A class has 12 boys and 18 girls.
One situation can be written several ways.
- Boys : girls = 12 : 18 = 2 : 3.
- Boys as a fraction of the class = 12/30 = 2/5.
- Boys as a decimal of the whole class = 0.4.
- Boys as a percentage of the whole class = 40%.
These statements are connected.
They are not interchangeable without checking the reference quantity.
Fluency across fractions, decimals, ratios and percentages means preserving the quantity and the comparison while changing the representation.
The difficult part is not moving symbols.
It is knowing what each symbol is comparing.
Singapore’s current Primary 6 syllabus explicitly includes the relationship between fraction and ratio, while percentage remains part of the Number and Algebra strand and decimal knowledge is already established from earlier primary levels. Treating all four forms as one connected proportional system is therefore a valuable integration task, even though “move fluently among four representations” is broader than a single named syllabus line.
The first question is always: compared with what?
If red : blue = 2 : 3, then:
red/blue = 2/3.
But red as a fraction of the total is:
2/(2+3) = 2/5.
As a percentage of the total:
2/5 = 40%.
So a ratio 2:3 does not become 2/3 of the whole.
The denominator 3 refers to the second part.
The denominator 5 refers to the combined whole.
A representation can only be converted correctly after the reference quantity is identified.
Fractions and decimals describe one number in different notation
Take:
3/4.
Divide numerator by denominator:
3 ÷ 4 = 0.75.
So:
3/4 = 0.75.
This is not a change in value.
It is a change in notation.
The fraction emphasises division and part-whole structure.
The decimal emphasises place value in powers of ten.
Decimals and percentages share a base-ten bridge
0.75 means 75 hundredths.
Percentage means per hundred.
Therefore:
0.75 = 75/100 = 75%.
Likewise:
0.08 = 8/100 = 8%.
And:
1.2 = 120/100 = 120%.
This last example is important.
Percentages can exceed 100% because the represented quantity can exceed one reference whole.
Fraction to percentage: choose the most transparent route
For 3/5:
make the denominator 100:
3/5 = 60/100 = 60%.
For 7/20:
7/20 = 35/100 = 35%.
For 3/8, denominator 100 is not reached by multiplying by a whole number.
Use division:
3 ÷ 8 = 0.375.
Then:
0.375 = 37.5%.
Fluency means choosing a route that preserves meaning and reduces unnecessary work.
Percentage to fraction: return to “out of 100”
45% means:
45/100.
Simplify:
45/100 = 9/20.
So:
45% = 9/20.
The simplification step exposes the underlying fraction structure hidden by the standard denominator 100.
Ratio to fraction: decide whether you want part-to-part or part-to-whole
Suppose A : B = 3 : 7.
A compared with B:
A/B = 3/7.
A as a fraction of the total:
3/(3+7) = 3/10.
B as a fraction of the total:
7/10.
As percentages of the total:
A = 30%.
B = 70%.
The original ratio 3:7 describes a part-to-part relationship.
The percentages describe each part relative to the whole.
Fraction to ratio: the target relationship matters
Suppose 2/5 of a class are boys.
Then 3/5 are girls.
Boys : girls = 2 : 3.
Boys : total = 2 : 5.
Girls : total = 3 : 5.
One fraction statement can therefore produce several valid ratios depending on the quantities named.
The “representation square”
For a part that is 3/4 of a whole:
- fraction of whole = 3/4;
- decimal = 0.75;
- percentage = 75%;
- part : whole = 3 : 4;
- part : remainder = 3 : 1.
The last two ratios are different because the comparison changed.
Part : whole uses 3:4.
Part : remainder uses 3:1 because the unselected remainder is 1/4.
This is a useful diagnostic check: if a learner can convert 3/4 to 75% but cannot distinguish 3:4 from 3:1, the conversion skill is stronger than the relational understanding.
Worked example 1: class composition
A class has 16 boys and 24 girls.
Total = 40.
Boys : girls = 16 : 24 = 2 : 3.
Boys as fraction of total:
16/40 = 2/5.
As decimal:
2/5 = 0.4.
As percentage:
40%.
