Percentage of a Quantity (harder)
This subtopic tackles harder percentage questions, such as 14% of 550 swimmers or a console with 25% off. In NSW Year 7 maths (NESA outcome MA4‑FRC‑C‑01) it adds percentage increase and decrease, discounts, GST and profit.
You will build awkward percentages from easy parts, use a multiplier to increase or decrease in one step, and work backwards with the unitary method: if 480 girls are 30% of a school, there are 1600 students. Multi-step questions, such as saving 15% of a wage and then paying rent, check that each percentage is taken of the right amount. This subtopic is part of Year 7 Percentages.
Every question with a fully worked solution.
- Percentage of a Quantity (harder) - Video - Percent word problem Watch
Theory
A percentage is a fraction out of \(100\): \(15\% = \dfrac{15}{100}\). To find a harder percentage of a quantity, multiply, or build it from easy parts such as \(10\%\), \(5\%\) and \(1\%\). Then add or take away to find a new amount after a discount, GST or a profit. This Year 7 maths page also shows decimal multipliers and how to find the whole when you know a part.
A percentage is a fraction whose denominator is always \(100\): \(15\% = \dfrac{15}{100}\). “Of” means multiply, so \(p\%\) of a quantity is \(\dfrac{p}{100} \times\) the quantity.
Percentage of a quantity in four steps: write the percentage as a fraction over \(100\), multiply it by the quantity, simplify (cancel) where you can, then work it out. \(75\%\) of \(40\) marks: \(\dfrac{75}{100} \times 40 = \dfrac{75}{10} \times 4 = 7.5 \times 4 = 30\) marks.
Build from easy parts. \(10\%\): divide by \(10\). \(5\%\): half of \(10\%\). \(1\%\): divide by \(100\). \(12\dfrac{1}{2}\% = \dfrac{1}{8}\): divide by \(8\). So \(45\% = 4 \times 10\% + 5\%\) and \(12\% = 10\% + 2 \times 1\%\).
Increase or decrease. Find the change, then add it on or take it away. Sale price \(=\) price \(-\) discount. Total \(=\) price \(+\) GST (GST is \(10\%\)). Selling price \(=\) cost price \(+\) profit, where the profit is a percentage of the cost price.
A multiplier does it in one step. To increase by \(6\%\) you end with \(100\% + 6\% = 106\%\), so multiply by \(1.06\). To decrease by \(12\%\) you end with \(100\% - 12\% = 88\%\), so multiply by \(0.88\).
Finding the whole. If you know a percentage of the whole (for example \(30\%\) is \(\$42\)), find \(10\%\) (or \(1\%\)) first, then \(100\%\).
Easy parts and multipliers:
| Change | You end with | Multiply by |
|---|---|---|
| up \(10\%\) (GST) | \(110\%\) | \(1.1\) |
| up \(6\%\) | \(106\%\) | \(1.06\) |
| down \(12\%\) | \(88\%\) | \(0.88\) |
| down \(25\%\) | \(75\%\) | \(0.75\) |
How to find a harder percentage of a quantity
- Change units first if the answer needs different units.
- Split the percentage into easy parts (\(10\%\), \(5\%\), \(1\%\)), or write it over \(100\) and multiply.
- Add the parts, then check with a multiplier such as \(0.26 \times 450\).
- Write the answer with units, and money to the cent.
How to increase or decrease by a percentage
- Find the change (the discount, GST or profit).
- Add it for an increase, or take it away for a decrease.
- Or multiply once by the percentage you end with, as a decimal.
How to find the whole from a part
- Divide the part to find \(10\%\) (or \(1\%\)).
- Multiply to find \(100\%\).
- Answer the question: the whole, or what is left.
| \(\dfrac{75}{100} \times 40\) | \(=\) | \(\dfrac{75}{10} \times 4\) |
| \(\) | \(=\) | \(7.5 \times 4\) |
| \(\) | \(=\) | \(30\) |
Tony scored \(30\) marks.
