Percentages and Money
Every question with a fully worked solution.
- Percentages and Money - Video - Solving percentage problems with the unitary method and equations Watch
Theory
A percentage of a quantity is a multiplication, and percentages turn up in money all the time: wages, prices, discounts and GST. This Year 9 maths page revises the Year 8 percentage work in financial contexts: finding a percentage of an amount, writing one amount as a percentage of another, adding GST, and working backwards to an original price.
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.
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 \(25\%\) | \(\times 0.75\) | \(\div 0.75\) |
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 \(1\%\).
(ii) Find \(4\dfrac{1}{2}\%\) of \(\$600\).
(iii) Find \(40\%\) of \(1.5\) L, in millilitres.
(i) “Of” means multiply. Write the percentage as a decimal:
| \(8\% \text{ of } 450\) | \(=\) | \(0.08 \times 450\) |
| \(\) | \(=\) | \(\mathbf{36}\ \textbf{m}\) |
Check with the unitary method: \(1\%\) of \(450\) is \(4.5\), and \(8 \times 4.5 = 36\).
(ii) Write the mixed-number percentage as a decimal first: \(4\dfrac{1}{2}\% = 4.5\% = 0.045\).
| \(0.045 \times 600\) | \(=\) | \(\mathbf{\$27}\) |
(iii) Change to millilitres first: \(1.5\) L \(= 1500\) mL.
| \(0.4 \times 1500\) | \(=\) | \(\mathbf{600}\ \textbf{mL}\) |
(ii) Jo earns \(\$720\) a week and spends \(\$95\) of it on transport. What percentage is that, correct to \(2\) decimal places?
(iii) A \(\$90\) shirt is on sale for \(\$72\). Find the percentage discount.
(i) Same units first: \(1\) hour \(= 60\) minutes. Part over whole, times \(100\%\):
| \(\dfrac{18}{60} \times 100\%\) | \(=\) | \(\mathbf{30\%}\) |
(ii) The part is \(\$95\) and the whole is \(\$720\):
| \(\dfrac{95}{720} \times 100\%\) | \(=\) | \(13.194\ldots\%\) |
| \(\) | \(\approx\) | \(\mathbf{13.19\%}\) |
(iii) Find the discount first, then compare it with the original price:
| \(\text{discount}\) | \(=\) | \(90 - 72 = \$18\) |
| \(\dfrac{18}{90} \times 100\%\) | \(=\) | \(\mathbf{20\%}\) |
(ii) A shop buys a jacket for \(\$50\) and marks it up by \(40\%\). Later the marked price is cut by \(15\%\). Find the new price.
(i) The GST is \(10\%\) of the price before tax. Add it on:
| \(\text{GST}\) | \(=\) | \(0.1 \times 480 = \mathbf{\$48}\) |
| \(\text{total}\) | \(=\) | \(480 + 48 = \mathbf{\$528}\) |
Check with one multiplier: \(480 \times 1.1 = 528\).
(ii) Use one multiplier for each change. A mark-up of \(40\%\) is \(\times 1.4\); a cut of \(15\%\) leaves \(85\%\), so \(\times 0.85\):
| \(50 \times 1.4\) | \(=\) | \(70\) |
| \(70 \times 0.85\) | \(=\) | \(\mathbf{\$59.50}\) |
The \(15\%\) is taken off the marked price of \(\$70\), not off the \(\$50\).
(ii) A \(15\%\) discount saves \(\$24\). What was the full price?
(i) The \(\$84\) is the old cost multiplied by \(1.2\), so divide to undo it:
| \(\text{original}\) | \(=\) | \(84 \div 1.2\) |
| \(\) | \(=\) | \(\mathbf{\$70}\) |
Unitary method: \(\$84\) is \(120\%\), so \(1\%\) is \(84 \div 120 = 0.7\), and \(100\%\) is \(\$70\).
(ii) The saving is \(15\%\) of the full price. So \(15\%\) is worth \(\$24\):
| \(1\%\) | \(=\) | \(24 \div 15 = 1.6\) |
| \(100\%\) | \(=\) | \(100 \times 1.6\) |
| \(\) | \(=\) | \(\mathbf{\$160}\) |
Check: \(15\%\) of \(\$160\) is \(\$24\).
Common pitfalls
Frequently asked questions
How do you find a percentage of a number?
Change the percentage to a decimal and multiply. For example, \(8\%\) of \(450\) is \(0.08 \times 450 = 36\).
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 \(20\%\) increase is \(\$84\), the original is \(84 \div 1.2 = \$70\).
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 add GST to a price?
GST in Australia is \(10\%\), so multiply the price by \(1.1\). A \(\$480\) charge plus GST is \(480 \times 1.1 = \$528\), which includes \(\$48\) of GST.