Girls as fraction of total:
3/5 = 0.6 = 60%.
Check:
40% + 60% = 100%.
Every representation is consistent because the same quantities and whole were preserved.
Worked example 2: recipe mixture
Syrup : water = 1 : 4.
Total ratio units = 5.
Syrup as fraction of drink:
1/5.
Decimal:
0.2.
Percentage:
20%.
Water:
4/5 = 0.8 = 80%.
If the final drink is 2.5 litres:
syrup = 20% of 2.5 L = 0.5 L.
water = 2.0 L.
The ratio gave the structure.
The percentage gave a convenient scale factor.
The decimal made multiplication by 2.5 direct.
Worked example 3: choose the most efficient representation
Find 25% of 84.
Possible forms:
- 25%;
- 0.25;
- 25/100;
- 1/4.
The fraction 1/4 is usually the most efficient here.
84 ÷ 4 = 21.
The best representation is not always the one in which the question was written.
Representation fluency is not conversion for its own sake. It is the ability to choose a form that makes the relationship easier to see or calculate.
Worked example 4: a non-benchmark percentage
Find 37.5% of 64.
Convert:
37.5% = 0.375 = 3/8.
Use the fraction:
64 ÷ 8 = 8.
8×3 = 24.
Answer:
24.
The decimal form is correct.
The fraction form is simply more efficient for this number.
Worked example 5: reverse percentage with fractions
30 students represent 60% of a cohort.
What is the whole cohort?
60% = 3/5.
3 ratio-like fraction units = 30.
1 unit = 10.
5 units = 50.
Answer:
50 students.
Converting 60% to 3/5 turns a reverse percentage problem into unitary reasoning.
A conversion chain should preserve value at every step
Take 7/20.
Fraction:
7/20.
Equivalent denominator 100:
35/100.
Decimal:
0.35.
Percentage:
35%.
If this is the fraction of a whole belonging to A, then:
A : whole = 7 : 20.
A : remainder = 7 : 13.
The ratio conversion requires context because ratio names both quantities being compared.
Do not turn every ratio into a percentage automatically
A ratio can compare:
- boys to girls;
- flour to sugar;
- map distance to real distance;
- one quantity to another quantity.
A percentage normally needs a chosen reference quantity treated as 100%.
If a recipe ratio is flour:sugar = 3:2, saying “flour is 150%” is incomplete unless we say 150% of what.
Flour is 150% of the sugar quantity.
But flour is 60% of the combined flour-and-sugar total.
Both can be true because the base changes.
Percentage base errors are representation errors
Suppose a price rises from $80 to $100.
Increase = $20.
Percentage increase uses the original $80 as the base:
20/80 = 1/4 = 25%.
If a learner divides by 100 instead, the result is 20%, which answers a different comparison:
the increase as a fraction of the new value.
The arithmetic may be accurate.
The reference quantity is wrong.
Ratio scale and percentage scale answer different kinds of questions
Ratio is often useful when several parts must be kept in relation.
Percentage is useful when a standard “per hundred” comparison is helpful.
Fraction is useful for exact part-whole and quotient structure.
Decimal is useful for place-value calculation, measurement and calculator-friendly arithmetic.
No form is universally superior.
The problem decides which representation earns its place.
Representation choice in multi-step problems
Suppose boys:girls = 3:5 and 40% of the boys wear glasses.
There are 64 students in total.
Total ratio units:
3 + 5 = 8.
1 unit = 64 ÷ 8 = 8.
Boys = 24.
40% = 2/5.
2/5 of 24 = 9.6, which is impossible if students must be counted as whole people.
This tells us the numerical givens are inconsistent with a whole-student context.
A strong learner should notice that representation conversion also provides a plausibility check.
Moving between representations should make contradictions easier to see, not merely produce more symbols.
Benchmark values worth knowing deeply
- 1/2 = 0.5 = 50%.
- 1/4 = 0.25 = 25%.
- 3/4 = 0.75 = 75%.
- 1/5 = 0.2 = 20%.
- 2/5 = 0.4 = 40%.
- 3/5 = 0.6 = 60%.
