Percentages in your head
Tonight's idea
"Percent" means "per hundred": a fraction with 100 on the bottom. Two building blocks do the work: 10% (move the decimal point one place) and 1% (move it two). 15% of 80 is 10% plus 5%: \(8 + 4 = 12\). 23% of 300 is \(2 \times 30 + 3 \times 3 = 69\).
A useful swap: \(a\%\) of \(b\) equals \(b\%\) of \(a\). So 8% of 25 is the same as 25% of 8, which is 2. Both equal \(\tfrac{a \times b}{100}\), and multiplication does not care about order, so swap whenever one side is easier.
Percentages are always of something, and that something changes. Go up 25% from 80 and you reach 100. Go down 25% from 100 and you reach 75, not 80. Any rise of \(x\%\) followed by a fall of \(x\%\) leaves you below where you started.
So treat a percentage change as a multiplier. Up 25% means multiply by 1.25. Down 25% means multiply by 0.75. Reversing a change means dividing by its multiplier.
Watch one
A jacket is 30% off, and at the till there is an extra 10% off the sale price. What is the total discount?
Multipliers: \(0.7 \times 0.9 = 0.63\). You pay 63%, so the total discount is 37%, not 40%. The second 10% was taken off an already-reduced price.
Reverse: after a 20% discount a game costs $48. Original price? The discount multiplier is 0.8, so divide: \(48 \div 0.8 = 60\). Check: 20% of 60 is 12, and \(60 - 12 = 48\).
Your turn
- In your head, using 10% and 1% blocks: 15% of 60, 35% of 80, 2.5% of 400, 12% of 50.
- Use the swap trick: 4% of 75, 16% of 25, 18% of 50, 36% of 25.
- A price goes up 25% and then down 25%. Start at $80. Where do you end up? Now start at $200. In each case, what percentage of the starting price has been lost overall?
- A phone costs $540 after a 10% price rise. What did it cost before? (Dividing by 1.1 is the move. Do not subtract 10%.)
- A town's population of 200 grows by 50%, then shrinks by 50%. What is it now? Write the chain as a single multiplier.
- Find the single percentage change equivalent to "up 10%, up 10%, up 10%". Is it more or less than 30%? Why?
Stretch: the rule of 70
Money growing at 7% a year does not take \(100 \div 7 \approx 14\) years to double. It takes about 10, because each year's 7% is taken on a bigger pile. The rule: doubling time ≈ 70 ÷ (percentage rate). At 7% that gives 10 years; at 10% about 7 years; at 5% about 14 years.
Check the rule for 10% by multiplying by 1.1 seven times (a calculator is allowed). Then: if $100 grows at 7% a year for 30 years, roughly how many doublings is that, and roughly how much money? Why is a loan at 20% so dangerous?
Answers
- 9 (6 + 3), 28 (24 + 4), 10 (8 + 2), 6 (5 + 1).
- 75% of 4 = 3. 25% of 16 = 4. 50% of 18 = 9. 25% of 36 = 9.
- $80 → $100 → $75. $200 → $250 → $187.50. Each time you lose \(\tfrac{1}{16}\), which is 6.25%, because \(1.25 \times 0.75 = 0.9375\).
- \(540 \div 1.1 = 490.909\ldots\), so about $490.91. Subtracting 10% of 540 would wrongly give $486.
- \(200 \times 1.5 \times 0.5 = 200 \times 0.75 = 150\).
- \(1.1^3 = 1.331\), a 33.1% rise. More than 30%, because the second and third 10% are taken on bigger amounts.
Stretch. \(1.1^7 \approx 1.95\), so seven years very nearly doubles the money. Thirty years at 7% is about three doublings: \(100 \to 200 \to 400 \to 800\). (Exactly, \(1.07^{30} \approx 7.6\), so about $760.) At 20% the doubling time is about 3.5 years, so a debt left alone for ten years grows roughly \(2^3 = 8\) times. (Exact: \(1.2^{10} \approx 6.2\); the rule is an approximation that gets rougher for big rates.)