Reverse percentages and percentage change

Work from a known percentage to the whole

Thirty percent of a collection is 24 cards. Ten percent is 24÷3 = 8 cards, so the whole collection is 8 × 10 = 80 cards.

Equivalently, 0.30 × whole = 24, so whole = 24÷0.30 = 80. Dividing 24 by 30 finds the amount for one percent, 0.8 card, and still needs multiplying by 100 to find the whole.

A final amount may represent more or less than 100%

A price after a 20% discount is $72. The final price is 80% of the original, so the original was 72÷0.80 = 90 dollars. Adding 20% of $72 gives $86.40, which uses the wrong reference whole.

After a 15% increase, a club has 92 members. Ninety-two represents 115% of the original. The original is 92÷1.15 = 80 members. Check that 15% of 80 is 12.

Calculate the change as a percentage of the original

A quantity rises from 50 to 65. The increase is 15, and the percentage increase is 15/50 × 100% = 30%. If it falls from 65 back to 50, the percentage decrease is 15/65 × 100% ≈ 23.1% to one decimal place. Keep the exact fraction until the final rounding. The same numerical change has a different base.

Check your understanding 1 Which amount is 100%?

A school club grows from 40 to 50 members. Another club shrinks from 50 to 40. Find the percentage change for each, naming the original whole. State whether each change is an increase or decrease.

Your percentage increase and decrease

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The first club gains 10 out of its original 40: 10/40 × 100% = 25% increase. The second loses 10 out of its original 50: 10/50 × 100% = 20% decrease. Both changes involve ten members, but each percentage compares with its own starting membership. Dividing by the final membership reverses the reference whole.

Check your understanding 2 Two successive changes

A $100 price rises by 10%, then the new price falls by 10%. Find the final price and explain whether it returns to $100.

Your answer or drawing on paper

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The rise gives $110. Ten percent of $110 is $11, so the final price is $99. The second percentage uses 110 as its whole, so the changes do not cancel.

Recover the starting amount

After a 25% decrease, a tank contains 18 ℓ. What was in it before the decrease?

Your original volume and a check