Percentage
The key question in a percentage problem is: percentage of which original amount? A percentage comparison uses the stated reference quantity as its base. A result above 100% is larger than its base: for example, 130% of a quantity is 1.3 times that quantity. For a percentage change, multiply the original by a change multiplier; for a reverse percentage, undo that multiplier to recover the amount before the change.
What this guide covers
Expressing and comparing quantities as percentages, percentages greater than 100%, percentage increase or decrease, and reverse percentages.
Syllabus reference: 2026 Mathematics 4052, N3, page 5. These are Mentora search groupings, not new official syllabus categories.
Worked example
After a 20% discount, a bag costs $72. Find its price before the discount.
After the discount, $72 is 80% of the original price. Let the original be P: 0.8P = 72, so P = 72 ÷ 0.8 = $90. Adding 20% of $72 would use the wrong base.
Compare against the named base
A club has 18 members in one group and 24 in another. By what percentage is the second group larger than the first?
The increase is 24 − 18 = 6. The base is the first group, 18, so the percentage increase is 6/18 × 100% = 33⅓%. Reversing the comparison changes the base and gives a different percentage.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
Increase $240 by 15%.
Show worked answer
Use 1.15 × 240 = $276. The increase is $36, which is 15% of $240.
Check 2
A value falls by 12% and then rises by 12%. Does it return to its starting value?
Show worked answer
No. Starting at 100, it becomes 88, then 88 × 1.12 = 98.56. The second 12% is based on 88, not 100.
Check 3
After a 25% increase, a quantity is 150. What was it before the increase?
Show worked answer
150 represents 125% of the original, so the original is 150 ÷ 1.25 = 120.
Mistakes to watch for
- For percentage change, divide the change by the original or reference quantity, not by the final quantity.
- Reverse a percentage change by dividing by its multiplier; subtracting or adding the same percentage of the new value usually uses the wrong base.
- For consecutive changes, apply each multiplier to the amount after the previous change. Equal rises and falls do not generally cancel.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
Related E-Math guides
Choose a useful next resource
To understand a recurring error rather than relearn the whole skill: Check the 4052 syllabus and revision order.
If you are unsure whether this or another topic needs attention, start with the Use the free E-Math diagnostic to choose a revision starting point.
Optional targeted practice: Optional printed practice: Ratio + Percentage + Proportion.
Published by Mentora · Sources checked 23 Sept 2026
Editorial and source details
Original public examples and answers were checked with automated mathematical/editorial QA. No individual author, human reviewer or independent expert review is recorded.