To recover an original amount, turn each percentage change into a multiplier and divide the final amount by the combined multiplier. Then check by applying the changes forwards.
This lesson is part of rates and percentages in multistep consumer questions. All amounts are invented.
What is the method?
Follow these steps in order.
- Write each change as a multiplier: a 20% fall is ×0.8, a 10% fall is ×0.9, a 25% rise is ×1.25.
- Multiply the multipliers to get the combined multiplier.
- Original = final ÷ combined multiplier.
- Check by applying the changes forwards to your answer.
Worked example 1: two discounts
A jacket is sold at RM126 after a 20% discount and then a further 10% off. Find the original amount.
Combined multiplier = 0.8 × 0.9 = 0.72. Original = 126 ÷ 0.72 = RM175.
Check forwards: 175 × 0.8 = 140, and 140 × 0.9 = 126.
Worked example 2: a rise then a fall
A quantity rises by 25% and then falls by 10%, ending at 450. Find the start.
Combined multiplier = 1.25 × 0.9 = 1.125. Start = 450 ÷ 1.125 = 400.
Check forwards: 400 × 1.25 = 500, and 500 × 0.9 = 450.
The mistake that adds the percentages
A common slip is to add 20% and 10% to get 30%, then divide 126 by 0.7. That gives RM180, which is the wrong answer.
Check it forwards. 180 × 0.8 = 144, and 144 × 0.9 = 129.6, which is not 126. The correct combined reduction is 28%, because the second 10% was taken from the smaller amount.
Check yourself
A number rises by 30% and then falls by 20%, ending at 312. Find the start.
Answer
Combined multiplier = 1.3 × 0.8 = 1.04.
Start = 312 ÷ 1.04 = 300.
Check forwards: 300 × 1.3 = 390, and 390 × 0.8 = 312.
What to study next
Next, see how percentage differences are described in distinguishing percentage points from a percentage change. Test the base yourself in the percentage base and index explorer.
If you want a teacher to go through your multipliers, see online one-to-one Mathematics tuition.