Percentage questions become difficult when the reference quantity is hidden. A 20% increase is measured from the original value. A 20% decrease after that increase is measured from a different base. Reverse percentages work backward from a changed amount. Percentage comparison depends on which quantity is used as the denominator.
This forty-third Secondary 4 Mathematics Learning Guide develops reverse percentages, comparison and repeated change as one multiplicative system. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
It deepens the earlier Ratio, Percentage, Rates and Financial Mathematics and Direct and Inverse Proportion, Scale and Rate Models guides.
Percent means per hundred of a stated base
A percentage is never complete without knowing what quantity the percentage is taken of.
percentage change = change / original value × 100%.
Worked Example 1 | Percentage increase
A price rises from $80 to $92.
Increase=$12. Original=$80.
Percentage increase=12/80×100%=15%.
Multipliers compress percentage change
An increase of p% multiplies by 1+p/100. A decrease of p% multiplies by 1−p/100.
- Increase 12% → multiplier 1.12.
- Decrease 12% → multiplier 0.88.
Worked Example 2 | Forward percentage change
A quantity of 450 increases by 8%.
450×1.08=486.
Reverse percentage means divide by the multiplier
If a final value is known after a percentage change, do not simply subtract the same percentage of the final value. Work backward through the multiplier.
Worked Example 3 | Reverse an increase
After a 25% increase, a value is 150. Find the original.
Final=original×1.25.
Original=150/1.25=120.
Worked Example 4 | Reverse a discount
After a 20% discount, an item costs $96. Find the original price.
A 20% discount leaves 80% of the original:
Original=96/0.80=$120.
Subtracting 20% of $96 would give the wrong original because $96 is not the base from which the discount was calculated.
Equal percentage increase and decrease do not cancel
Suppose 100 increases by 20% and then decreases by 20%.
100×1.20=120.
120×0.80=96.
Net result=4% decrease.
The two percentages act on different bases.
Repeated change is multiplier multiplication
Two successive increases of 10% give overall multiplier 1.10×1.10=1.21.
Overall increase=21%, not 20%.
Worked Example 5 | Two different successive changes
A price increases by 15% and then decreases by 8%. Find the net percentage change.
Combined multiplier=1.15×0.92=1.058.
Net change=5.8% increase.
Percentage comparison is directional
If A=120 and B=100, A is 20% greater than B because the comparison uses B as the base:
(120−100)/100×100%=20%.
But B is not 20% less than A:
(120−100)/120×100%=16.67%.
The base changes with the wording.
Worked Example 6 | “More than” versus “less than”
Class X has 72 students and Class Y has 60.
- X has (12/60)×100%=20% more students than Y.
- Y has (12/72)×100%=16.67% fewer students than X.
Percentage points are not the same as percentage change
If a rate rises from 40% to 50%, the increase is 10 percentage points.
Relative percentage increase is:
(50−40)/40×100%=25%.
Both statements can be correct, but they answer different questions.
Worked Example 7 | Percentage points versus relative increase
A success rate rises from 64% to 72%.
- Increase=8 percentage points.
- Relative increase=8/64×100%=12.5%.
Original value can be recovered after several changes
If successive multipliers are known, divide the final value by their product.
Worked Example 8 | Reverse repeated change
A value increases by 10%, then by 20%, and ends at 660. Find the original.
Combined multiplier=1.10×1.20=1.32.
Original=660/1.32=500.
Repeated annual change follows exponential structure
If a quantity grows by r% each year for n years:
final = initial × (1+r/100)n.
For repeated decrease, use a multiplier below 1.
Worked Example 9 | Repeated growth
A quantity of 800 grows by 6% per year for 3 years.
800(1.06)³≈952.81.
Worked Example 10 | Repeated depreciation
An asset valued at $20,000 decreases by 12% each year for 2 years.
20000(0.88)²=$15,488.
The total decrease is 22.56%, not 24%, because the second 12% is taken from a smaller base.
Find the percentage needed to restore a decrease
If 100 falls by 20%, it becomes 80. To return from 80 to 100 requires an increase of 20 on a base of 80:
20/80×100%=25%.
A 20% loss therefore requires a 25% increase to recover the original value.
Worked Example 11 | Recovery percentage
A value falls from 250 to 200. What percentage increase is needed to return to 250?
(50/200)×100%=25%.
A multiplier-first workflow
- Identify the original reference quantity.
- Translate each percentage change into a multiplier.
- Multiply for forward repeated change.
- Divide for reverse percentage problems.
- Use the wording to select the comparison base.
- Distinguish percentage points from relative percentage change.
- Check whether the final answer should be a value, a rate or a percentage.
Common failure modes
| Error | Cause | Repair |
|---|---|---|
| Subtracts a percentage of the final value to reverse an increase | Wrong base | Divide by the original multiplier |
| Adds successive percentages directly | Multiplicative structure ignored | Multiply the change factors |
| Says 120 is 20% more than 100 and 100 is 20% less than 120 | Comparison base unchanged incorrectly | Recalculate denominator from wording |
| Confuses 10 percentage points with 10% increase | Rates compared without a base | Separate absolute point change from relative change |
| Uses 1.15 for a 15% decrease | Direction ignored | Decrease multiplier is 0.85 |
| Rounds after each repeated step | Precision lost early | Use one combined multiplier and round at the end |
Independent practice
- After a 15% increase, a value is 230. Find the original.
- After a 30% discount, an item costs $84. Find the original price.
- A value rises by 12% and then falls by 5%. Find the overall percentage change.
- 80 is what percentage greater than 64?
- 64 is what percentage less than 80?
- A rate rises from 50% to 58%. State the percentage-point increase and relative percentage increase.
Explained answers
1. 230/1.15=200.
2. 84/0.70=$120.
3. 1.12×0.95=1.064, so 6.4% increase.
4. 16/64×100%=25%.
5. 16/80×100%=20%.
6. 8 percentage points; relative increase=8/50×100%=16%.
Final thought
Percentage control is really base control. Once the reference quantity is identified, multipliers make forward and reverse change systematic and prevent the common illusion that equal-looking percentages must cancel.
Find the base, convert the percentage to a multiplier, and let the direction of the problem decide whether to multiply or divide.
Return to the Secondary Mathematics Hub.
