Proportional Reasoning with Percents
A percent is a rate out of 100. Turn it into a multiplier and every percent problem becomes one multiplication.
A jacket costs $80. The sign says 30% off. What do you pay?
Most people find 30% of 80, get 24, and then subtract. That works — but there is a faster way to see it.
Think of the full price as 100 little squares. Thirty of them are taken away.
So you are paying for the 70 that remain — whatever the price happens to be.
70% of 80 is 56, in one step rather than two.
That single number, 0.70, is called a multiplier, and it is what makes percent problems stack. Two successive 10% rises are 1.10 × 1.10 = 1.21 — a 21% rise, not 20%.
Percent means "out of 100"
The word comes from per centum — for each hundred. So 45% means 45 out of every 100.
That gives you three forms of the same thing, and you should be able to slide between them instantly: 45% = 45100 = 0.45.
To go from percent to decimal, move the point two places left. To go back, move it two places right.
The multiplier trick
Most students compute the change and then add or subtract. That works, but it is two steps and two chances to slip.
Instead, fold it into one number. A 30% discount leaves 70% of the price, so multiply by 0.70. An 8% tax gives 108% of the price, so multiply by 1.08.
Multipliers
Increase by p%: multiply by 1 + p100.
Decrease by p%: multiply by 1 - p100.
Why multipliers matter: changes stack
A store raises a price 50%, then puts it 50% off. Back to where it started? No.
Multipliers: 1.50 × 0.50 = 0.75. The final price is 75% of the original — a 25% drop.
The reason is that the two changes are measured from different bases. The increase is on the small original; the decrease is on the bigger new number, so it takes away more.
Percent changes never simply cancel
Up 50% then down 50% is not a wash. Multiply the multipliers and see what you actually get.
Percent change
To find how much something changed in percent terms: new - oldold, then convert to a percent.
The denominator is always the original amount. Going from 40 to 50 is a 25% increase (1040). Going from 50 back to 40 is a 20% decrease (1050). Same 10, different percents, because the starting point changed.
Finding the original
If a jacket is $63 after a 30% discount, the $63 is 70% of the original. So original × 0.70 = 63, which means original = 63 ÷ 0.70 = $90.
When you know the result and want the start, divide by the multiplier instead of multiplying.
Worked examples
Tax then tip
A meal costs $40. Tax is 8%, and a 20% tip is added to the after-tax total. What is the final cost?
- Tax multiplier: 1.08 You pay 100% plus 8% more.
- 40 × 1.08 = 43.20 The after-tax total.
- Tip multiplier: 1.20 The tip is on the after-tax amount, as stated.
- 43.20 × 1.20 = 51.84 Or do it in one shot: 40 × 1.08 × 1.20.
$51.84
Find the original price
After a 35% discount, a bike costs $286. What was the original price?
- The sale price is 65% of the original. 100% - 35% = 65%.
- So original × 0.65 = 286 Write the sentence as an equation.
- 286 ÷ 0.65 = 440 Undo the multiplication.
- Check: 440 × 0.65 = 286 ✓ Always check by running it forward.
$440
Successive changes
A population grows 20% one year and falls 20% the next. What is the overall percent change?
- Multipliers: 1.20 and 0.80 Up 20%, then down 20%.
- 1.20 × 0.80 = 0.96 Chained changes multiply.
- 0.96 means 96% of the start. That is 4% less than 100%.
A 4% decrease overall.
Words to know
- Percent
- Parts per hundred. 37% = 37100 = 0.37.
- Multiplier
- The single number you multiply by. A 20% increase has multiplier 1.20.
- Percent change
- new - oldold × 100% — always divided by the original.
- Markup / discount
- An increase / a decrease applied to a starting price.
Mistakes to watch for
Adding percent changes instead of multiplying multipliers. Up 10% then up 10% is 1.10 × 1.10 = 1.21, a 21% rise — because the second rise is taken on the already larger amount.
Using the new amount as the denominator in percent change. Always divide by the original: going from 40 to 50 is a 25% rise, not 20%, because the change is measured against where you started.
Multiplying when you should divide. If you are given the sale price and want the original, you must undo the multiplier: $63 after 30% off is 63 ÷ 0.70 = $90.
Thinking a 20% fall undoes a 20% rise. $100 up 20% is $120; down 20% from there is $96. The two percentages are taken on different amounts.
The word percent is from the Latin per centum, "by the hundred." Roman emperors taxed goods at rates like 1100, so hundredths were already the natural unit.
The % sign evolved from a scribal shorthand: Italian clerks wrote per cento as "per 100", then "p 100", and the 100 slowly collapsed into %.
You can see the two zeros still sitting in it, one either side of the slash.