Expressing Amounts as Fractions and Percentages
This NSW Year 8 maths subtopic (NESA outcome MA4‑FRC‑C‑01) writes one amount as a fraction or a percentage of another. Both amounts are changed to the same unit first, then the part is written over the whole and simplified.
You will turn test marks such as 24 out of 32 into 75%, compare 850 g with a 3.4 kg packet of rice, round a percentage that does not stop, and find a percentage increase or a loss as a percentage of the cost price. Netball shooting records, a guitar solo’s share of a song and the free space on a phone show how often one quantity is compared with another. It is part of Year 8 Fractions and Percentages.
Every question with a fully worked solution.
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Theory
To write one amount as a fraction of another, put the part over the whole in the same units and simplify; multiply by \(100\%\) for a percentage. \(28\) out of \(40\) is \(\dfrac{28}{40} = \dfrac{7}{10} = 70\%\). This Year 8 maths page shows how.
Step 1: match the units. Change both amounts to the same unit, usually the smaller one so no decimals are left: cents, grams, millimetres, millilitres, seconds or minutes.
Step 2: part over whole. If the whole is made of two or more groups, add them first: \(14\) red and \(21\) blue beads make \(35\), so red is \(\dfrac{14}{35} = \dfrac{2}{5}\).
Step 3: simplify by dividing the top and the bottom by their HCF.
Step 4: for a percentage, multiply by \(100\%\). \(\dfrac{13}{25} \times 100\% = \dfrac{1300}{25}\% = 52\%\). If the denominator divides into \(100\), make it \(100\): \(\dfrac{13}{25} = \dfrac{52}{100} = 52\%\).
Rounding. When the percentage does not stop, round to the accuracy asked for and write \(\approx\). An exact answer keeps a fractional percentage: \(\dfrac{1}{6} \times 100\% = 16\dfrac{2}{3}\%\).
Choose the right whole. A change, a profit or a loss is compared with the amount before the change: the original price, or the cost price of something bought and sold.
Comparing results out of different totals: change each to a percentage, then compare. The difference between two percentages is measured in percentage points.
Part over whole, then times 100%:
| Amounts | Fraction | Percentage |
|---|---|---|
| \(18\) of \(30\) | \(\dfrac{18}{30} = \dfrac{3}{5}\) | \(60\%\) |
| \(40\) cm of \(3\) m | \(\dfrac{40}{300} = \dfrac{2}{15}\) | \(13\dfrac{1}{3}\%\) |
| \(105\) min of \(5\) h | \(\dfrac{105}{300} = \dfrac{7}{20}\) | \(35\%\) |
| \(\$2.40\) of \(\$8\) | \(\dfrac{240}{800} = \dfrac{3}{10}\) | \(30\%\) |
Expressing one amount as a percentage of another
- Match the units (change to the smaller unit).
- Write part over whole; add groups first if the whole is made of several.
- Simplify the fraction.
- Multiply by \(100\%\), and round with \(\approx\) only at the end.
Percentage change, profit or loss
- Find the change (new \(-\) old, or the profit or loss).
- Divide by the original amount (the cost price).
- Multiply by \(100\%\).
(i) What fraction are girls?
(ii) What percentage are boys?
(iii) Annie scored \(28\) out of \(40\) in a test. What percentage is this?
| \(\dfrac{18}{30}\) | \(=\) | \(\dfrac{6 \times 3}{6 \times 5} = \dfrac{3}{5}\) |
| \(\dfrac{12}{30} \times 100\%\) | \(=\) | \(\dfrac{2}{5} \times 100\%\) |
| \(\) | \(=\) | \(40\%\) |
| \(\dfrac{28}{40} \times 100\%\) | \(=\) | \(\dfrac{7}{10} \times 100\%\) |
| \(\) | \(=\) | \(70\%\) |
(i) \(\dfrac{3}{5}\)
(ii) \(40\%\)
(iii) \(70\%\)
(i) What fraction is this, in simplest form?
(ii) What percentage is this?
| \(\dfrac{52}{80}\) | \(=\) | \(\dfrac{13}{20}\) |
| \(\dfrac{13}{20}\) | \(=\) | \(\dfrac{65}{100} = 65\%\) |
(i) \(\dfrac{13}{20}\) (HCF \(4\))
(ii) \(65\%\)
(ii) A \(5\) h trip includes \(1\) h \(45\) min of driving on dirt roads. What percentage of the trip is that?
(i) \(2.4\) m \(= 240\) cm.
(ii) \(5\) h \(= 300\) min and \(1\) h \(45\) min \(= 105\) min.
| \(\dfrac{90}{240} \times 100\%\) | \(=\) | \(\dfrac{3}{8} \times 100\%\) |
| \(\) | \(=\) | \(37.5\%\) |
| \(\dfrac{105}{300} \times 100\%\) | \(=\) | \(\dfrac{7}{20} \times 100\%\) |
| \(\) | \(=\) | \(35\%\) |
(i) \(37.5\%\). Check: \(0.375 \times 240 = 90\).
(ii) \(35\%\)
| \(\dfrac{19}{31} \times 100\%\) | \(=\) | \(\dfrac{1900}{31}\%\) |
| \(\) | \(=\) | \(61.29\ldots\%\) |
| \(\) | \(\approx\) | \(61.3\%\) |
About \(61.3\%\). Estimate: \(19\) is a bit over half of \(31\).
(ii) A price rises from \(\$40\) to \(\$50\). Find the percentage increase.
| \(120 - 78\) | \(=\) | \(42\) |
| \(\dfrac{42}{120} \times 100\%\) | \(=\) | \(\dfrac{7}{20} \times 100\% = 35\%\) |
| \(\dfrac{10}{40} \times 100\%\) | \(=\) | \(25\%\) |
(i) A \(35\%\) loss. Check: \(0.35 \times 120 = 42\).
(ii) A \(25\%\) increase (the change \(\$10\) over the original \(\$40\)).
(ii) A \(4000\) L tank is \(75\%\) full. Then \(1000\) L is used. What percentage full is it now?
| \(\dfrac{33}{40} \times 100\%\) | \(=\) | \(82.5\%\) |
| \(\dfrac{41}{50} \times 100\%\) | \(=\) | \(82\%\) |
| \(0.75 \times 4000 - 1000\) | \(=\) | \(2000\) |
| \(\dfrac{2000}{4000} \times 100\%\) | \(=\) | \(50\%\) |
(i) Kim did better by \(0.5\) percentage points.
(ii) \(50\%\) full
Common pitfalls
Frequently asked questions
How do I write one amount as a percentage of another?
Write the part over the whole in the same units, then multiply by \(100\%\): \(15\) out of \(60\) is \(\dfrac{15}{60} \times 100\% = 25\%\).
Why must the units be the same?
A fraction compares like with like. 40 cm out of 3 m only makes sense when both are in centimetres.
How do I find a percentage increase?
Divide the increase by the original amount and multiply by \(100\%\).
What is the cost price?
The amount paid for something. A profit or loss percentage is worked out on the cost price.
What is a percentage point?
The difference between two percentages: \(82.5\%\) and \(82\%\) differ by \(0.5\) percentage points.