PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 4 · GUIDE 15
Percentage becomes useful when it leaves the isolated calculation and enters a real quantity. A discount is a percentage of an original price. GST-style school questions add a stated percentage to a price. Annual interest examples calculate a stated percentage of a principal under the conditions given. In every case, the essential question is the same: what quantity represents 100%?
The current Primary 5 Mathematics syllabus explicitly includes expressing a part of a whole as a percentage, using %, finding a percentage part of a whole, and finding discount, GST and annual interest. This guide develops those relationships through original teaching examples while keeping real-world rates separate from the mathematics. When a problem supplies a rate, use the supplied rate unless the task specifically asks for a current external rate.
For the official framework, see the MOE Primary Mathematics Syllabus.
Series route: return to the Primary 5 Mathematics Learning Hub. Earlier: Fraction Word Problems, Reference Wholes & Unknown Parts. Continue to Rate Word Problems, Unitary Method & Constant-Rate Reasoning.
1. Percentage means per hundred
35% means 35 out of 100. Therefore 35% = 35/100 = 0.35.
A percentage is a scaled comparison. It tells how large a part is relative to a whole of 100 equal percentage units.
2. Identify the 100% quantity first
If an item costs $240 before a discount, the original $240 is 100%.
If 35% of 240 is required, the percentage applies to that whole:
0.35 × 240 = 84.
If the whole is misidentified, correct arithmetic still produces the wrong answer.
3. Find percentage parts using fractions
25% = 1/4, so 25% of 320 = 320 ÷ 4 = 80.
50% = 1/2, 20% = 1/5 and 10% = 1/10. Familiar fraction equivalents are powerful mental routes.
4. Build unfamiliar percentages from benchmarks
Find 35% of 260:
- 30% = 78
- 5% = 13
- 35% = 91
This decomposition makes the percentage scale visible and provides an independent check against calculator input.
5. Express a part as a percentage of a whole
If 45 of 180 students choose an activity, the fraction is 45/180 = 1/4. Therefore the percentage is 25%.
The structure is part ÷ whole × 100%.
Do not reverse the fraction unless the question asks how many times the whole is compared with the part.
6. Same part, different percentage
30 is 30% of 100 but only 20% of 150.
The percentage changes because the reference whole changes. This is why “30 is what percentage?” is incomplete without specifying “of what?”
7. Discount amount versus sale price
An item costs $240 and receives a 15% discount.
Discount = 15% × 240 = $36.
Sale price = 240 − 36 = $204.
$36 and $204 are both mathematically correct numbers, but they answer different questions.
8. Sale price can be found using the remaining percentage
After a 15% discount, 85% remains.
Sale price = 85% × 240 = 0.85 × 240 = $204.
The subtraction route and remaining-percentage route should agree.
9. Reverse a discount problem
After a 20% discount, a bag costs $72. The sale price is 80% of the original.
80% = $72. One percent = $0.90. Therefore 100% = $90.
Directly, original = 72 ÷ 0.8 = 90.
Subtracting 20% of $72 would use the wrong reference whole.
10. Successive percentage changes use successive wholes
A price of $200 receives a 10% discount, then a further 20% discount on the reduced price.
After first discount: 90% × 200 = $180.
Second discount: 20% of $180 = $36.
Final price = $144.
The total discount is not simply 30% of $200 because the second percentage uses a different whole.
11. Why successive percentages do not usually add
Ten percent off followed by twenty percent off leaves 90% × 80% = 72% of the original. The overall reduction is 28%.
Percentage factors multiply because each stage acts on the state produced by the previous stage.
12. GST-style problems: use the stated rate
For an invented classroom problem, suppose a stated 9% tax is applied to a $500 pre-tax price.
Tax = 9% × 500 = $45.
Total = $545.
The purpose is the mathematics of adding a stated percentage. Real tax rules and rates can change, so do not assume an unstated live rate in a school exercise.
13. Find the pre-tax price from a tax-inclusive total
If a problem states that a total of $545 represents 109% of the pre-tax price, then:
109% = 545, so 1% = 5 and 100% = $500.
This is reverse percentage reasoning.
14. Do not subtract the tax percentage from the final total
If $545 already includes 9% tax, subtracting 9% of $545 does not recover the original $500. Nine percent was calculated from the pre-tax whole, not from the final total.
Instead, treat $545 as 109% of the original.
15. Annual interest as a percentage problem
Suppose a classroom problem states that $2,000 earns 3% simple annual interest for one year on the original principal.
Interest = 3% × 2000 = $60.
Total after adding that interest = $2,060.
This is a simplified school mathematics model, not a description of every real financial product.
16. Interest over several stated years
If the same simplified problem states 3% simple interest on the original $2,000 for 4 years, yearly interest is $60 and total interest is 4 × 60 = $240.
Total = $2,240.
Do not apply compounding unless the question specifies a compound process.
