For successive changes, write a multiplier for each change and multiply them in order. An increase of r% is ×(1 + r/100) and a decrease of r% is ×(1 − r/100). The product is the single overall multiplier.
This lesson builds on reversing a percentage and leads to why equal rises and falls do not cancel in percentages and changing bases.
What changes at each step?
Every percentage change is taken from the amount that exists just before that change. After the first change the amount is different, so the second percentage works on a new base.
This is why 10% up then 20% down is not a 10% fall. The 20% is taken from a larger amount than the original.
How do I set it up?
- Convert each change to a multiplier. Up 10% is 1.1. Down 20% is 0.8.
- Multiply the amount by the first multiplier, then the result by the second. Or multiply the multipliers together first.
- Read the overall multiplier. Above 1 is an increase, below 1 is a decrease. Subtract 1 and convert to a percentage.
- Check the order of the stages in a quick table.
Multiplication is commutative, so the final amount does not depend on the order of the changes. The amount at the middle stage does.
Worked example
An item costs RM500. Its price is increased by 10%, then the new price is decreased by 20%. Find the final price and the overall percentage change.
Step 1, multipliers: up 10% is 1.1, down 20% is 0.8.
Step 2, apply the first change: 500 × 1.1 = 550.
Step 3, apply the second change: 550 × 0.8 = 440.
Step 4, overall multiplier: 1.1 × 0.8 = 0.88, which is a decrease of 12%.
The final price is RM440, an overall decrease of 12%.
Check: 12% of 500 is 60, and 500 − 60 = 440. ✓ Both routes give the same answer.
The mistake to watch for
The slip is to add or subtract the percentages as if they shared a base.
Mistaken answer: +10% and −20% gives −10% overall, so 500 × 0.9 = RM450.
The student treated both percentages as parts of the original RM500.
The forward stage table shows what went wrong.
After the first change the price is RM550, and 20% of 550 is 110, not 100. So the second change removes RM110, leaving 440. The correct habit is to write the new base after every stage.
Check yourself
Try these without a calculator where you can, then open each answer.
1. A quantity of 200 is increased by 50%, then decreased by 10%. Find the final value and the overall percentage change.
Show answer
200 × 1.5 = 300. Then 300 × 0.9 = 270. Overall multiplier 1.5 × 0.9 = 1.35, so a 35% increase.
Check: 35% of 200 is 70, and 200 + 70 = 270. ✓
2. A price of RM80 is reduced by 25%, then reduced by a further 10%. Find the final price and the overall percentage reduction.
Show answer
80 × 0.75 = 60. Then 60 × 0.9 = RM54. Overall multiplier 0.75 × 0.9 = 0.675, so the reduction is 1 − 0.675 = 0.325, which is 32.5%.
Check: 32.5% of 80 is 26, and 80 − 26 = 54. ✓
3. RM2,000 is invested at 5% interest per year, added each year to the balance. What is the balance after 3 years?
Show answer
Multiplier 1.05 each year. 1.05² = 1.1025 and 1.05³ = 1.1025 × 1.05 = 1.157625. Balance = 2000 × 1.157625 = RM2,315.25.
Check: Year 1 gives 2100, year 2 gives 2205, year 3 gives 2205 × 1.05 = 2315.25. ✓
Where this leads next
Next, see why an equal rise and fall do not cancel, which uses the same multiplier idea in its simplest form. The percentage-base explorer traces the amount through each stage, and the non-calculator working trainer supports the arithmetic.
If stacked percentages still feel unpredictable in timed conditions, online one-to-one Mathematics tuition gives you a teacher who can watch your working as it happens.