Percentage of a Quantity (easy)
Finding a percentage of a quantity answers questions such as ‘what is 45% of 90 kg?’. In NSW Year 7 maths (NESA outcome MA4‑FRC‑C‑01) you learn that ‘of’ means multiply and that per cent means out of 100.
You will use benchmark fractions (50% is a half, 25% is a quarter), build answers from 10% and 1%, and use the unitary method for values such as \(3\frac{1}{2}\%\) of $1200 ($42). You will also add GST of 10% to an electrician’s bill, work out how a farm’s land is shared between wheat, canola and grazing, and always give the answer with its units. This subtopic is part of Year 7 Percentages.
Every question with a fully worked solution.
- Percentage of a Quantity (easy) - Video - Percentage of a Quantity Watch
Theory
To find a percentage of a quantity, remember that “of” means multiply and per cent means out of \(100\): \(35\%\) of \(600\) L \(= \dfrac{35}{100} \times 600\) L. This Year 7 maths page shows three ways to find it: benchmark fractions, building from \(10\%\), and the unitary method (find \(1\%\) first).
Method 1. Write the percentage as a fraction and multiply: \(15\%\) of \(300\) m \(= \dfrac{15}{100} \times 300\) m \(= 15 \times 3 = 45\) m.
Method 2. Use a benchmark. \(50\% = \dfrac{1}{2}\) (divide by \(2\)), \(25\% = \dfrac{1}{4}\) (divide by \(4\)), \(10\% = \dfrac{1}{10}\) (divide by \(10\)). Build others from \(10\%\): \(30\%\) is \(3\) lots of \(10\%\); \(5\%\) is half of \(10\%\).
Method 3. The unitary method. Find \(1\%\) (divide by \(100\)), then multiply by the percentage. For \(18\%\) of \(\$450\): \(1\%\) is \(4.5\) and \(18 \times 4.5 = 81\), so the answer is \(\$81.00\). It works for any percentage, even \(4\dfrac{1}{2}\%\) (multiply by \(4.5\)).
Change units first when the answer needs different units. GST (goods and services tax) is \(10\%\) of a price and is added on. Always write the answer with units, and money to the cent.
Three ways:
| Percentage | Fraction | Do this |
|---|---|---|
| \(50\%\) | \(\dfrac{1}{2}\) | \(\div\,2\) |
| \(25\%\) | \(\dfrac{1}{4}\) | \(\div\,4\) |
| \(10\%\) | \(\dfrac{1}{10}\) | \(\div\,10\) |
| \(1\%\) | \(\dfrac{1}{100}\) | \(\div\,100\) |
How to find a percentage of a quantity
- Change units first if the answer needs different units.
- Choose a method: benchmark, build from \(10\%\), or find \(1\%\).
- Calculate, then check the size.
- Write the answer with units (money to the cent).
(i) \(50\%\) of \(64\)
(ii) \(25\%\) of \(120\)
(iii) \(10\%\) of \(350\)
| \(64 \div 2\) | \(=\) | \(32\) |
| \(120 \div 4\) | \(=\) | \(30\) |
| \(350 \div 10\) | \(=\) | \(35\) |
(i) \(32\)
(ii) \(30\)
(iii) \(35\)
| \(\dfrac{15}{100} \times 300\) | \(=\) | \(15 \times 3\) |
| \(\) | \(=\) | \(45\) |
\(45\) m
| \(350 \div 100\) | \(=\) | \(3.5\) |
| \(24 \times 3.5\) | \(=\) | \(84\) |
\(84\) kg. Check: \(25\%\) is \(350 \div 4 = 87.5\); take away \(1\%\): \(87.5 - 3.5 = 84\).
| \(40 \div 100\) | \(=\) | \(0.4\) |
| \(5.5 \times 0.4\) | \(=\) | \(2.2\) |
\(2.2\) kg. (As fractions: \(\dfrac{11}{200} \times 40 = \dfrac{22}{10} = 2.2\).)
| \(3 \times 60\) | \(=\) | \(180\) |
| \(180 \div 10\) | \(=\) | \(18\) |
| \(3 \times 18\) | \(=\) | \(54\) |
\(54\) minutes
(i) What is the total cost of Quote A?
(ii) Which quote is cheaper, and by how much?
| \(420 \div 10\) | \(=\) | \(42\) |
| \(420 + 42\) | \(=\) | \(462\) |
| \(462 - 455\) | \(=\) | \(7\) |
(i) \(\$462.00\)
(ii) Quote B, by \(\$7.00\)
| \(480 \div 100\) | \(=\) | \(4.8\) |
| \(45 \times 4.8\) | \(=\) | \(216\) |
| \(12.5 \times 4.8\) | \(=\) | \(60\) |
| \(480 - 216 - 60\) | \(=\) | \(204\) |
\(204\) students are driven. Check: \(100\% - 45\% - 12.5\% = 42.5\%\) and \(42.5 \times 4.8 = 204\).
Common pitfalls
Frequently asked questions
How do you find a percentage of an amount?
Write the percentage over \(100\) and multiply: \(\dfrac{15}{100} \times 300 = 45\), so \(15\%\) of \(300\) is \(45\).
How do you find 10% of a number?
Divide by \(10\). \(10\%\) of \(350\) is \(35\).
What is the unitary method?
Find \(1\%\) by dividing by \(100\), then multiply by the percentage.
What is GST?
A \(10\%\) tax added to the price.
Is 8% of 50 the same as 50% of 8?
Yes. Both are \(4\).