To undo a percent change, divide by the multiplier that made it
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.
20% more than 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.
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.
Percentages · A Camera Marked Up, Then Sold at a Discount: Working Back to the Cost
QuestionA shop bought a camera and marked up its cost price by 25% to set the marked price. In a sale, the shop took 16% off the marked price, and a customer bought the camera for $840. (a) What was the marked price of the camera? (b) How much profit did the shop make on the camera?
1.The 16% is off the marked price, so draw the marked price as 100 parts. The sale took 16 parts, and the customer paid the other 84 parts, so 84 parts are $840.
The 16% is off the marked price, so the marked price is 100 parts. The customer paid the other 84 parts: $840. 2.1 part is 840 ÷ 84 = $10.
1 part is 840 ÷ 84 = $10. 3.(a) The marked price is 100 parts: 100 × $10 = $1000.
(a) The marked price is 100 parts: 100 × $10 = $1000. 4.The 25% markup is of the cost price, so draw the cost as 100 units. The marked price is 100 + 25 = 125 units, and 125 units are $1000, so 1 unit is 1000 ÷ 125 = $8 and the cost is 100 × $8 = $800.
The cost is 100 units and the marked price 100 + 25 = 125 units, so 1 unit is 1000 ÷ 125 = $8 and the cost is $800. 5.(b) The profit is the price the camera sold for less its cost: 840 − 800 = $40. Check: 1.25 × 800 = 1000, and 0.84 × 1000 = 840.
(b) The profit is the sale price less the cost: 840 − 800 = $40.
Answer: (a) $1000; (b) $40
Watch out for
- Adding 16% of $840 back on to get the marked price: 840 + 134.40 = $974.40. The 16% is a percentage of the marked price, not of the sale price, so $840 is 84% of the marked price, and the marked price is 840 ÷ 0.84 = $1000.
- Taking the profit as the marked price less the cost, 1000 − 800 = $200. The shop was paid the sale price, $840, not the marked price, so the profit is 840 − 800 = $40.
Percentages · A School's Enrollment Rise, Worked Back and Carried a Year On
QuestionAfter 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%
Watch out for
- 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.
Your turn
1.After a 25% increase the price is $75. What was it before?