Expressing Quantities as Fractions
Expressing one quantity as a fraction of another means writing the part over the whole. In NSW Year 7 maths (NESA outcome MA4‑FRC‑C‑01) both amounts must be in the same units before the fraction is simplified.
You will convert units first (350 g of 2 kg is \(\frac{350}{2000} = \frac{7}{40}\)), simplify using the HCF, find the part left over and work back from a fraction to the whole. You will also write one result three ways: 18 minutes of an hour is \(\frac{3}{10}\), 0.3 or 30%. Lunch as a share of the school day is a typical question. This subtopic is part of Year 7 Percentages.
Every question with a fully worked solution.
- Expressing Quantities as Fractions - Video - Expressing one quantity as a fraction of another Watch
Theory
To write one quantity as a fraction of another, write the part (the quantity) on top and the whole (the total amount) on the bottom, in the same units, then simplify. \(50\) minutes of \(2\) hours is \(\dfrac{50}{120} = \dfrac{5}{12}\). This Year 7 maths page shows the steps, the units to change, the part left over and working backwards.
\(\text{Fraction} = \dfrac{\text{part}}{\text{whole}}\). The part (the quantity) goes in the numerator; the whole amount (the total) goes in the denominator.
Same units first. Both numbers must be in the same units (minutes, grams, metres ...). \(50\) minutes of \(2\) hours: \(2 \times 60 = 120\) minutes, so the fraction is \(\dfrac{50}{120}\).
Simplify by dividing the top and the bottom by the HCF (highest common factor). The HCF of \(50\) and \(120\) is \(10\): \(\dfrac{50}{120} = \dfrac{5}{12}\). If the HCF is \(1\), the fraction is already in simplest form: \(9\) cm of \(1\) m is \(\dfrac{9}{100}\).
Check: \(\dfrac{5}{12}\) of \(120\) is \(120 \div 12 \times 5 = 50\).
The part left over is \(1\) take away the part: \(1 - \dfrac{5}{12} = \dfrac{7}{12}\).
Working backwards. If \(200\) mL is \(\dfrac{1}{5}\) of a bottle, the whole bottle is \(5\) parts: \(5 \times 200 = 1000\) mL, which is \(1\) L.
The rule, and the units to change:
| Change to the same units | |
|---|---|
| \(1\) m \(= 100\) cm | \(1\) km \(= 1000\) m |
| \(1\) kg \(= 1000\) g | \(1\) g \(= 1000\) mg |
| \(1\) L \(= 1000\) mL | \(1\) h \(= 60\) min |
| \(1\) day \(= 24\) h | \(1\) dollar \(= 100\) cents |
How to write a quantity as a fraction of another
- Change both quantities to the same units (use the smaller unit).
- Write the part in the numerator and the whole in the denominator.
- Simplify: divide the top and bottom by the HCF.
- Check: the fraction of the whole should give the part back.
(ii) A netball team won \(12\) of its \(16\) games. What fraction of its games did it win?
| \(\dfrac{18}{45}\) | \(=\) | \(\dfrac{2}{5}\) |
| \(\dfrac{12}{16}\) | \(=\) | \(\dfrac{3}{4}\) |
(i) \(\dfrac{2}{5}\) (HCF \(9\); check: \(45 \div 5 \times 2 = 18\))
(ii) \(\dfrac{3}{4}\) (HCF \(4\))
| \(3 \times 1000\) | \(=\) | \(3000\) |
| \(\dfrac{600}{3000}\) | \(=\) | \(\dfrac{6}{30}\) |
| \(\) | \(=\) | \(\dfrac{1}{5}\) |
\(\dfrac{1}{5}\)
| \(28 \times 1000\) | \(=\) | \(28000\) |
| \(\dfrac{140}{28000}\) | \(=\) | \(\dfrac{14}{2800}\) |
| \(\) | \(=\) | \(\dfrac{1}{200}\) |
\(\dfrac{1}{200}\) (the HCF of \(140\) and \(28\,000\) is \(140\))
(i) Express \(450\) g as a fraction of \(2.5\) kg.
(ii) Express \(75\) cents as a fraction of \(\$6\).
| \(2.5 \times 1000\) | \(=\) | \(2500\) |
| \(\dfrac{450}{2500}\) | \(=\) | \(\dfrac{9}{50}\) |
| \(6 \times 100\) | \(=\) | \(600\) |
| \(\dfrac{75}{600}\) | \(=\) | \(\dfrac{1}{8}\) |
(i) \(\dfrac{9}{50}\) (HCF \(50\))
(ii) \(\dfrac{1}{8}\) (HCF \(75\))
(i) What fraction of the trip does she read?
(ii) What fraction does she not read?
| \(60 + 20\) | \(=\) | \(80\) |
| \(\dfrac{50}{80}\) | \(=\) | \(\dfrac{5}{8}\) |
| \(1 - \dfrac{5}{8}\) | \(=\) | \(\dfrac{3}{8}\) |
(i) \(\dfrac{5}{8}\)
(ii) \(\dfrac{3}{8}\). Check: \(80 - 50 = 30\) min, and \(\dfrac{30}{80} = \dfrac{3}{8}\).
| \(8 \times 300\) | \(=\) | \(2400\) |
| \(2400 \div 1000\) | \(=\) | \(2.4\) |
The bottle holds \(2.4\) L.
| \(\dfrac{300}{2000}\) | \(=\) | \(\dfrac{3}{20} = \dfrac{9}{60}\) |
| \(\dfrac{40}{240}\) | \(=\) | \(\dfrac{1}{6} = \dfrac{10}{60}\) |
\(10 > 9\), so \(40\) min of \(4\) h is the greater fraction.
Common pitfalls
Frequently asked questions
How do you write one quantity as a fraction of another?
Write both in the same units, put the part over the whole, then simplify.
Why must the units be the same?
Otherwise the fraction is wrong: \(9\) cm of \(1\) m is \(\dfrac{9}{100}\), not \(9\).
How do you simplify the fraction?
Divide the top and bottom by their highest common factor. \(\dfrac{50}{120} = \dfrac{5}{12}\).
How do you find the part that is left?
Take the fraction away from \(1\). \(1 - \dfrac{5}{12} = \dfrac{7}{12}\).
If 200 mL is 1/5 of a bottle, how much does the bottle hold?
\(5 \times 200 = 1000\) mL, which is \(1\) L.