To reverse a percentage change, treat the new amount as a known percentage of the original, then divide. After a 15% increase the new amount is 115% of the original, so original = new amount ÷ 1.15.
This lesson follows finding a percentage of an amount and is where many students lose marks in percentages and changing bases.
What does “reverse” really mean?
A percentage change is always taken from a starting amount, called the base (or the original). If a price rises by 15%, the new price is the original plus 15% of the original. That equals the original × 1.15.
The question now hands you the new price and asks for the original. You are working backwards, so you undo the multiplication with a division.
How do I set it up?
- Write the multiplier for the change: increase r% gives 1 + r/100, decrease r% gives 1 − r/100.
- State what the new amount equals: new = original × multiplier.
- Divide both sides by the multiplier: original = new ÷ multiplier.
- Check forwards by applying the change to your answer.
An alternative that many students like is the unitary method. If 115% stands for the new amount, find 1% by dividing by 115, then multiply by 100.
Worked example
After a 15% increase, a pair of shoes costs RM92. Find the original price.
Step 1, multiplier: an increase of 15% gives 1 + 0.15 = 1.15.
Step 2, relate the amounts: original × 1.15 = 92.
Step 3, divide: original = 92 ÷ 1.15 = 80.
The original price was RM80.
Unitary check: 115% = 92, so 1% = 92 ÷ 115 = 0.8. Then the original, which is 100 percent, is 0.8 × 100 = 80. ✓
Forward check: 80 × 1.15 = 92. ✓ The working holds both ways.
The mistake to watch for
The most common slip is to take the percentage of the new amount and subtract it.
Mistaken answer: 15% of 92 = 13.80, so the original was 92 − 13.80 = RM78.20.
The student used 92 as the base. But the 15% was taken from the original price, not from 92.
The forward check exposes this at once: 78.20 × 1.15 = 89.93, which is not 92. The correction is to ask “what is the base?” before any arithmetic. The base is the original, so the new amount is 115% of the base, and you divide by 1.15.
A good rule: if the answer has to be found from the new amount, you divide. You do not take a percentage of the new amount.
Check yourself
Try these without a calculator where possible, then open each answer.
1. After a 20% increase, a bill is RM138. What was the bill before the increase?
Show answer
Multiplier 1.2. Original = 138 ÷ 1.2 = RM115.
Check: 115 × 1.2 = 138. ✓
2. A town’s population rose by 8% to 4.05 million. Find the population before the rise.
Show answer
Multiplier 1.08. Original = 4.05 ÷ 1.08 = 3.75 million.
Check: 3.75 × 1.08 = 3.75 + 0.3 = 4.05. ✓
3. After a 25% decrease, a jacket costs RM60. Find the original price.
Show answer
Multiplier 1 − 0.25 = 0.75. Original = 60 ÷ 0.75 = RM80.
Check: 80 × 0.75 = 60. ✓ The original was bigger, as it must be after a decrease.
Where this leads next
Reverse percentages set you up for comparing percentage change with percentage-point change and then for successive increases and decreases. Use the percentage-base explorer to see how the base shifts at each step, and try the non-calculator working trainer for division practice.
If reverse questions are where your marks slip even when the forward ones are fine, that pattern is something our teachers can find quickly in online one-to-one Mathematics tuition.