Per cent means out of 100. Turn every percentage into a multiplier and percentage increases, decreases, profit, discount and even reverse percentages become one multiplication or division.
Use a multiplier: an increase of \(r\%\) multiplies by \(1 + \dfrac{r}{100}\), a decrease of \(r\%\) by \(1 - \dfrac{r}{100}\). To go back to the original (reverse percentage), DIVIDE by the multiplier.
Fig. 1: a double number line for a reverse percentage. The sale price $56 matches 80% of the original price, not 100%
Section 01
Percentages of amounts.
In the lab, choose Percentage of and drag the percentage slider along the double number line: the top shows the percentage, the bottom shows the amount. Then switch to As a percentage and see one amount written out of the other.
Percentage of an amount: \(35\%\) of \(\$240 = 0.35 \times 240 = \$84\).
One quantity as a percentage of another: \(\dfrac{\text{part}}{\text{whole}} \times 100\), with both in the same units.
Over 100%: \(125\%\) of an amount is more than the amount: \(1.25 \times 80 = 100\).
Worked example As a percentage
Express 42 cm as a percentage of 1.2 m.
Same units: \(1.2\) m \(= 120\) cm.
\(\dfrac{42}{120} \times 100 = 35\).
Answer: \(35\%\)
Your turn
Write 45 as a percentage of 36.
\(\dfrac{45}{36} \times 100 = 125\%\). It is over 100% because 45 is more than 36.
Examiner trap
Divide by the WHOLE, not by the bigger number out of habit. 'What percentage of 36 is 45?' has 36 on the bottom, giving 125%, not 80%.
Section 02
Increase, decrease and change.
A multiplier does an increase or decrease in one step. For a percentage change, divide the change by the ORIGINAL amount.
Change
Multiplier
increase by 15%
1.15
decrease by 12%
0.88
increase by 4.5%
1.045
decrease by 3%
0.97
Worked example Increase and decrease
(a) Increase \(\$80\) by 15%. (b) Decrease 450 kg by 12%.
Total paid: \(180 + 792 = \$972\), so the extra is \(\$72\).
\(\dfrac{72}{900} \times 100 = 8\).
Answer: \(8\%\) more
Examiner trap
Percentage change uses the ORIGINAL value on the bottom. From \(\$64\) to \(\$52\) is \(\dfrac{12}{64}\), not \(\dfrac{12}{52}\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Find 15% of 60.
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Which multiplier decreases an amount by 6%?
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What is 12 as a percentage of 48?
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Increase 200 by 35%.
Section 03
Reverse and repeated percentages.
(Extended) When you know the amount AFTER a change, the new amount is a percentage of the original. Divide by the multiplier to undo the change (Fig. 1). For repeated changes, multiply the multipliers.
Worked example Reverse percentage (Extended)
After a 20% discount a coat costs \(\$56\). Find the original price.
The sale price is \(100\% - 20\% = 80\%\) of the original.
So original \(\times 0.8 = 56\).
Original \(= 56 \div 0.8 = 70\).
Answer: \(\$70\)
Worked example Cost price from selling price (Extended)
A shop sells a sofa for \(\$1495\), making a profit of 15%. Find the cost price.
Selling price \(= 115\%\) of the cost price, multiplier 1.15.
Cost price \(= 1495 \div 1.15 = 1300\).
Answer: \(\$1300\)
Worked example Repeated change (Extended)
A price rises by 10% and then falls by 10%. What is the overall percentage change?
Overall multiplier \(= 1.1 \times 0.9 = 0.99\).
\(0.99\) means \(99\%\) of the original, a fall of \(1\%\).
Answer: A \(1\%\) decrease (not zero)
Examiner trap
For a reverse percentage, do NOT take 20% of the sale price and add it on: \(56 + 0.2 \times 56 = 67.20\), not 70. The 20% was of the ORIGINAL price, so divide by 0.8.
Stage 2 · Exam
Exam-style questions
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After a 15% increase, the price of a ticket is \(\$69\). What was the price before the increase?
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A trader buys a lamp for \(\$40\) and sells it for \(\$46\). Find the percentage profit.
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A town's population increases by 20% one year and decreases by 20% the next. What is the overall change?
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A shirt marked \(\$48\) is reduced by 25% in a sale. A further 10% is then taken off the sale price. What is the final price?
Skill unlocked
Percentages, sorted.
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