What percent? Work backward
Find the percent, the whole, or the price before a discount.
In your head, jot if needed · no calculator why?
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the idea
Every percent question links three numbers: the whole, the part, and the percent. The part is that percent of the whole, and most mistakes come from using the wrong number as the whole.
more of the idea
When the percent is missing, divide the part by the whole, then multiply by 100. When the whole is missing, work backward from the part. In a change, the whole is the starting amount. After a discount, it is the original price.
techniques
Part over whole
- Name the whole: the number after "of", or the starting amount in a change.
- In a change, the part is the difference between the two amounts.
- Divide the part by the whole, then multiply by 100.
A price rises from $60 to $69. What is the percent increase?
- The change is $69 − $60 = $9.
- Divide by the start: 9 ÷ 60 = 0.15 = 15%.
Answer: 15%
Work back from the part
- Write the percent as a fraction: 25% = 1/4, and 60% = 3/5.
- If the top is 1, multiply the part by the bottom.
- Otherwise, divide the part by the top to get one piece, then multiply by the bottom.
48 is 80% of what number?
- 80% = 4/5.
- One fifth is 48 ÷ 4 = 12.
- The whole is 5 × 12 = 60.
Answer: 60
Divide by what is left
- Say what share of the original is left: 20% off leaves 80%.
- Divide the sale price by that share, written as a decimal.
- Check: the discount taken off your answer should give the sale price.
After 25% off, a lamp costs $60. What was the original price?
- $60 is 75% of the original.
- $60 ÷ 0.75 = $80.
- Check: 25% of $80 is $20, and $80 − $20 = $60.
Answer: $80
watch out for
- Dividing the whole by the part. 18 is 30% of 60, but dividing 60 by 18 gives over 300%.
- Dividing a change by the new amount. A rise from $80 to $92 is $12 on a start of $80, so 15%.
- Adding the discount back on. After 20% off, $96 is 80% of the original: $120, not $115.20.
- Taking the percent of the part. If 30 is 25% of a number, that number is 4 × 30 = 120.
practice
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What percent?
worked example
70 is what percent of 200?
- Part ÷ whole: 70/200 = 7/20 = 0.35 = 35%.
Answer: 35%
Find the whole
worked example
19 is 20% of what number?
- 20% is one fifth, so the whole is 5 × 19 = 95.
Answer: 95
Percent change
worked example
A price rises from $500 to $525. What is the percent increase?
- Change ÷ starting value: 25 ÷ 500 = 0.05 = 5%.
Answer: 5%
Price before the discount
worked example
After 25% off, a lamp costs $198. What was the original price?
- $198 is 75% of the original, so divide by 0.75: 198 ÷ 0.75 = $264.
Answer: $264.00