TACHS: Percent change: sale prices, increases and working backward
Learn to find a sale price, a percent increase or decrease, and an original price, using one idea: every percent is taken of a starting amount.
How do you solve percent problems without mixing up the discount, the new price and the original price?
Short answer
Name the starting amount, turn the percent change into a multiplier, and multiply. To find an original price, divide by that same multiplier.
Know first: Multiplying decimals, Common fractions such as 1/4 and 3/4
Step 1 of 7
A percent is a part of a starting amount
Percent means per hundred, so 15% is 15/100, or 0.15. Every percent question has a starting amount, and the percent is taken of that amount. In a sale, the starting amount is the original price. In a price increase, it's the old price.
| Percent | Decimal | Fraction |
|---|---|---|
| 10% | 0.1 | 1/10 |
| 20% | 0.2 | 1/5 |
| 25% | 0.25 | 1/4 |
| 50% | 0.5 | 1/2 |
| 75% | 0.75 | 3/4 |
Knowing these lets you do many percent questions in your head.
Step 2 of 7
Find a sale price
Worked example: A $60 game at 20% off
A video game costs $60. It is on sale for 20% off. What is the sale price?
- A$12
- B$40
- C$48
- D$72
- 1
Name the starting amount
The original price, $60.
- 2
Part route
20% of 60 is 0.2 × 60 = 12. Subtract the discount: 60 − 12 = 48.
- 3
Multiplier route
20% off leaves 80%. 0.8 × 60 = 48. Same answer in one step.
- 4
See why the other choices are there
$12 is the discount, not the price you pay. $72 adds the discount instead of subtracting it. $40 subtracts 20 dollars instead of 20 percent.
Answer
$48
Check yourself · Question 1
A $25 book is on sale for 40% off. What is the sale price?
Answer: B
40% of 25 is 0.4 × 25 = 10. The sale price is 25 − 10 = $15. Or use the multiplier: 40% off leaves 60%, and 0.6 × 25 = 15.
Step 3 of 7
Percent increase and percent change
Prices can go up too. A 10% increase means the new price is 110% of the old one, so you multiply by 1.1.
To find the percent change between two amounts, divide the change by the starting amount, then turn the decimal into a percent.
From $50 to $58
Change: 58 − 50 = 8
Divide by the starting amount: 8 ÷ 50 = 0.16
Percent change: a 16% increase
Wrong: 8 ÷ 58 is about 13.8%
Right: 8 ÷ 50 = 16%
Always divide by the amount you started with. Dividing by the new amount is the most common wrong answer.
Check yourself · Question 2
A ticket price rises from $40 to $46. By what percent did it increase?
Answer: C
The change is 46 − 40 = 6. Divide by the starting price: 6 ÷ 40 = 0.15, which is 15%.
Step 4 of 7
Work backward to the original price
Sometimes you know the price after a change and need the price before it. Write the change as a multiplier, then undo it by dividing.
Worked example: A shirt after a 10% discount
A shirt costs $36 after a 10% discount. What was the original price?
- 1
Write the multiplier
10% off leaves 90%, so sale price = 0.9 × original.
- 2
Undo it
Divide the sale price by 0.9: 36 ÷ 0.9 = 40.
- 3
Check
10% of 40 is 4, and 40 − 4 = 36. It works.
Answer
The original price was $40.
Common mistake
Adding 10% of the sale price back on: 36 + 3.60 = $39.60.
- Why it's tempting
- It feels like the natural way to undo a 10% discount.
- Do this instead
- The 10% was taken from the original price, not the sale price. Divide by the multiplier, 0.9, to undo it.
Step 5 of 7
Work backward on your own
Check yourself · Question 3
A lamp costs $54 after a 10% discount. What was the original price?
Answer: C
10% off leaves 90%, so 0.9 × original = 54. Divide: 54 ÷ 0.9 = 60. Check: 10% of 60 is 6, and 60 − 6 = 54.
Trap answer: $59.40
Undoing on the wrong base
- Why it looks right
- Adding 10% back feels like the exact reverse of taking 10% off.
- Why it's wrong
- The discount was 10% of the original price, which is larger than $54. So 10% of $54 is too small, and the result comes up short.
- The right answer
- $60, because 0.9 × 60 = 54.
Step 6 of 7
What to remember
Remember
- Find the starting amount first. The percent is taken of it.
- A p% discount multiplies by (100 − p)%. A p% increase multiplies by (100 + p)%.
- Percent change is the change divided by the starting amount. To find an original amount, divide by the multiplier.
Step 7 of 7
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what was the starting amount, and which wrong choice would you get by stopping after finding the discount?