Percentage Increase and Decrease
Every question with a fully worked solution.
- Percentage Increase and Decrease - Video - Percentage Increase and Decrease (Multiplier Method) Watch
Theory
A percentage increase or decrease can be done in one step with a multiplier. This Year 9 maths page revises the Year 8 work: increasing and decreasing by a percentage, finding the percentage change between two amounts, applying changes one after another, and working backwards to the original amount.
To find a percentage of a quantity, write the percentage as a decimal (or a fraction over \(100\)) and multiply. "Of" means multiply: \(15\%\) of \(80\) is \(0.15 \times 80 = 12\).
In an increase, the new amount is the whole \(100\%\) plus the extra. Up \(12\%\) makes \(112\%\), so multiply by \(1.12\).
In a decrease, the new amount is what is left: \(100\%\) minus the cut. Down \(40\%\) leaves \(60\%\), so multiply by \(0.6\).
The unitary method finds what \(1\%\) is worth first, by dividing, then multiplies up to the percentage you want. It is handy for checking, and for working backwards.
Working backwards: if you know the amount after a change and want the original, divide by the multiplier instead of multiplying.
The percentage change compares the change with the original amount: change \(\div\) original \(\times 100\%\).
Percentage of a quantity, increase and decrease:
Going forwards multiplies; going backwards divides:
| Change | Multiplier | To undo it |
|---|---|---|
| up \(5\%\) | \(\times 1.05\) | \(\div 1.05\) |
| up \(20\%\) | \(\times 1.2\) | \(\div 1.2\) |
| down \(10\%\) | \(\times 0.9\) | \(\div 0.9\) |
| down \(25\%\) | \(\times 0.75\) | \(\div 0.75\) |
Percentage change:
How to solve a percentage problem
- Decide what is asked: a part, a new amount after a change, or the original amount.
- Part: multiply the amount by the percentage as a decimal.
- New amount: multiply by \(1 +\) the rate for an increase, or \(1 -\) the rate for a decrease.
- Original amount: divide the new amount by the multiplier, or use the unitary method.
- Check by going the other way, or with \(1\%\).
How to find a percentage change
- Subtract to find the change: new minus original.
- Divide by the original amount, then multiply by \(100\%\).
- Say whether it is an increase or a decrease.
(i) \(\$60\) by \(12\%\)
(ii) \(80\) kg by \(150\%\).
(i) The new amount is \(100\% + 12\% = 112\%\) of the old one, so the multiplier is \(1.12\):
| \(60 \times 1.12\) | \(=\) | \(\mathbf{\$67.20}\) |
Or in two steps: \(12\%\) of \(\$60\) is \(0.12 \times 60 = \$7.20\), and \(\$60 + \$7.20 = \$67.20\). Both ways give the same answer.
(ii) The new amount is \(100\% + 150\% = 250\%\) of the old one, so the multiplier is \(2.5\):
| \(80 \times 2.5\) | \(=\) | \(\mathbf{200}\ \textbf{kg}\) |
An increase of more than \(100\%\) more than doubles the amount.
(i) \(70\) kg by \(40\%\)
(ii) \(3\) tonnes by \(18\%\). Give the answer in kilograms.
(i) What is left is \(100\% - 40\% = 60\%\), so the multiplier is \(0.6\):
| \(70 \times 0.6\) | \(=\) | \(\mathbf{42}\ \textbf{kg}\) |
Multiplying by \(0.4\) instead gives \(28\) kg. That is the amount taken off, not the amount left. Check: \(42 + 28 = 70\).
(ii) Change to kilograms first: \(3\) t \(= 3000\) kg. What is left is \(100\% - 18\% = 82\%\):
| \(3000 \times 0.82\) | \(=\) | \(\mathbf{2460}\ \textbf{kg}\) |
(ii) A town's population falls from \(8400\) to \(7980\). Find the percentage decrease.
(i) Find the change, then divide by the original price:
| \(\text{increase}\) | \(=\) | \(2.80 - 2.50 = \$0.30\) |
| \(\dfrac{0.30}{2.50} \times 100\%\) | \(=\) | \(\mathbf{12\%}\) |
(ii) The change is a fall, so it is a percentage decrease:
| \(\text{decrease}\) | \(=\) | \(8400 - 7980 = 420\) |
| \(\dfrac{420}{8400} \times 100\%\) | \(=\) | \(\mathbf{5\%}\) |
Dividing by the new amount instead would give the wrong answer.
(ii) After a \(15\%\) decrease, a number is \(306\). What was the original number?
(i) Use one multiplier for each change. The second \(10\%\) comes off the reduced price:
| \(400 \times 0.8\) | \(=\) | \(320\) |
| \(320 \times 0.9\) | \(=\) | \(\mathbf{\$288}\) |
That is \(\$112\) off, which is \(28\%\) of \(\$400\), not \(30\%\).
(ii) The \(306\) is the original times \(0.85\), so divide to undo it:
| \(\text{original}\) | \(=\) | \(306 \div 0.85\) |
| \(\) | \(=\) | \(\mathbf{360}\) |
Check: \(360 \times 0.85 = 306\).
Common pitfalls
Frequently asked questions
How do you increase an amount by a percentage?
Multiply by \(1\) plus the percentage as a decimal. To increase \(\$60\) by \(12\%\), multiply by \(1.12\) to get \(\$67.20\).
How do you decrease an amount by a percentage?
Multiply by \(1\) minus the percentage as a decimal. To decrease \(70\) kg by \(40\%\), multiply by \(0.6\) to get \(42\) kg.
How do you find the original price before a percentage change?
Divide the new price by the multiplier. If a price after a \(25\%\) increase is \(\$75\), the original is \(75 \div 1.25 = \$60\).
What is the unitary method for percentages?
Find what \(1\%\) is worth by dividing, then multiply to get the percentage you need. If \(15\%\) is \(\$24\), \(1\%\) is \(\$1.60\), so \(100\%\) is \(\$160\).
How do you calculate a percentage increase?
Subtract to find the increase, divide by the original amount and multiply by \(100\%\). From \(\$2.50\) to \(\$2.80\) is an increase of \(\$0.30\), and \(\dfrac{0.30}{2.50} \times 100\% = 12\%\).
Is 20 per cent off then 10 per cent off the same as 30 per cent off?
No. The second discount is taken off the already reduced price: \(0.8 \times 0.9 = 0.72\), so the total saving is \(28\%\), not \(30\%\).