Percentage Problems
Theory
Every percentage problem starts by naming the whole (\(100\%\)), then choosing the move: find a part, write a part as a percentage, or find the whole. \(4\) absent out of \(24\) is \(\dfrac{4}{24} \times 100\% = 16\dfrac{2}{3}\%\). This Year 8 maths page shows how.
Name the whole. Decide which quantity is \(100\%\) before anything else. In a two-step problem the whole often changes partway through.
Move 1: find a part. Multiply the whole by the percentage as a decimal: \(24\%\) of \(600\) is \(0.24 \times 600 = 144\).
Move 2: write a part as a percentage. \(\dfrac{\text{part}}{\text{whole}} \times 100\%\).
Move 3: find the whole. Divide the known amount by its percentage to get \(1\%\), then multiply by \(100\). After a discount of \(p\%\), the price you know is \((100 - p)\%\) of the original: \(\$72\) after \(10\%\) off is \(90\%\), so \(1\%\) is \(\$0.80\) and the original is \(\$80\).
The rest. Take the given percentages off \(100\%\), then use Move 1 (or Move 3) on what is left.
A percentage of a percentage. Multiply the decimals: \(40\%\) of a group, then \(25\%\) of those, is \(0.4 \times 0.25 = 0.1\), which is \(10\%\) of the whole group.
Commission is a percentage of sales; a retainer is a fixed amount paid as well. Pay \(=\) retainer \(+\) commission.
Reaching a target. Work out the total needed, then take away what you already have. To find how many weeks it takes, divide the goal by the amount each week. A year has \(52\) weeks or \(26\) fortnights.
Choose the move:
| You know | You want | Move |
|---|---|---|
| the whole and a percentage | the part | whole \(\times\) decimal |
| the part and the whole | the percentage | \(\dfrac{\text{part}}{\text{whole}} \times 100\%\) |
| the part and its percentage | the whole | part \(\div\) percentage \(\times 100\) |
| some percentages of the whole | the rest | \(100\% -\) the others |
| \(a\%\) of \(b\%\) | one percentage | multiply the decimals |
Solving a percentage problem
- Name the whole (\(100\%\)) for each step.
- Choose the move: part, percentage or whole.
- Work one step at a time; change units first.
- Check the parts add to the whole, and answer in a sentence.
Comparing two groups
- Write each result as a percentage of its own group.
- Compare the percentages, not the counts.
(ii) A basketball team scored \(600\) points in a season. Tony shot \(24\%\) of the points. How many points did he score?
| \(\dfrac{4}{24} \times 100\%\) | \(=\) | \(\dfrac{1}{6} \times 100\%\) |
| \(\) | \(=\) | \(16\dfrac{2}{3}\%\) |
| \(\dfrac{24}{100} \times 600\) | \(=\) | \(24 \times 6 = 144\) |
(i) \(16\dfrac{2}{3}\%\)
(ii) \(144\) points
(i) What percentage are senior players?
(ii) How many senior players are there?
| \(100\% - 55\% - 15\%\) | \(=\) | \(30\%\) |
| \(0.3 \times 320\) | \(=\) | \(96\) |
(i) \(30\%\)
(ii) \(96\). Check: \(176 + 48 + 96 = 320\).
| \(\dfrac{17}{20} \times 100\%\) | \(=\) | \(85\%\) |
| \(\dfrac{33}{40} \times 100\%\) | \(=\) | \(82.5\%\) |
8B had the better rate, \(85\%\) against \(82.5\%\), even though more students improved in 8F.
(i) What mass of almonds is in the bag, in grams?
(ii) What percentage of the bag is almonds?
| \(0.3 \times 2500\) | \(=\) | \(750\) |
| \(0.4 \times 750\) | \(=\) | \(300\) |
| \(0.3 \times 0.4\) | \(=\) | \(0.12 = 12\%\) |
(i) \(300\) g
(ii) \(12\%\). Check: \(\dfrac{300}{2500} \times 100\% = 12\%\).
(ii) Zoe saves \(15\%\) of her \(\$120\) weekly pay. How many weeks will it take to save \(\$270\)?
| \(431 - 95\) | \(=\) | \(336\) |
| \(\dfrac{336}{4800} \times 100\%\) | \(=\) | \(7\%\) |
| \(0.15 \times 120\) | \(=\) | \(18\) |
| \(270 \div 18\) | \(=\) | \(15\) |
(i) \(7\%\). Check: \(0.07 \times 4800 + 95 = 431\).
(ii) \(15\) weeks
(ii) Ali made \(60\%\) of her first \(40\) shots. She takes \(10\) more. How many must she make to have made \(64\%\) of all \(50\)?
| \(100\% - 65\%\) | \(=\) | \(35\%\) |
| \(70 \div 35 \times 100\) | \(=\) | \(200\) |
| \(0.4 \times 200\) | \(=\) | \(80\) |
| \(0.64 \times 50\) | \(=\) | \(32\) |
| \(0.6 \times 40\) | \(=\) | \(24\) |
| \(32 - 24\) | \(=\) | \(8\) |
(i) \(200\) students were surveyed and \(80\) walk.
(ii) She needs \(32\) in total and has \(24\), so she must make \(8\) of the \(10\).
Common pitfalls
Frequently asked questions
How do I start a percentage problem?
Decide which quantity is the whole, 100%, then decide whether you want a part, a percentage or the whole.
How do I find the rest as a percentage?
Take all the given percentages away from \(100\%\).
What is a percentage of a percentage?
Multiply the two decimals: \(50\%\) of \(40\%\) is \(0.5 \times 0.4 = 0.2\), so \(20\%\) of the whole.
How do I compare results from groups of different sizes?
Write each result as a percentage of its own group, then compare the percentages.
How do I find the commission rate?
Subtract any retainer from the pay, then write the commission as a percentage of the sales.