Did you know? A price that rises by 20% and then falls by 20% does not return to where it started. The percentages use different reference amounts. That single idea explains why many Secondary 1 G2 students make mistakes in discounts, price changes and reverse-percentage questions.
For families searching for a Secondary 1 G2 Mathematics tutor in Bukit Timah, the quick method is identify the original amount, convert the change to a multiplier, calculate the final amount, and reverse by division when the original is unknown. It is easier to understand than memorising separate formulas for each shopping story.
The examples follow the MOE G2 Mathematics syllabus and the eduKateSG Secondary 1 small-group tutorial model, which describes three-student weekly 90-minute classes near Sixth Avenue MRT. The worked questions are for learning and are not predictions of a particular school’s assessment.
Percentage change in one minute
- Percentage of an amount: multiply by percentage as a decimal.
- Increase by p%: multiply original by 1 + p/100.
- Decrease by p%: multiply original by 1 – p/100.
- Percentage change: amount changed divided by the original, multiplied by 100%.
- Reverse a discount: divide final price by the percentage remaining as a decimal.
- Reverse an increase: divide final value by the increase multiplier.
- For multiple changes, multiply successive factors instead of adding rates.
- Verify reverse answers by applying the original stated change.
The base changes—and so does the result
Imagine a $100 item discounted by 20%. The sale price is $80. Now imagine the shop increases that $80 price by 20%. The increase is $16, not $20, so the new price is $96. The student has used 20% twice, but not on the same amount.
A good tutor makes the two reference quantities visible in a before-and-after table. Once the child understands that each change is applied to its current base, the calculations become more reliable.
Seventeen principles that explain percentage changes
1. A percentage always has a reference whole
The word percentage means parts per hundred. But a percentage of a price, score or quantity is meaningful only after the reference amount is clear. This is why two students can use the same percent and obtain different answers.
Example: Twenty percent of $80 is $16. Twenty percent of $120 is $24. The percentage is the same, but the base is different.
Common error check: Before multiplying, underline the quantity after ‘of’ or name the original amount being compared.
2. Percentages are multipliers in another notation
To calculate a percentage of a value, write the percentage as a decimal or fraction and multiply. This connects the topic to fractions learned earlier.
Example: 15% of 240 = 0.15 × 240 = 36. A mental check is ten percent equals twenty-four and five percent equals twelve.
Common error check: If a student writes 15 × 240, remind them that the percent symbol requires division by one hundred.
3. A percentage increase adds a fraction of the original
An increase by 25% means retain the original 100% and add another 25%, leaving 125% of the original.
Example: $80 increased by 25% becomes 80 × 1.25 = $100. The change is $20; the new total is $100.
Common error check: Distinguish the amount of increase from the final value. The question may ask for either one.
4. A percentage decrease leaves a remaining fraction
A decrease by 25% means 75% of the original remains. Multiplying by the remaining fraction often makes the relationship easier to track.
Example: $240 reduced by 25% becomes 240 × 0.75 = $180. The discount itself is $60.
Common error check: Subtracting 25 as a dollar amount rather than 25% of the price misreads the units.
5. Percentage change compares the change with the start
The general calculation for percentage increase or decrease is absolute amount of change divided by original value, times one hundred percent. The starting amount is the denominator.
Example: A price from $80 to $100 rises by $20, so the percentage increase is 20/80 × 100% = 25%.
Common error check: Using 100 as the denominator gives 20%, which is the percentage that the change represents of the final price, not the original.
6. The reverse journey uses a different base
A rise from $80 to $100 is 25%, while a fall from $100 to $80 is 20%. The number twenty is the same change, but the reference amount differs.
Example: For the decrease, 20/100 × 100% = 20%. For the increase, 20/80 × 100% = 25%.
Common error check: Never assume an increase and a decrease of the same dollar amount have equal percentages.
7. Reverse percentage begins with the remaining portion
If a sale price is known after a discount, it represents a stated percentage of the unknown original. Working backwards means dividing by that remaining decimal multiplier.
Example: A bag costs $64 after a 20% discount. The sale price is 80% of original: 0.8p = 64, so p = 64 ÷ 0.8 = $80.
Common error check: Adding 20% of the sale price does not reverse the original discount because it uses a changed base.
8. Reversing an increase means dividing by more than one
When a final quantity includes a percentage increase, the final value is the original multiplied by 1 plus the increase rate. Divide by that factor to recover the start.