- 4/5 = 0.8 = 80%.
- 1/10 = 0.1 = 10%.
- 1/8 = 0.125 = 12.5%.
- 3/8 = 0.375 = 37.5%.
- 5/8 = 0.625 = 62.5%.
- 7/8 = 0.875 = 87.5%.
These should not be memorised as isolated pairs.
They should be understood well enough that each form can reconstruct the others.
Common misconception 1: ratio a:b means a/b of the whole
If A:B = 2:3, A is 2/5 of the total, not 2/3.
Repair: add ratio units when converting from part-to-part to part-to-whole.
Common misconception 2: every decimal-to-percent conversion is just “move the point two places”
The shortcut works because percentage means per hundred.
Repair: occasionally rewrite 0.37 as 37/100 before using the shortcut.
Common misconception 3: 25% and 0.25 are different amounts
They are two notations for the same number.
Repair: place them on the same number line.
Common misconception 4: a percentage cannot exceed 100%
150% = 1.5 = 3/2.
It means one and a half times the reference quantity.
Common misconception 5: always convert into the form requested first
A problem written with percentages may be easier as fractions.
A ratio problem may be easier with part-whole fractions.
Representation is a tool, not a loyalty test.
A diagnostic ladder for representation fluency
- Can the learner convert simple fractions to decimals?
- Can the learner connect decimals with percentages through hundredths?
- Can the learner simplify percentages to fractions?
- Can the learner distinguish part-to-part from part-to-whole ratio?
- Can the learner convert a ratio into fractions of the whole?
- Can the learner convert a fraction of the whole into a part-to-part ratio?
- Can the learner choose an efficient form for calculation?
- Can the learner identify the correct percentage base?
- Can the learner preserve units and whole-number constraints during conversion?
- Can the learner explain why two representations are equivalent rather than merely execute a conversion rule?
A five-minute transfer exercise
Start with:
60%.
Ask the learner to produce:
- a decimal;
- a simplified fraction;
- a part:whole ratio;
- a part:remainder ratio;
- a concrete example using 50 objects.
Expected:
- 0.6;
- 3/5;
- 3:5;
- 3:2;
- 30 selected and 20 remaining.
Then reverse the task.
Start from 3:2 and ask for the fraction and percentage of the total represented by the first part.
That tests relationship understanding rather than one-way conversion memory.
What parents and teachers should listen for
- “I need to know what the denominator refers to.”
- “2:3 means 2/3 compared with the second part, but 2/5 of the total.”
- “I changed 37.5% to 3/8 because the fraction is easier with 64.”
- “The value stayed the same even though the notation changed.”
- “This percentage uses the original quantity as 100%.”
- “The ratio tells me which quantities are being compared, so I cannot convert it without naming them.”
How this fits current Singapore Primary 6 Mathematics
The Primary Mathematics syllabus applicable to Primary 6 from 2026 explicitly includes the relationship between fraction and ratio, equivalent ratios, and Primary 6 percentage work such as finding the whole from a part and percentage and finding percentage increase or decrease. Fraction and decimal knowledge built earlier remains prerequisite.
The title of this article describes a broader integration skill rather than a single official syllabus heading. That distinction matters. A learner may meet the four forms in separate lessons, but difficult problems often require recognising that the same proportional structure can be represented in several ways.
The 2026 PSLE Mathematics assessment objectives include interpretation, application in varied contexts, reasoning and strategy selection. Representation fluency supports all of these because it gives learners more than one route into a problem.
The deeper lesson: representation is a choice about what to make visible
A fraction makes part-whole structure visible.
A decimal makes base-ten place value visible.
A percentage makes a common hundred-part scale visible.
A ratio makes multiplicative comparison between named quantities visible.
The strongest learner is not the one who prefers one form.
It is the one who can change form without changing meaning.
Mathematical fluency is partly the ability to keep the same reality steady while viewing it through a more useful representation.
Final thought
Do not teach fractions, decimals, ratios and percentages as four rooms with locked doors.
Teach the doors.
Then teach the learner to notice which room makes the next piece of reasoning easiest.