SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 27
Percentages compare a quantity with a reference base of 100. Percentage points compare two percentage values directly. Percentage change compares the change with the original quantity. These are related ideas, but they are not interchangeable.
This guide develops percentage points, absolute change, percentage change, relative comparison, repeated comparisons, reverse checks, discounts, mark-ups, population-style comparisons and common base-value errors. It extends Percentages and Reverse Percentages by focusing on comparison structure.
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1. A percentage always has a reference quantity
25% means 25 out of every 100 of a stated base.
Without knowing the base, a percentage can be incomplete for comparison.
Example
25% of 200 = 50, while 25% of 80 = 20.
2. Percentage points compare percentage values directly
If a rate rises from 40% to 55%, the increase is 15 percentage points.
This is a direct subtraction of percentage values: 55−40=15.
3. Percentage-point change is not the same as percentage change
A rise from 40% to 55% is 15 percentage points.
Relative to the original 40%, the percentage increase is:
(15/40)×100% = 37.5%.
The two answers describe different comparisons.
4. Percentage change uses the original value as the denominator
Percentage change = change ÷ original × 100%.
Worked example
A value rises from 80 to 100.
Change = 20.
Percentage increase = 20/80×100% = 25%.
5. A decrease uses the same structure
A value falls from 150 to 120.
Decrease = 30.
Percentage decrease = 30/150×100% = 20%.
6. The denominator decides the story
Going from 80 to 100 is a 25% increase.
Going from 100 back to 80 is a 20% decrease.
The numerical change is 20 both ways, but the starting bases differ.
Core lesson
Percentage change is directional.
7. Relative comparison answers “how much larger than”
A is 60 and B is 50.
A is 10 larger than B.
Relative to B, A is 10/50×100% = 20% larger.
Relative to A, B is 10/60×100% ≈ 16.7% smaller.
8. “More than” and “of” are not the same comparison
120 is 20% more than 100.
But 120 is 120% of 100.
The first describes change beyond the base; the second describes total relative size.
9. Percentage multipliers compress increase and decrease
Increase by 18% means multiply by 1.18.
Decrease by 18% means multiply by 0.82.
Worked example
$250 increased by 18% becomes 250×1.18 = $295.
10. Reverse checking confirms the base
If 250 increased by 18% gives 295, the increase is 45.
45/250×100%=18%, confirming the model.
Checking against 295 would answer a different question.
11. Discounts are percentage decreases from the listed price
A $160 item discounted by 25% costs:
160×0.75 = $120.
The listed price is the original base.
12. Mark-ups use the original cost as the base
An item costing $80 is marked up by 30%.
Mark-up = 0.30×80 = 24.
Selling price = $104.
13. A discount followed by an equal percentage increase does not return to the start
Start at 100.
Decrease 20% → 80.
Increase 20% → 96.
The second 20% uses 80 as its base.
This is a changed-base effect.
14. Percentage-point comparisons are useful for rates and proportions
If participation rises from 62% to 68%, the change is 6 percentage points.
Relative increase = 6/62×100% ≈ 9.68%.
Use the description the question actually requests.
15. Percentages can compare groups of different sizes
Class A: 18 of 30 students choose an option → 60%.
Class B: 24 of 48 choose it → 50%.
Although more students chose the option in Class B, the proportion is higher in Class A.
Count versus rate
Absolute count and percentage answer different questions.
16. Percentage difference between two values may use a different convention
Some contexts use “percentage difference” with the average of two values as the denominator, especially when neither is a natural original baseline.
This is different from percentage change. Use only the convention specified by the task or field.
For ordinary school increase/decrease questions, percentage change uses the original value.
17. Percentage points can cross zero or exceed 100 in some contexts
Percentages representing shares usually lie between 0% and 100%.
Percentage changes, however, can exceed 100% when a value more than doubles.
Example
20 to 50 is an increase of 30, so percentage increase = 30/20×100% = 150%.
18. Estimate before calculating
A rise from 400 to 460 is a change of 60. Since 10% of 400 is 40 and 5% is 20, the increase is 15%.
Benchmark percentages can make exact calculation transparent.
19. Common comparison errors
| Error | Why it fails | Repair prompt |
|---|---|---|
| 40%→50% called a 10% increase | Percentage points confused with relative change | 10 is what fraction of the original 40? |
| Uses new value as denominator | Original base lost | What value did the change start from? |
| 20% down then 20% up assumed equal | Base changed between steps | What is the second percentage applied to? |
| Compares counts only across unequal groups | Group-size difference ignored | Should the comparison use proportions? |
| Writes “120% more” when meaning “120% of” | Total and increase confused | Is 100% the original amount itself? |
20. Practice laboratory
- A rate rises from 35% to 47%. Find the percentage-point increase.
- For the same change, find the percentage increase relative to 35%.
- A value rises from 240 to 300. Find the percentage increase.
- A value falls from 500 to 425. Find the percentage decrease.
- By what percentage is 72 larger than 60?
- By what percentage is 60 smaller than 72?
- Increase $180 by 15%.
- Decrease $180 by 15%.
- A $240 item is discounted by 30%. Find the sale price.
- A cost of $90 is marked up by 25%. Find the selling price.
- Start with 200, decrease by 10%, then increase the result by 10%. Find the final value.
- Class A has 24 of 40 choosing option X. Class B has 30 of 60. Which class has the higher percentage?
- A value increases from 25 to 70. Can the percentage increase exceed 100%?
- Explain the difference between 80% of a quantity and 80% more than a quantity.
21. Explained answers
1. 12 percentage points.
2. 12/35×100% ≈ 34.3%.
3. 60/240×100%=25%.
4. 75/500×100%=15%.
5. 12/60×100%=20%.
6. 12/72×100%≈16.7%.
7. 180×1.15=$207.
8. 180×0.85=$153.
9. 240×0.70=$168.
10. 90×1.25=$112.50.
11. 200×0.9×1.1=198.
12. A=60%, B=50%, so Class A.
13. Yes. Increase=45; 45/25×100%=180%.
14. 80% of means 0.8 times the base; 80% more means 1.8 times the base.
22. Complete mixed problem
A fictional club’s attendance rises from 120 to 150, while the share of members arriving before 9am rises from 40% to 52%.
Total-attendance increase = 30.
Percentage increase = 30/120×100% = 25%.
Early-arrival rate increase = 12 percentage points.
Relative increase in the early-arrival rate = 12/40×100% = 30%.
Three different comparisons appear in the same scenario; each uses a different base.
23. Teaching percentage comparison through base questions
Before calculating, require the learner to complete the sentence: “The change is being compared with ______.”
If the blank is not clear, the percentage setup is not ready.
Changed-case test
Compare 50→60 and 100→110. Both rise by 10, but the percentage increases are 20% and 10%. This exposes why absolute change alone is insufficient.
24. Questions students often ask
What is a percentage point?
It is the direct difference between two percentage values.
Why is 40% to 50% a 25% increase?
The increase is 10, and 10 is one-quarter of the original 40.
Can percentage increase exceed 100%?
Yes, when the increase is greater than the original value.
Why does 20% down then 20% up not cancel?
The two percentages are applied to different bases.
25. Return path and sources
Percentage comparison depends on reference-base control. Revisit Percentages and Reverse Percentages for core percentage structure and Ratio and Proportion for multiplicative comparison.
Official curriculum reference: MOE Secondary Syllabus Directory. Exact applications and financial-context depth vary by subject level and school.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Identify the original base, separate absolute change from relative change, distinguish percentage points from percentage change, verify with a multiplier and return the comparison to its context.
