Non-Calculator Percentages
Every question with a fully worked solution.
- Non-Calculator Percentages - Video - Non-Calculator Percentages 1 Watch
Theory
Every percentage can be built in your head from a few benchmarks: \(10\%\) is a division by \(10\), \(1\%\) a division by \(100\), and the rest come from halving and adding. \(65\%\) of \(80\) is \(40 + 8 + 4 = 52\) (\(50\% + 10\% + 5\%\)). This Year 8 maths page shows how.
Ten and one. For \(10\%\), divide by \(10\). For \(1\%\), divide by \(100\). Almost everything is built from those two.
Halving. \(50\%\) is half, \(25\%\) is half of that, and \(12\dfrac{1}{2}\%\) (\(12.5\%\)) is half again. Halving \(10\%\) gives \(5\%\), then \(2\dfrac{1}{2}\%\).
Fraction benchmarks. \(20\% = \dfrac{1}{5}\), \(40\% = \dfrac{2}{5}\), \(60\% = \dfrac{3}{5}\), \(75\% = \dfrac{3}{4}\), \(33\dfrac{1}{3}\% = \dfrac{1}{3}\), \(66\dfrac{2}{3}\% = \dfrac{2}{3}\), \(37.5\% = \dfrac{3}{8}\), \(62.5\% = \dfrac{5}{8}\).
Add or subtract the parts. \(65\% = 50\% + 10\% + 5\%\), \(17.5\% = 10\% + 5\% + 2.5\%\) and \(9\% = 10\% - 1\%\). Work out each part, then add or take away.
Change units first when needed: \(2\) h \(30\) min is \(150\) minutes, so \(30\%\) of it is \(3 \times 15 = 45\) minutes.
Increase and decrease. Find the change, then add it on or take it off. A second change is worked out on the new amount.
Backwards. If \(20\%\) of a number is \(14\), then \(10\%\) is \(7\) and the whole is \(70\).
Estimate. Round to a friendly number and a benchmark percentage, then write \(\approx\).
Building percentages of \(\$80\):
| Percentage | Built from | Of \(\$80\) |
|---|---|---|
| \(10\%\) | \(80 \div 10\) | \(\$8\) |
| \(5\%\) | half of \(10\%\) | \(\$4\) |
| \(1\%\) | \(80 \div 100\) | \(\$0.80\) |
| \(85\%\) | \(8 \times 10\% + 5\%\) | \(\$68\) |
| \(9\%\) | \(10\% - 1\%\) | \(\$7.20\) |
| \(12\dfrac{1}{2}\%\) | \(\dfrac{1}{8}\) | \(\$10\) |
Finding a percentage in your head
- Split the percentage into benchmarks (\(50\%\), \(25\%\), \(10\%\), \(5\%\), \(1\%\) or a fraction).
- Find each benchmark by dividing or halving.
- Add or subtract the parts.
- Write money to the cent and check it is a sensible size.
Increase, decrease or backwards
- Find the change from benchmarks, then add it on or take it off.
- Backwards: scale the known percentage down to a benchmark, then up to \(100\%\).
(ii) What is \(75\%\) of \(48\) g?
| \(2 \times 60 + 30\) | \(=\) | \(150~\text{minutes}\) |
| \(\dfrac{30}{100} \times 150\) | \(=\) | \(3 \times 15 = 45\) |
| \(\dfrac{75}{100} \times 48\) | \(=\) | \(\dfrac{3}{4} \times 48 = 36\) |
(i) \(45\) minutes
(ii) \(36\) g
(i) \(30\%\) of \(70\) kg
(ii) \(8\%\) of \(\$180\)
(iii) \(17.5\%\) of \(\$160\)
(ii) \(8\% = 10\% - 2\%\).
(iii) \(17.5\% = 10\% + 5\% + 2.5\%\).
| \(3 \times 7\) | \(=\) | \(21\) |
| \(18 - 2 \times 1.8\) | \(=\) | \(14.4\) |
| \(16 + 8 + 4\) | \(=\) | \(28\) |
(i) \(21\) kg
(ii) \(\$14.40\)
(iii) \(\$28\)
(i) \(75\%\) of \(36\) m
(ii) \(12\dfrac{1}{2}\%\) of \(56\) g
(iii) \(33\dfrac{1}{3}\%\) of \(42\) m
| \(18 + 9\) | \(=\) | \(27\) |
| \(56 \div 2 \div 2 \div 2\) | \(=\) | \(7\) |
| \(42 \div 3\) | \(=\) | \(14\) |
(i) \(27\) m (\(50\% + 25\%\))
(ii) \(7\) g (one eighth)
(iii) \(14\) m (one third)
(i) Estimate the number of scarves.
(ii) The club counted \(18\,725\). How good was the estimate?
| \(38\,000 \div 2\) | \(=\) | \(19\,000\) |
| \(19\,000 - 18\,725\) | \(=\) | \(275\) |
(i) About \(19\,000\) (\(38\,214 \approx 38\,000\) and \(49\% \approx 50\%\)).
(ii) \(275\) too high, less than \(2\%\) out (\(2\%\) of \(19\,000\) is \(380\)).
(ii) A surf club had \(250\) members. Membership grew by \(20\%\), then fell by \(5\%\). How many members now?
(iii) \(20\%\) of a number is \(14\). Find the number.
| \(60 + 6 + 3\) | \(=\) | \(69\) |
| \(250 + 50\) | \(=\) | \(300\) |
| \(300 - 15\) | \(=\) | \(285\) |
| \(7 \times 10\) | \(=\) | \(70\) |
(i) \(\$69\)
(ii) \(285\), because the \(5\%\) comes off \(300\), not \(250\).
(iii) \(10\%\) is \(7\), so the number is \(70\).
Common pitfalls
Frequently asked questions
How do I find 10% of a number?
Divide by \(10\): \(10\%\) of \(\$74\) is \(\$7.40\).
How do I find 5% without a calculator?
Find \(10\%\), then halve it: \(5\%\) of \(\$74\) is \(\$3.70\).
How do I find 15%?
Add \(10\%\) and \(5\%\): \(15\%\) of \(\$40\) is \(\$4 + \$2 = \$6\).
How do I find 12.5% in my head?
It is one eighth, so halve three times: \(12.5\%\) of \(48\) is \(6\).
How do I estimate a percentage?
Round the amount to a friendly number and the percentage to a benchmark, then work it out and write approximately equal.