17. Percentage of money remaining
A student has $180 and spends 35%.
Spent = 0.35 × 180 = $63.
Remaining = $117.
Alternatively, 65% remains: 0.65 × 180 = 117.
18. Percentage after a fraction stage
A shop has 600 bottles. Two fifths are blue. Twenty-five percent of the blue bottles are sold.
Blue = 2/5 × 600 = 240.
Sold blue = 25% × 240 = 60.
Blue remaining = 180.
The percentage whole is the blue subset, not all 600 bottles.
19. Fraction after a percentage stage
A tank contains 500 ℓ. Twenty percent is drained. One quarter of the remaining water is then used.
After 20% drain, 80% remains = 400 ℓ.
One quarter of 400 = 100 ℓ.
Final amount = 300 ℓ.
20. Percentage and rate together
A machine produces 180 items per hour for 8 hours. Fifteen percent are rejected.
Total = 180 × 8 = 1440.
Rejected = 15% × 1440 = 216.
Accepted = 1224 items.
The percentage applies to the total production, not the hourly rate.
21. Percentage comparison language
If A = 120 and B = 100, A is 20 more than B. A is also 20% more than B because the difference 20 is compared with B’s 100.
But B is not 20% less than A. The decrease from 120 to 100 is 20/120 = 16 2/3%.
Percentage comparisons depend on which quantity is the reference.
22. Increase and decrease are not symmetric
If $100 increases by 20%, it becomes $120. Reducing $120 by 20% gives $96, not $100.
The two 20% calculations use different wholes.
This is an important runway concept for later percentage-change work.
23. Money answers require cents control
When a money calculation produces a decimal, preserve cents appropriately. $12.5 is usually written $12.50 in ordinary currency notation.
Follow the conventions and precision required by the question.
24. Estimate money percentages
49% of $398 should be close to half of $400 = $200. The exact value $195.02 is plausible.
A result of $19.50 is approximately ten times too small and should be rejected.
25. Error map
| Visible error | Likely cause | Repair question |
|---|---|---|
| Discount amount reported as sale price | Target lost | What quantity did you just calculate? |
| Tax subtracted from tax-inclusive total using same percentage | Reference whole wrong | What percentage of the original does the final total represent? |
| Two successive discounts added | Changing whole ignored | What is 100% at the second stage? |
| Percentage part larger than whole | Scale error | Should a percentage below 100% exceed the whole? |
| 20% rise followed by 20% fall assumed to return to start | Reference symmetry assumed | Are the two percentage wholes the same? |
26. Practice laboratory
- Find 35% of 280.
- Express 45 out of 180 as a percentage.
- An item costs $320 and receives a 15% discount. Find the discount and sale price.
- After a 25% discount, an item costs $90. Find the original price.
- A $250 item receives 10% off, then another 20% off the reduced price. Find the final price.
- A classroom problem applies a stated 9% tax to $600. Find tax and total.
- A total of $654 represents 109% of a pre-tax price. Find the pre-tax price.
- A simple-interest problem uses $3,000 at 4% per year for 3 years on the original principal. Find total interest.
- A student spends 35% of $240. Find the remainder.
- Two fifths of 750 items are Category A. Twenty percent of Category A is removed. How many Category A items remain?
- A machine makes 250 items/hour for 6 hours. Twelve percent are rejected. Find accepted items.
- A quantity rises from 100 to 120 then falls by 20%. Find the final value.
27. Explained answers
1. 98.
2. 45/180 = 1/4 = 25%.
3. Discount = $48; sale price = $272.
4. $90 is 75%; original = $120.
5. After 10% off: $225. After 20% off: $180.
6. Tax = $54; total = $654.
7. $600.
8. Yearly interest = $120; three years = $360.
9. Spent = $84; remain = $156.
10. Category A = 300; removed = 60; remain = 240.
11. Total = 1500; rejected = 180; accepted = 1320.
12. 120 × 0.8 = 96.
28. Full mixed money problem
A shop lists a bag at $400. A 15% discount is applied. A classroom problem then applies a stated 9% tax to the discounted price.
Discount = 15% × 400 = $60.
Discounted price = $340.
Tax = 9% × 340 = $30.60.
Final stated total = $370.60.
The 9% applies to the discounted $340 because that is the stated pre-tax price at the second stage.
29. Final checkpoint
A strong Primary 5 percentage learner can identify the reference whole, move among fraction-decimal-percentage forms, distinguish percentage part from final amount, solve direct and reverse discount questions, handle stated GST-style and interest problems, preserve changing wholes and estimate whether a result has sensible scale.
Continue to Primary 5 Mathematics Learning Guide | Rate Word Problems, Unitary Method & Constant-Rate Reasoning.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Declare the 100% state, apply each percentage only to its rightful whole, update the whole after every change, and reject any result that cannot reconstruct the stated money relationship.
