The change, out of what?
A sunflower was 40 cm tall, and a week later it is 50 cm tall. It grew by 50 − 40 = 10 cm. To give that change as a percent, the 10 has to be a share of something. But 10 out of what?
Each part is 10 cm. The sunflower was 4 parts tall, and now it is 5 parts tall.
Out of where you started
A percentage change is always measured against the amount you started with, called the original. The sunflower started at 40 cm, so the change is 10 out of 40. 40 splits into four 10s, so 10 out of 40 is .
The original 40 is cut into four 10s, and the change is one of them: a quarter.
Write it out of 100
A quarter of 100 is 25, so . The sunflower grew by 25%.
In one line: divide the change by the original, then multiply by 100%. Here 10 ÷ 40 = 0.25, and 0.25 × 100% = 25%.
A quarter of the hundred square is 25 squares: 10 out of 40 is 25 out of 100.
The same gap, the other way
Now go the other way. A toy costs $50, and its price falls to $40. The change is 10 again, but this time the original is 50. 10 out of 50 is , and , so the price fell by 20%.
So a change of 10 is 25% of 40 and only 20% of 50. Whenever a percent looks wrong, ask: a percent of what? That is also why dividing by the new amount is a mistake. For 40 to 50, 10 ÷ 50 gives 20%, which measures the rise against the wrong amount.
Up and down again
Because each percent is a share of a different amount, going up by a percent and then down by the same percent does not bring you back. 100 up by 10% is 110. 10% of 110 is 11, so 110 down by 10% is 99, not 100.
up P% takes 100 to 120; down P% takes 120 to 96 < 100, because the second P% is a share of the bigger bar
Set P = 50 and read where the bar lands
The first bar is 100. Up 20% makes it 120, and down 20% takes 20% of 120 away, which lands on 96. Set P, the percent, to 50: up 50% gives 150, and down 50% takes half of 150, leaving 75.
Percentage points
A rate is already a percent, and a change in a rate can be given in two ways. At a school, 20% of the pupils walked to school last year and 25% walk this year.
The gap between the two rates is 25 − 20 = 5 percentage points. Measured against the 20% it started at, the rise is 5 out of 20, which is , or 25%. So the rate went up by 5 percentage points, and it also went up by 25%. Percentage points give the gap; the percent gives the gap as a share of where the rate started.
The 20 gold squares are last year's 20%. The 5 fainter squares are the 5 percentage points added, and 5 is a quarter of the 20.
Worked example: Profit and Loss on Cost vs Discount on Marked Price
Question A merchant bought an electric scooter for a certain cost price. He marked up the cost price by 40% to set the display marked price. During a festival sale, he offered a 15% discount on the marked price. If he sold the scooter at this discounted price and made a profit of $114: (a) What was the cost price of the electric scooter? (b) What was the marked price of the scooter?
1.Cost price = 100 units.
The cost price is 100 units. 2.Marked price = 100 + 40 = 140 units.
Marked up 40%: 140 units. 3.Calculate 15% discount on 140 units: 0.15 × 140 = 21 units.
15% of 140 units is 21 units. 4.Sale price = 140 − 21 = 119 units.
Sale price: 140 − 21 = 119 units. 5.Profit = 119 units − 100 units = 19 units.
Profit: 119 − 100 = 19 units. 6.19 units = $114 ⟹ 1 unit = 114 ÷ 19 = $6.
19 units is $114, so 1 unit is $6. 7.(a) Cost price: 100 × 6 = $600.
(a) Cost: $600. 8.(b) Marked price: 140 × 6 = $840.
(b) Marked price: 140 units, $840.
Answer: (a) $600; (b) $840
Common mistakes
- Subtracting the 15% discount directly from the 40% markup (40% − 15% = 25%) and assuming the profit is 25% of cost.
- Calculating the 15% discount on the cost price rather than on the marked price.