Example: A fee becomes $132 after a 10% increase. The original is 132 ÷ 1.10 = $120.
Common error check: Subtracting ten percent of the final $132 does not recover the original exactly.
9. ‘Increase by’ and ‘increase to’ are not interchangeable
‘Increase by twenty dollars’ describes the size of the change; ‘increase to twenty dollars’ describes the new final value. Percentage questions often hide the same distinction in their wording.
Example: If $50 increases by 20%, the increase is $10 and the final price is $60. If a price increases to $60, the increase cannot be found without its starting value.
Common error check: Read the requested final quantity before selecting an operation.
10. Percentage points are different from relative percentage change
A rate moving from 40% to 50% rises by ten percentage points. Relative to its starting rate of 40%, however, that change is 25%.
Example: The difference in rate values is 50% – 40% = 10 percentage points. The relative percentage increase is 10/40 × 100% = 25%.
Common error check: Use the term percentage points only for subtraction of percentage levels; specify the original denominator for relative change.
11. Successive discounts apply to changing bases
A 20% discount followed by a 10% discount does not simply mean a 30% discount. The second rate applies to the price after the first reduction.
Example: Starting at $100, a 20% discount leaves $80. A further 10% reduction leaves $72. The total reduction is 28%.
Common error check: Use the product 0.8 × 0.9 = 0.72 of the original instead of adding the advertised percentages.
12. Equal percentage up and down do not cancel
A 20% increase followed by a 20% decrease acts on two different base values. Multipliers reveal the combined change.
Example: $100 increased by 20% becomes $120. Decreasing $120 by 20% leaves $96, which is 4% below the starting value.
Common error check: The multipliers 1.2 and 0.8 multiply to 0.96, not one.
13. Repeated increases compound rather than add
When a value rises by a fixed percentage each period, each increase is calculated from the new amount, not always from the starting amount.
Example: A mathematical savings balance of $200 growing 10% each year becomes $220 after one year and $242 after two years. Two 10% increases give a total increase of 21%.
Common error check: Do not add 10% and 10% to obtain a 20% result when the second increase uses the new base.
14. A reverse discount can be checked forward
After finding the original price, reapply the stated discount to see whether it produces the known sale price. This check directly tests the original relationship.
Example: If $90 is the price after a 25% discount, original = 90 ÷ 0.75 = $120. Twenty-five percent of $120 is $30, and $120 – $30 = $90.
Common error check: Checking a reverse-percent solution with the same reverse division is less revealing than reconstructing the final value.
15. A percentage can exceed 100%
Percentages describe a ratio relative to the base, and there is no mathematical requirement that the ratio stay below one hundred. A 150% value is one and a half times the base.
Example: 150% of 80 is 1.5 × 80 = 120. A 150% increase, however, leaves 250% of the base: 80 × 2.5 = 200.
Common error check: Distinguish ‘150% of’ from ‘increased by 150%’. The first is the new amount; the second adds the original as well.
16. The final answer needs its context and precision
A question may ask for the original amount, final amount, percentage change or discount. Each has a different unit or form. Do not copy the last calculator display without interpreting it.
Example: If a product originally costs $80 and is discounted 15%, the discount is $12 and the sale price is $68. A question asking for the sale price needs sixty-eight dollars, not fifteen percent or twelve dollars.
Common error check: Return to the final sentence to confirm what quantity is requested, and round money or measurements only as instructed.
17. An estimate can reveal a wrong denominator
If an original price is above a discounted price, the reverse-calculated starting value must be greater than the sale price. If it is smaller, the model has almost certainly been reversed.
Example: A product sold for $75 after 25% off should have an original price near $100. Dividing by 0.75 gives exactly $100.
Common error check: Use this relationship check before accepting the displayed number, especially after a long multi-step question.
30 worked percentage clinics
Attempt each question with the original amount underlined. For reverse percentage questions, write ‘final = multiplier × original’ before solving.
Clinic 1: Find 10% of $240
Answer: $24. Reason: 0.10 × 240.
Verify: One tenth of 240. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 2: Find 15% of $240
Answer: $36. Reason: 0.15 × 240.
Verify: Ten percent plus five percent. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 3: Find 25% of $80
Answer: $20. Reason: A quarter of eighty.
