Every percentage refers to a whole
A percentage expresses a quantity in relation to one hundred equal parts of a stated whole. Twenty-five per cent means twenty-five out of every hundred of that whole. The actual amount depends on which whole the question uses.
Twenty-five per cent of 80 is 20. Twenty-five per cent of 200 is 50. The percentage has not changed, but the reference amount has.
For primary families in Punggol, the most useful first step is often to label the base, which is the quantity treated as 100%. This is especially important when a question contains an original amount, an increase, a reduced amount or two groups being compared. Percentage work generally builds on fraction and decimal understanding; use it at the stage appropriate to the learner’s current school work.
Find the amount by connecting the percentage to the base
Suppose 30% of 150 counters are yellow. The whole collection of 150 counters represents 100%.
Find 10% by dividing the base by ten:
150 ÷ 10 = 15 counters.
Then find 30%:
15 × 3 = 45 counters.
The same calculation can be written 30/100 × 150 = 45. Both routes express the same relationship.
A useful model labels the complete bar 150 and 100%. The selected part is 30% and 45. The unselected part is 70% and 105. These pairs keep the percentage and the actual amount connected.
Do not write “30% equals 30” unless the whole is 100 units. Thirty per cent describes a share, not a fixed count.
Worked example: a percentage reduction
A set of classroom materials originally costs $80. Its price is reduced by 25%. Find the reduction and the new price.
The original $80 is the base, so it represents 100%.
Find 25%, which is one quarter:
$80 ÷ 4 = $20.
The reduction is $20. Find the new price:
$80 − $20 = $60.
The new price is 75% of the original. Check this relationship:
75/100 × $80 = $60.
Now suppose a separate question asks, “The new price is $60. What percentage increase would return it to $80?”
The increase in dollars is still $20, but the new comparison uses $60 as the base. Calculate:
20/60 × 100% = 33⅓%.
The return increase is therefore not 25%. The amount changed by $20 in both directions, but the bases differ.
A 25% reduction from $80 and a 25% increase from $60 are different relationships. The second would add 25% of $60, which is $15, producing $75 rather than $80.
Correct a wrong-base solution
An incorrect solution might reason, “The price fell by 25%, so increasing it by 25% restores the original price.” This treats the percentage as a fixed amount and ignores the changed base.
The corrected reasoning identifies what represents 100% in each question. During the reduction, $80 is 100%. During the increase from the reduced price, $60 is 100%.
A table can make the change clear.
| Comparison | Base treated as 100% | Amount of change | Percentage of that base |
|---|---|---|---|
| Reduction from original price | $80 | $20 | 25% |
| Increase from reduced price | $60 | $20 | 33⅓% |
This example is about a mathematical price relationship. It does not claim that a particular shop offers the reduction or that a given discount is a good purchase. The value of a real decision also depends on what is needed and what is being compared.
Distinguish a percentage from a percentage-point difference
Suppose the share of pupils choosing an activity increases from 40% to 50% of the same group. The difference between the two percentages is 10 percentage points.
The share has increased by 10/40 × 100% = 25% relative to its previous share. These statements describe different comparisons.
This distinction is useful enrichment when a learner is ready for it. It need not become a new rule to memorise before basic percentages are secure. Begin by asking what each comparison measures: the gap between percentage labels, or the change relative to the earlier value.
Also check whether the total group stayed the same. Forty per cent of one group and fifty per cent of another do not determine the change in pupil count without knowing both group sizes.
Use percentage comparisons with appropriate evidence
The Science article Compare Solutions Using Evidence and Tradeoffs asks readers to state a purpose and compare relevant evidence. Percentage comparisons need similarly clear conditions.
If a hypothetical design uses 20% less material, identify the original material amount and whether the designs perform the same stated job. A percentage can summarise a numerical difference, but it does not establish that the lighter design is stronger, safer or preferable for every purpose.
Mathematics tells us what the quantities mean relative to a base. A broader conclusion needs evidence about the actual task.
Independent practice with full explanations
A collection contains 240 cards. Thirty-five per cent are picture cards. How many are picture cards, and how many are other cards?
The complete collection of 240 represents 100%.
Find 10%: 240 ÷ 10 = 24.
Find 30%: 24 × 3 = 72.
Find 5%, half of 10%: 24 ÷ 2 = 12.
Find 35%: 72 + 12 = 84 picture cards.
The number of other cards is 240 − 84 = 156. Checking gives 84 + 156 = 240. The remaining share is 65%, which also gives 65/100 × 240 = 156.
For a second task, a club’s collection increases from 50 books to 60 books. Find the percentage increase.
The increase is 60 − 50 = 10 books. The original 50 books is the base. Therefore the percentage increase is 10/50 × 100% = 20%.
Using 60 as the denominator would answer a different question: what fraction of the final collection the ten additional books represent. It would not calculate the increase relative to the starting collection.
A parent prompt for choosing the base
Ask, “Which quantity is 100% in this sentence?” Have the child label it before finding any percentage.
After the answer is complete, change the question so it refers to the new total rather than the original total. Ask what must change in the working. That small variation checks whether the child understands the reference whole instead of repeating a percentage procedure.
Continue learning
Return to the Mathematics in Punggol study guide.
Related practice: Fractions: Keep Track of the Whole · Compare Choices: Unit Price, Total Cost and Practical Limits.
