The Unitary Method
The unitary method finds one unit first, then any amount you need. In NSW Year 8 maths (NESA outcome MA4‑FRC‑C‑01) it finds a whole from a known percentage: if 8% of an amount is $56, then 1% is $7 and the whole is $700.
You will find an original price from a sale price, last year’s pay from this year’s after a rise, a car’s price before a loss and the amount invested from one year’s interest. The same idea works with fractions (\(\frac{3}{8}\) of a tank is 27 L) and with rates, such as the charging time left on a phone. You will also compare it with dividing by a multiplier and handle two discounts in a row. It is part of Year 8 Fractions and Percentages.
Theory
The unitary method finds the value of one unit first (one per cent, or one equal part) and then multiplies up. If \(55\%\) of a crowd is \(20\,119\), then \(1\%\) is \(20\,119 \div 55 = 365.8\) and the whole crowd is \(36\,580\). This Year 8 maths page shows how.
The four steps. Write what you know, find \(1\%\) by dividing by the percentage, find \(100\%\) by multiplying by \(100\), then check by working forwards.
Any percentage, not only \(100\%\). Once \(1\%\) is known, multiply it by the percentage the question asks for.
Fractions work the same way. If \(\dfrac{3}{7}\) of an amount is known, divide by \(3\) for \(\dfrac{1}{7}\), then multiply by \(7\).
Rates too. If \(30\%\) of a job takes \(12\) minutes, then \(1\%\) takes \(0.4\) minutes and the whole job takes \(40\) minutes.
After a change, name the percentage you actually know. A discount or a loss of \(p\%\) leaves \((100 - p)\%\); a rise or a profit of \(p\%\) gives \((100 + p)\%\).
One step instead. Dividing by the multiplier does the same job: after \(20\%\) off, divide by \(0.8\); after an \(8\%\) rise, divide by \(1.08\).
Two changes in a row. Combine them into one percentage first: \(20\%\) off then \(10\%\) off leaves \(0.8 \times 0.9 = 0.72\), so \(72\%\) of the original.
Name the percentage you know:
| Situation | The known amount is | Divide by |
|---|---|---|
| \(18\%\) of a hall is full | \(18\%\) | \(18\) (for \(1\%\)) |
| after \(30\%\) off | \(70\%\) of the original | \(0.7\) |
| after a \(15\%\) rise | \(115\%\) of the original | \(1.15\) |
| after a \(20\%\) loss | \(80\%\) of the cost price | \(0.8\) |
| \(\dfrac{5}{9}\) of a book | \(5\) ninths | \(5\) (for one ninth) |
The unitary method
- Write what you know: \(p\%\) of the whole \(=\) amount.
- Find \(1\%\): divide the amount by \(p\).
- Multiply by \(100\) for the whole, or by any other percentage asked for.
- Check by working forwards.
After a discount or a rise
- Name the percentage you know: \((100 - p)\%\) after a discount, \((100 + p)\%\) after a rise.
- Find \(1\%\), then \(100\%\); or divide by the multiplier.
After \(30\%\) off, the \(\$63\) is \(70\%\) of the original price.
| \(\text{70\%}\) | \(=\) | \(\$63\) |
| \(\text{1\%}\) | \(=\) | \(63 \div 70 = 0.90\) |
| \(\text{100\%}\) | \(=\) | \(0.90 \times 100 = 90\) |
The original price was \(\$90\). Check: \(0.7 \times 90 = 63\).
(ii) A school hall is \(18\%\) full; that is \(63\) people. How many seats does the hall have?
| \(\text{1\%}\) | \(=\) | \(20\,119 \div 55 = 365.8\) |
| \(\text{100\%}\) | \(=\) | \(365.8 \times 100 = 36\,580\) |
| \(\text{1\%}\) | \(=\) | \(63 \div 18 = 3.5\) |
| \(\text{100\%}\) | \(=\) | \(3.5 \times 100 = 350\) |
(i) \(36\,580\) people
(ii) \(350\) seats. Check: \(0.18 \times 350 = 63\).
| \(\text{1\%}\) | \(=\) | \(168 \div 24 = 7\) |
| \(\text{45\%}\) | \(=\) | \(7 \times 45 = 315\) |
\(315\) pear trees. Check: the orchard has \(700\) trees and \(0.45 \times 700 = 315\).
(ii) \(30\%\) of a download takes \(12\) minutes. How long does the whole download take?
| \(\text{one ninth}\) | \(=\) | \(240 \div 5 = 48\) |
| \(\text{the whole}\) | \(=\) | \(48 \times 9 = 432\) |
| \(\text{1\%}\) | \(=\) | \(12 \div 30 = 0.4~\text{min}\) |
| \(\text{100\%}\) | \(=\) | \(0.4 \times 100 = 40~\text{min}\) |
(i) \(432\) pages
(ii) \(40\) minutes
(ii) After an \(8\%\) pay rise, Sam earns \(\$1080\) a fortnight. What did he earn before?
| \(\text{1\%}\) | \(=\) | \(91 \div 65 = 1.4\) |
| \(\text{100\%}\) | \(=\) | \(1.4 \times 100 = 140\) |
| \(140 - 91\) | \(=\) | \(49\) |
| \(1080 \div 1.08\) | \(=\) | \(1000\) |
(i) \(\$140\), saving \(\$49\). In one step: \(91 \div 0.65 = 140\).
(ii) \(\$1000\)
| \(0.8 \times 0.9\) | \(=\) | \(0.72\) |
| \(54 \div 0.72\) | \(=\) | \(75\) |
The final price is \(72\%\) of the original, so the original price was \(\$75\). Check: \(75 \times 0.8 = 60\), then \(60 \times 0.9 = 54\).
Common pitfalls
Frequently asked questions
What is the unitary method?
A method that finds the value of one unit (such as 1%) first, then multiplies to find any other amount.
How do I find the whole amount from a percentage?
Divide the amount by the percentage to get \(1\%\), then multiply by \(100\).
How do I find the original price after a discount?
Write the sale price as \((100 - p)\%\) of the original, find \(1\%\), then \(100\%\). Or divide by the multiplier.
Can I use the unitary method with fractions?
Yes. If \(\dfrac{2}{5}\) is \(30\), then \(\dfrac{1}{5}\) is \(15\) and the whole is \(75\).
How do I check my answer?
Work forwards: take the percentage of your answer and see whether you get the amount you were given.