Verify: Four quarters make eighty. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 4: Increase $80 by 25%
Answer: $100. Reason: 80 × 1.25.
Verify: An increase must exceed eighty. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 5: Decrease $240 by 25%
Answer: $180. Reason: 240 × 0.75.
Verify: A decrease must be below 240. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 6: Find percentage increase from 80 to 100
Answer: 25%. Reason: 20/80 × 100%.
Verify: Original amount is eighty. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 7: Find percentage decrease from 100 to 80
Answer: 20%. Reason: 20/100 × 100%.
Verify: Original amount is one hundred. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 8: Find percentage increase from 50 to 65
Answer: 30%. Reason: 15/50 × 100%.
Verify: Check 50 × 1.30 = 65. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 9: Find percentage decrease from 120 to 90
Answer: 25%. Reason: 30/120 × 100%.
Verify: Check 120 × 0.75 = 90. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 10: A $100 price falls by 15%. Sale price?
Answer: $85. Reason: 100 × 0.85.
Verify: Discount amount is $15. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 11: A $64 sale price follows 20% off. Original?
Answer: $80. Reason: 64 ÷ 0.80.
Verify: Check 80 minus 16 = 64. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 12: A $90 sale price follows 25% off. Original?
Answer: $120. Reason: 90 ÷ 0.75.
Verify: Check 120 × 0.75 = 90. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 13: A $132 value follows a 10% increase. Original?
Answer: $120. Reason: 132 ÷ 1.10.
Verify: Check 120 × 1.10 = 132. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 14: A price becomes $150 after a 25% increase. Original?
Answer: $120. Reason: 150 ÷ 1.25.
Verify: Check 120 + 30 = 150. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 15: A $200 price rises 15%. New price?
Answer: $230. Reason: 200 × 1.15.
Verify: Increase $30. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 16: A $200 price falls 15%. New price?
Answer: $170. Reason: 200 × 0.85.
Verify: Decrease $30. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 17: $100 reduced 20%, then 10%. Final?
Answer: $72. Reason: 100 × 0.8 × 0.9.
Verify: Total decrease is 28%. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 18: Effective discount for 20% then 10%
Answer: 28%. Reason: Remaining 72%, so reduction 28%.
Verify: Not 30%. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 19: $100 increased 20%, then reduced 20%. Final?
Answer: $96. Reason: 100 × 1.2 × 0.8.
Verify: Net 4% decrease. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 20: $100 rises 10% then falls 10%. Final?
Answer: $99. Reason: 100 × 1.1 × 0.9.
Verify: Net 1% decrease. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 21: $200 increases 10% for two periods. Final?
Answer: $242. Reason: 200 × 1.1².
Verify: 220 after first period, 242 second. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 22: A rate moves 40% to 50%. Percentage-point increase?
Answer: 10 percentage points. Reason: Subtract rate values.
Verify: Not the same as relative increase. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 23: A rate moves 40% to 50%. Relative percentage increase?
Answer: 25%. Reason: 10/40 × 100%.
Verify: Starting rate is 40%. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 24: Find 150% of 80
Answer: 120. Reason: 1.5 × 80.
Verify: One and a half times eighty. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 25: Increase 80 by 150%
Answer: 200. Reason: Original 100% plus 150% gives 250% of eighty.
Verify: Do not confuse with 150% of eighty. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 26: After 15% discount a bag costs $85. Original?
Answer: $100. Reason: 85 ÷ 0.85.
Verify: Check 15% of 100 is 15. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 27: A sale price of $48 represents 60% of original. Original?
Answer: $80. Reason: 48 ÷ 0.60.
Verify: Check 80 × 0.6 = 48. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 28: A fee changes $75 to $90. Percentage increase?
Answer: 20%. Reason: 15/75 × 100%.
Verify: Check 75 × 1.2 = 90. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 29: A fee changes $90 to $81. Percentage decrease?
Answer: 10%. Reason: 9/90 × 100%.
Verify: Check 90 × 0.9 = 81. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
Clinic 30: A value goes 75 to 90 to 81. Net change from start?
Answer: 8% increase. Reason: (81 – 75)/75 × 100% = 8%.
Verify: Do not merely add the two individual percentages. When an answer is wrong, locate which base was used. Retest the same relationship with fresh amounts rather than copying a correct solution immediately.
