Expressing Amounts as Fractions and Percentages
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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.