Working back to the old price
A price went up by 20%, and now it is $60. What was it before the rise? The old price is the whole, 100%. The rise added 20% of the old price, so $60 is 100% + 20% = 120% of the old price.
The old price is 10 parts, 100%. The new price is 12 of the same parts, 120%, and it is $60.
The rise was a multiplication
120% is a multiplier of 1.2, so the rise did this: old price × 1.2 = 60. Try an old price of $40: 40 × 1.2 = 48, which is short of 60. Try $50: 50 × 1.2 = 60 exactly. The old price was $50.
40 × 1.2 = 48, short of 60, so the old price is more than 40
Drag the old price until 20% more is $60
The top bar is the old price, and the bar under it is 120% of it. Drag the old price until the lower bar ends on the dashed line at $60. Only $50 lands on it.
Undo it by dividing
Trying prices works, but dividing is quicker. Dividing undoes multiplying, so undo the × 1.2 with ÷ 1.2: 60 ÷ 1.2 = 50. The old price was $50.
The bar gives the same answer. $60 is 12 parts, so one part is 60 ÷ 12 = 5 dollars, and the old price is 10 parts: 10 × 5 = 50.
Check by going forward: 20% of 50 is 10, and 50 + 10 = 60.
Each part is $5. The old price is 10 parts, $50, and the new price is 12 parts, $60.
Why taking 20% off does not work
The tempting route is to take 20% off the $60: 20% of 60 is 12, and 60 − 12 = 48. But the rise was 20% of the old price, $50, which is $10. 20% of $60 is a bigger amount, $12, so taking it away goes back too far.
Check 48 going forward: 48 × 1.2 = 57.60, not 60.
Each part is $2. The new price, $60, is 30 parts, and the old price, $50, is 25 parts. Taking $12 off $60 removes 6 parts and leaves 24, which is $48, one part short of the old price.
After a discount
A fall is undone the same way, with a multiplier less than 1. A coat costs $60 after 20% off. The sale price is 80% of the original price, and 80% is × 0.8, so the original was 60 ÷ 0.8 = $75.
Adding 20% of 60 back gives 60 + 12 = 72, which is wrong for the same reason: the 20% was a share of $75, not of $60.
60 is 75% of the original, so the original is 60 ÷ 0.75 = 80; adding the percentage back gives 75, which is wrong because the discount was a share of the original, not of 60
Set A = 60 after 20% off and read the original
It opens at a price of 60 after 25% off, which is 75% of the original: 60 ÷ 0.75 = 80. Set the discount to 20%. The original becomes 60 ÷ 0.8 = 75, and adding 20% of 60 back lands short of it, at 72.
A price with the tax in it
A bill of $108 includes a sales tax of 8%. The tax was added on top of the price, so the bill is 108% of the price before tax, and 108% is × 1.08. The price before tax was 108 ÷ 1.08 = $100.
Worked example: Reverse Percentage with Shifting Markup and Markdown
Question A camera retailer marked up the cost price of a camera by 25% to set the display marked price. During a clearance sale, the retailer offered a 16% discount on the marked price. A customer purchased the camera at the sale price for $840. (a) What was the marked price of the camera? (b) How much profit did the retailer make from selling the camera?
1.Represent marked price as 100 parts: after 16% discount, 84 parts = $840.
After 16% off, 84 parts of the marked price remain. 2.1 part = 840 ÷ 84 = $10.
1 part is 840 ÷ 84 = 10. 3.100 parts (Marked Price) = 100 × 10 = $1000.
(a) Marked price: 100 parts, $1000. 4.Represent cost price as 100 units. The marked price is 100 + 25 = 125 units.
The marked price is the cost plus 25%: 125 units. 5.125 units = $1000 ⟹ 1 unit = 1000 ÷ 125 = $8.
The marked price is the cost plus 25%: 125 units. 6.Cost price (100 units) = 100 × 8 = $800.
1 unit is $8, so the cost is $800. 7.Profit = Sale Price − Cost = 840 − 800 = $40.
(b) Profit: $840 − $800 = $40.
Answer: (a) $1000; (b) $40
Common mistakes
- Assuming the marked price can be found by adding 16% to the $840 sale price (840 × 1.16 = $974.40, which incorrectly uses $840 as the base).
- Calculating profit as marked price minus cost price (1000 − 800 = $200) instead of sale price minus cost price (840 − 800 = $40).
Worked example: A School's Enrollment Rise, Worked Back and Carried a Year On
Question After a new wing opened, a school's enrollment rose by 25% from last year to this year, and the school now has 1000 students. The school expects its enrollment to rise next year by the same number of students as it rose this year. (a) How many students did the school have last year? (b) By what percentage of this year's enrollment is the school expected to grow next year?
1.25% = 14, so draw last year's enrollment as 4 units and this year's rise as 1 more unit.
25% is a quarter: last year is 4 units and the rise is 1 unit. 2.This year's enrollment is 4 + 1 = 5 units, and 5 units are 1000 students.
This year: 5 units are 1000 students. 3.1 unit = 1000 ÷ 5 = 200 students.
1 unit is 1000 ÷ 5 = 200 students. 4.(a) Last year: 4 units = 4 × 200 = 800 students.
(a) Last year: 4 × 200 = 800 students. 5.Next year the school expects another rise of 1 unit, or 200 students, on top of this year's 5 units: 1000 + 200 = 1200 students.
Next year adds another 200 students: 1200. 6.(b) The rise is 1 unit out of this year's 5 units: 15 = 20100 = 20%. Check: 2001000 × 100% = 20%.
(b) The rise is 1 unit of this year's 5: 15 = 20%.
Answer: (a) 800 students; (b) 20%
Common mistakes
- Taking 25% off 1000 to get 750 students for last year. The 25% was a percentage of last year's enrollment, not of this year's, so this year's 1000 students are 125% of last year's.
- Answering 25% for next year because the number of new students is the same. The same 200 students are now compared with 1000 students instead of 800, so they make a smaller percentage.