How to teach the topic over four weeks
Week 1: find the reference whole
Convert simple percentages to fractions and decimals. Compute ‘p% of x’ using mental benchmarks and multiplication. Make students name the base before calculating.
Keep practice brief enough for students to explain their reasoning and check a changed question independently. A precise repair of one mistaken denominator is more useful than many identical questions completed with the wrong base.
Week 2: increase, decrease and relative change
Compare $80 to $100 and $100 to $80. Explain why the percentage changes differ. Distinguish the amount changed from the final amount and practise multipliers above and below one.
Keep practice brief enough for students to explain their reasoning and check a changed question independently. A precise repair of one mistaken denominator is more useful than many identical questions completed with the wrong base.
Week 3: reverse percentages
Start with a $100 original and a 20% reduction, then reverse from the $80 sale price. Vary the numbers, practise dividing by a remaining multiplier and check answers in the forward direction.
Keep practice brief enough for students to explain their reasoning and check a changed question independently. A precise repair of one mistaken denominator is more useful than many identical questions completed with the wrong base.
Week 4: successive changes and mixed stories
Use short two-step changes, distinguishing percentage points from relative percentage change where appropriate. Mix discounts with ratio and algebra questions so students must select a method without chapter hints.
Keep practice brief enough for students to explain their reasoning and check a changed question independently. A precise repair of one mistaken denominator is more useful than many identical questions completed with the wrong base.
What a parent should ask before correcting the answer
Ask three questions: ‘Which amount was the original?’, ‘What percent of that original remains?’ and ‘If we apply your answer in the forward direction, do we recover the sale price?’ A child who can answer all three is doing more than remembering a discount formula.
For successive changes, write a row of values: start, after first change, after second change. Then compare the end with the start. If the student simply adds the percentages, point to the second row and ask which value the second rate is applied to.
The role of small-group tutoring
During a small-group session, one student might solve with multipliers, another with a bar diagram and a third with a backward equation. The tutor can compare approaches, inspect the first wrong reference base and ask each student to check an unfamiliar version alone.
The eduKateSG Secondary 1 Mathematics tutorial reference describes weekly 90-minute sessions in groups of three near Sixth Avenue MRT. Parents can discuss school-topic fit and learning needs; the format is not a promise of a particular exam result.
Frequently asked questions
What is reverse percentage?
Working backwards from a final amount after a stated percentage change to find the original amount. Divide the final amount by the remaining or increased multiplier.
Why does adding the percentage back to the discounted price fail?
The original discount was based on the original price, while an addition to the sale price would use a smaller base. Dividing by the remaining percentage reverses the relationship correctly.
How can my child tell increase from decrease?
Identify the starting amount and final amount. An increase multiplier is above one; a decrease multiplier is between zero and one for ordinary positive prices.
Why do equal percentage increases and decreases not cancel?
Because the second percentage applies to a different base. For example 100 rises to 120 with a 20% increase, then falls to 96 with a 20% decrease.
Do two successive discounts simply add together?
No. Each discount uses the latest amount as its base. A 20% discount then 10% discount leaves 72% of the original, or a 28% total reduction.
What is the difference between percentage points and percentage change?
Percentage points directly subtract rates; relative percentage change divides that difference by the original rate. A rate from 40% to 50% rises ten points but 25% relatively.
Does a percentage above 100% make sense?
Yes, a quantity can exceed its base. Distinguish ‘150% of’ from ‘increase by 150%’ because the second includes the original 100% as well.
How can we check reverse percentage answers?
Take the recovered original and apply the stated increase or decrease to reconstruct the given final amount.
Does a Secondary 1 G2 student need to study compound interest?
Not to understand successive percentage changes. The same multiplier reasoning can illustrate repeated changes without requiring an advanced financial formula or off-syllabus assumptions.
What helps with percentage word problems at tuition?
A tutor can identify the reference-base error, use a before-and-after table and check a fresh problem independently. The source eduKateSG tutorial model describes small groups of three with weekly 90-minute sessions near Sixth Avenue MRT.
Further reading
For topic coverage, consult the official G2 Mathematics syllabus. For tutorial context, read the eduKateSG small-group Maths reference. The following articles develop related skills:
- Ratio, percentage and speed word problems
- Fractions, decimals and percent conversions
- How to read Maths questions precisely
The key question in percentage work is: ‘Percentage of which amount?’ Once that is settled, the choice of multiplier—and the answer—becomes much easier to justify.