(i) \(26\%\) of \(\$450\)
(ii) \(45\%\) of \(3\) hours, in minutes
| \(10\% \text{ of } 450\) | \(=\) | \(45\) |
| \(5\% \text{ of } 450\) | \(=\) | \(22.5\) |
| \(1\% \text{ of } 450\) | \(=\) | \(4.5\) |
| \(45 + 45 + 22.5 + 4.5\) | \(=\) | \(117\) |
| \(3 \times 60\) | \(=\) | \(180\) |
| \(4 \times 18 + 9\) | \(=\) | \(81\) |
(i) \(26\% = 10\% + 10\% + 5\% + 1\%\), so the answer is \(\$117\). Check: \(0.26 \times 450 = 117\).
(ii) \(10\%\) of \(180\) min is \(18\) min and \(5\%\) is \(9\) min, so \(45\% = 4 \times 10\% + 5\%\) is \(81\) minutes.
| \(\dfrac{2.5}{100} \times 1500\) | \(=\) | \(2.5 \times 15\) |
| \(\) | \(=\) | \(37.5\) |
| \(1500 + 37.5\) | \(=\) | \(1537.5\) |
Her new pay is \(\$1537.50\) a week.
(ii) A phone repair costs \(\$85\), then \(10\%\) GST is added. Find the total cost.
| \(36 + 18\) | \(=\) | \(54\) |
| \(360 - 54\) | \(=\) | \(306\) |
| \(85 \div 10\) | \(=\) | \(8.5\) |
| \(85 + 8.5\) | \(=\) | \(93.5\) |
(i) The sale price is \(\$306\). Check: you pay \(85\%\), and \(0.85 \times 360 = 306\).
(ii) The total is \(\$93.50\). Check: \(1.1 \times 85 = 93.5\).
(ii) Decrease \(\$150\) by \(12\%\).
| \(320 \times 1.06\) | \(=\) | \(339.2\) |
| \(150 \times 0.88\) | \(=\) | \(132\) |
(i) \(\$339.20\)
(ii) \(\$132.00\)
(i) What is the full cost?
(ii) How much does she still have to pay?
| \(42 \div 3\) | \(=\) | \(14\) |
| \(10 \times 14\) | \(=\) | \(140\) |
| \(140 - 42\) | \(=\) | \(98\) |
(i) \(\$140\) (see the bar model in Diagrams)
(ii) \(\$98\). Check: \(70\%\) is left, and \(7 \times 14 = 98\).
(i) Find the selling price.
(ii) Find the sale price.
(iii) Find the profit now.
| \(14 + 14 + 14 + 7\) | \(=\) | \(49\) |
| \(140 + 49\) | \(=\) | \(189\) |
| \(2 \times 18.9\) | \(=\) | \(37.8\) |
| \(189 - 37.8\) | \(=\) | \(151.2\) |
| \(151.2 - 140\) | \(=\) | \(11.2\) |
(i) \(\$189.00\)
(ii) \(\$151.20\) (the \(20\%\) is taken from the new price, \(\$189\))
(iii) \(\$11.20\), not \(15\%\) of \(\$140\)
Common pitfalls
Frequently asked questions
How do you find a hard percentage of an amount?
Build it from \(10\%\), \(5\%\) and \(1\%\). \(26\%\) of \(450\) is \(45 + 45 + 22.5 + 4.5 = 117\).
How do you increase an amount by a percentage?
Find the increase and add it on, or multiply by \(1\) plus the percentage as a decimal: up \(6\%\) means multiply by \(1.06\).
How do you decrease an amount by a percentage?
Find the decrease and take it away, or multiply by what is left: down \(12\%\) means multiply by \(0.88\).
How do you find the whole from a percentage?
If \(30\%\) is \(\$42\), then \(10\%\) is \(\$14\) and \(100\%\) is \(\$140\).
Is profit a percentage of the cost price or the selling price?
Of the cost price, unless the question says otherwise.