Fractions of a Quantity
Fractions of a Quantity in NSW Year 8 maths (NESA outcome MA4‑FRC‑C‑01) finds a fraction of an amount by dividing by the denominator and multiplying by the numerator, so \(\frac{5}{6}\) of 216 km is \(216 \div 6 \times 5 = 180\) km.
You will change units first (\(\frac{5}{12}\) of an hour is 25 minutes), use mixed numerals such as \(1\frac{2}{3}\) of 45 L, increase an amount by a fraction, take a fraction of what is left and work backwards from a part to the whole. You will also judge whether the answer should be more or less than the quantity before working it out. Bus fares, planks of wood and bottles of juice are typical contexts. It is part of Year 8 Fractions and Percentages.
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.