Percentage of a Quantity
Theory
To find a percentage of a quantity, divide the percentage by \(100\) to get a decimal multiplier, then multiply: \(62\%\) of \(250\) is \(0.62 \times 250 = 155\). Per cent means out of \(100\) and ‘of’ means times. This Year 8 maths page shows how.
The decimal multiplier. Divide the percentage by \(100\): \(9\% = 0.09\), \(74\% = 0.74\), \(118\% = 1.18\), \(3.5\% = 0.035\), \(0.4\% = 0.004\). Then multiply the quantity by it.
Fractional percentages. Change the mixed numeral to an improper fraction and divide by \(100\): \(7\dfrac{1}{2}\% = \dfrac{15}{2} \div 100\).
Size check. A multiplier below \(1\) gives less than the quantity; above \(1\) gives more. \(118\%\) of an amount is bigger than the amount; \(0.4\%\) of it is tiny.
The part left is \((100 - p)\%\) of the quantity: if \(72\%\) is used, \(28\%\) is left.
Change the unit first when the answer needs a different one, then multiply.
GST (Goods and Services Tax) is \(10\%\) of the price before tax, added on top. The price including GST is \(110\%\) of it, so multiply by \(1.1\).
Two percentages in a row. The second is taken of the new amount. In one step, multiply the multipliers: \(15\%\) off then \(10\%\) off is \(0.85 \times 0.9 = 0.765\), so \(76.5\%\) is left, a discount of \(23.5\%\), not \(25\%\).
Commission is a payment worked out as a percentage of a sale, so it is found the same way.
Comparing deals. A percentage discount and a dollar discount can be compared only when both are in dollars. Money is rounded to the nearest cent (two decimal places). Better value. Divide each price by its quantity to get a price per kilogram (or per litre), round to the cent, and choose the lower one.
Percentages as multipliers:
| Percentage | Multiplier | Answer is |
|---|---|---|
| \(9\%\) | \(0.09\) | smaller |
| \(74\%\) | \(0.74\) | smaller |
| \(118\%\) | \(1.18\) | bigger |
| \(3.5\%\) | \(0.035\) | smaller |
| \(0.4\%\) | \(0.004\) | tiny |
| Situation | Multiplier |
|---|---|
| \(p\%\) of an amount | \(p \div 100\) |
| \(p\%\) is used, the rest is left | \((100 - p) \div 100\) |
| add 10% GST | \(1.1\) |
| \(a\%\) off, then \(b\%\) off | multiply the two multipliers |
Finding a percentage of a quantity
- Change the quantity to the unit the answer needs.
- Write the percentage as a decimal (divide by \(100\)).
- Multiply the quantity by the decimal.
- Round money to the nearest cent and check the size.
Percentages in a row
- Write each change as a multiplier (\(20\%\) off \(\to 0.8\), \(10\%\) on \(\to 1.1\)).
- Apply them one after another, or multiply them together first.
(ii) Find \(3\dfrac{1}{3}\%\) of \(\$300\).
(i) \(3\) hours \(= 180\) minutes.
| \(7\dfrac{1}{2}\% \times 180\) | \(=\) | \(\dfrac{15}{2} \times \dfrac{1}{100} \times 180\) |
| \(\) | \(=\) | \(15 \times \dfrac{90}{100}\) |
| \(\) | \(=\) | \(13.5\) |
| \(3\dfrac{1}{3}\% \times 300\) | \(=\) | \(\dfrac{10}{3} \times \dfrac{300}{100}\) |
| \(\) | \(=\) | \(\dfrac{10}{3} \times 3 = 10\) |
(i) \(13.5\) minutes
(ii) \(\$10\)
| \(1.8 \times 1000\) | \(=\) | \(1800~\text{mL}\) |
| \(0.45 \times 1800\) | \(=\) | \(810\) |
\(810\) mL is poured. Check: \(55\%\) is left, \(0.55 \times 1800 = 990\), and \(810 + 990 = 1800\).
| \(1.1 \times 58\) | \(=\) | \(63.8\) |
| \(58 + 0.1 \times 58\) | \(=\) | \(58 + 5.8 = 63.8\) |
\(\$63.80\) a month.
| \(0.75 \times 400\) | \(=\) | \(300\) |
| \(1.2 \times 300\) | \(=\) | \(360\) |
| \(0.75 \times 1.2\) | \(=\) | \(0.9\) |
\(360\) magazines. May is \(90\%\) of March: down \(10\%\) overall, not down \(5\%\).
| \(11.5\%\) | \(=\) | \(0.115\) |
| \(0.115 \times 726.40\) | \(=\) | \(83.536\) |
\(83.536 \approx 83.54\), so \(\$83.54\). Estimate: \(10\%\) is about \(\$73\), so \(11.5\%\) is a little over \(\$80\).
(ii) A school has \(800\) students. \(60\%\) are in the junior school and \(35\%\) of those play a sport. How many junior students play a sport?
(iii) An agent earns \(3\%\) commission on a \(\$450\,000\) sale. Find the commission.
| \(0.25 \times 300\) | \(=\) | \(75\) |
| \(0.35 \times 0.6 \times 800\) | \(=\) | \(168\) |
| \(0.03 \times 450\,000\) | \(=\) | \(13\,500\) |
(i) Shop A saves \(\$75\), which is \(\$5\) more than Shop B.
(ii) \(168\) students
(iii) \(\$13\,500\)
Common pitfalls
Frequently asked questions
How do I find a percentage of an amount?
Write the percentage as a decimal and multiply: \(35\%\) of \(80\) is \(0.35 \times 80 = 28\).
How do I add GST to a price?
Multiply the price before GST by \(1.1\), because the price with GST is \(110\%\) of it.
Why is 25% off then 10% off not 35% off?
The second discount is taken from the smaller sale price, so it removes less than 10% of the original price.
How do I find a percentage like 12 1/2%?
Change it to an improper fraction and divide by \(100\): \(12\dfrac{1}{2}\% = \dfrac{25}{2} \div 100 = \dfrac{1}{8}\).
How do I round money?
Round to two decimal places, the nearest cent: \(\$83.536 \approx \$83.54\).