An equal percentage rise and fall do not cancel because the second percentage is taken from a different base. A 20% rise then a 20% fall multiplies by 1.2 × 0.8 = 0.96, so the result is 4% below the start.
This lesson is the clearest case of the method in successive increases and decreases, and it closes the reasoning route in percentages and changing bases.
What is the real reason?
A percentage is always “a part of something”. The something changes after the first step.
If you start with 100 and add 20%, you add 20. The new total is 120.
Now take 20% of 120, which is 24. The fall removes 24, more than the 20 that was added. You end at 96.
How do I explain it in an exam?
A full answer usually has three parts.
- Use an example or a letter. Start with 100, or call the original amount x.
- Write the multipliers. Up 20% is ×1.2, down 20% is ×0.8.
- State the result and the reason. 1.2 × 0.8 = 0.96, which is less than 1, because the percentage fall is taken from a larger amount.
For a general statement, use r as a decimal. A rise of r followed by a fall of r multiplies by (1 + r)(1 − r) = 1 − r². Since r² is positive, the product is always below 1.
Worked example
The price of a bag is RM100. It is increased by 20%, then the new price is decreased by 20%. Explain why the final price is not RM100.
Step 1, apply the rise: 100 × 1.2 = 120.
Step 2, apply the fall: 120 × 0.8 = 96.
Step 3, find the net change: 96 − 100 = −4, so the price is 4% lower than the start.
Step 4, use multipliers: 1.2 × 0.8 = 0.96, which is 4% below 1.
Explanation: the 20% fall is taken from RM120, not RM100. It removes RM24, but the rise added only RM20.
Check with the other order. Fall first: 100 × 0.8 = 80, then rise: 80 × 1.2 = 96. The end result is the same, 96. ✓
The mistake to watch for
The slip is to assume the two changes are “opposites”.
Mistaken answer: +20% and −20% cancel, so the price returns to RM100.
The student treated the two percentages as equal amounts of money. They are equal percentages of different amounts.
The correction is to write the amount after each stage.
Percentages cancel only if they are taken from the same base. A 20% fall needs a 25% rise to recover, because 80 × 1.25 = 100. The recovery rise is larger since it acts on a smaller amount.
Check yourself
Try these without a calculator, then open each answer.
1. RM50 is increased by 10% and then decreased by 10%. Find the final amount and the overall percentage change.
Show answer
50 × 1.1 = 55. Then 55 × 0.9 = RM49.50. Overall multiplier 1.1 × 0.9 = 0.99, so a 1% decrease.
Check: 1% of 50 is 0.50, and 50 − 0.50 = 49.50. ✓
2. A share price falls by 50%. What percentage rise is needed to return it to its original value?
Show answer
The multiplier after the fall is 0.5. To reach 1 you need 1 ÷ 0.5 = 2. A multiplier of 2 is a rise of one hundred percent.
Check: start at 80, fall to 40, then double to 80. ✓
3. RM240 is decreased by 25%, then increased by 25%. Find the final amount, and the overall percentage change.
Show answer
240 × 0.75 = 180. Then 180 × 1.25 = RM225. Overall multiplier 0.75 × 1.25 = 0.9375, so the change is 6.25% decrease.
Check: 225 ÷ 240 = 0.9375. ✓
Where this leads next
Try the percentages practice set to mix all five lessons, and use the percentage-base explorer to watch the base change on screen. The non-calculator working trainer helps with the decimal steps.
A good explanation is a skill of its own, separate from getting the number. In online one-to-one Mathematics tuition, a teacher can read your written reasoning and help you tighten it.