doing math

courses › Fractions, Decimals & Percents

What percent? Work backward

level 13 course

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 lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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.

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

What percent one number is of another, or a percent change.

  1. Name the whole: the number after "of", or the starting amount in a change.
  2. In a change, the part is the difference between the two amounts.
  3. Divide the part by the whole, then multiply by 100.
worked example

Example: A price rises from $60 to $69. What is the percent increase?

  1. The change is $69 − $60 = $9.
  2. Divide by the start: 9 ÷ 60 = 0.15 = 15%.

Answer: 15%

Work back from the part

Finding the whole from a part and its percent.

  1. Write the percent as a fraction: 25% = 1/4, and 60% = 3/5.
  2. If the top is 1, multiply the part by the bottom.
  3. Otherwise, divide the part by the top to get one piece, then multiply by the bottom.
worked example

Example: 48 is 80% of what number?

  1. 80% = 4/5.
  2. One fifth is 48 ÷ 4 = 12.
  3. The whole is 5 × 12 = 60.

Answer: 60

Divide by what is left

Finding the price before a discount.

  1. Say what share of the original is left: 20% off leaves 80%.
  2. Divide the sale price by that share, written as a decimal.
  3. Check: the discount taken off your answer should give the sale price.
worked example

Example: After 25% off, a lamp costs $60. What was the original price?

  1. $60 is 75% of the original.
  2. $60 ÷ 0.75 = $80.
  3. Check: 25% of $80 is $20, and $80 − $20 = $60.

Answer: $80

Tips by skill

  • TipWhat percent?: Divide the part by the whole, the number after "of", then multiply by 100.
  • TipFind the whole: Write the percent as a fraction. Divide the part by its top, then multiply by its bottom.
  • TipPercent change: Find the change, divide it by the starting amount, not the new one, then multiply by 100.
  • TipPrice before the discount: The sale price is the share left after the discount, such as 80% of the original. Divide by that share.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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