Stage 10 · Percentages & speed
Finding the original
Original = what you know ÷ its %.
How to do it
Method
Original = what you know ÷ its %
$32 after 20% off: 32 ÷ 80% = 40
Another way
As a fraction, one part first
80% = 45: 32 ÷ 4 = 8, 8 × 5 = 40
In full
Knowing the number after a percentage, work back to the one before. First find what percentage of the original the number you know is: 40% of a number is 18 → 40%; 20% off (八折) → 80%; up 25% → 100% + 25% = 125%; down 20% → 100% − 20% = 80%. Then the original = the number you know ÷ that percentage, that is times 100, then ÷ the percentage: $32 after 20% off, 32 ÷ 80% → 32 × 100 = 3200, 3200 ÷ 80 = 40, so it was $40. You can also write the percentage as a fraction in simplest form and find one part first: 80% = 45, 4 parts are $32, one part is 32 ÷ 4 = 8, 5 parts are 8 × 5 = 40 — the same answer. Careful: if 150 is 25% more, don't take 25% of 150 off 150, because the 25% added was 25% of the original: 150 ÷ 125% = 120. Check by working forwards: 40 × 80% = 32.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
5% of ? = 2
Show the answer and working
Answer: 5% of 40 = 2
2 is 5% of the number, so the number = 2 ÷ 5%: 2 × 100 = 200, 200 ÷ 5 = 40
Example 2
50% off is $100: was $?
Show the answer and working
Answer: 50% off is $100: was $200
50% off leaves 50% of the price, so the price = 100 ÷ 50%: 100 × 100 = 10000, 10000 ÷ 50 = 200
Example 3
? decreased by 15% = 340
Show the answer and working
Answer: 400 decreased by 15% = 340
Down 15% leaves 100% − 15% = 85% of the number, so the number before = 340 ÷ 85%: 340 × 100 = 34000, 34000 ÷ 85 = 400
Example 4
? increased by 5% = 168
Show the answer and working
Answer: 160 increased by 5% = 168
Up 5% leaves 100% + 5% = 105% of the number, so the number before = 168 ÷ 105%: 168 × 100 = 16800, 16800 ÷ 105 = 160
Practise finding the original in the game →Stage 10 · Percentages & speed · Free to start, unlimited questions, no download