{4}\) of \(36\), because \(\dfrac{5}{4}\) is the only fraction greater than \(1\).
Common pitfalls
Frequently asked questions
How do you find a fraction of a number?
Divide the number by the denominator, then multiply by the numerator: \(\dfrac{2}{3}\) of \(27\) is \(27 \div 3 \times 2 = 18\).
What does of mean in maths?
‘Of’ means multiply: \(\dfrac{1}{4}\) of \(12\) is \(\dfrac{1}{4} \times 12 = 3\).
How do I find a fraction of an hour?
Change the hour to \(60\) minutes first: \(\dfrac{5}{6}\) of an hour is \(60 \div 6 \times 5 = 50\) minutes.
What if the fraction is more than 1?
Change it to an improper fraction and multiply as usual. The answer will be bigger than the quantity.
How do I find the whole from a fraction of it?
Divide the known part by the numerator to get one part, then multiply by the denominator.