Ali has 8 marbles.
Ben has 12 marbles.
How do they compare?
Subtraction says Ben has 4 more marbles.
Ratio says something different:
Ali : Ben = 8 : 12 = 2 : 3.
A ratio is a multiplicative comparison. It tells us how quantities scale relative to one another, not merely how far apart they are.
The difference 4 is additive.
The ratio 2:3 is multiplicative.
That distinction becomes one of the most important transitions in upper-primary mathematics.
Under the current Singapore Primary Mathematics syllabus, ratio is explicit Primary 5 content. It builds naturally from multiplication, division, fractions and comparison bar models, but it introduces a new question:
How many units of one quantity correspond to how many units of another?
The quick answer: ratio compares by scale
If red beads : blue beads = 2 : 3, then for every 2 red units there are 3 blue units.
Possible actual quantities include:
- 2 red and 3 blue;
- 4 red and 6 blue;
- 6 red and 9 blue;
- 20 red and 30 blue.
All preserve the same multiplicative relationship.
That is why 2:3, 4:6 and 20:30 are equivalent ratios.
Ratio is not difference
Compare two pairs.
Pair A:
- 4 and 6;
- difference = 2;
- ratio = 2:3.
Pair B:
- 20 and 30;
- difference = 10;
- ratio = 2:3.
The differences are different.
The ratios are the same.
This is the essence of multiplicative comparison.
Ratio preserves relative scale even when absolute size changes.
Part-to-part ratio
A basket contains 8 apples and 12 oranges.
Apples : oranges = 8 : 12 = 2 : 3.
This compares one part with another part.
The total number of fruits is:
8 + 12 = 20.
That total is not written directly in the part-to-part ratio 2:3.
Part-to-whole ratio
Using the same basket:
Apples : total fruit = 8 : 20 = 2 : 5.
Oranges : total fruit = 12 : 20 = 3 : 5.
This is a different comparison.
Part-to-part and part-to-whole ratios may use the same quantities but answer different questions.
This distinction is essential because learners often see the same numbers and assume the ratio must stay the same regardless of what is being compared.
Ratio and fraction are related but not identical
From apples : oranges = 2 : 3, we can say:
apples/oranges = 2/3.
That fraction expresses the ratio of one part to another.
But the fraction of the whole that is apples is:
2/(2+3) = 2/5.
The same ratio gives two different useful fractions depending on the reference quantity.
So ratio and fraction connect through division, but they should not be collapsed into one concept.
Equivalent ratios are multiplicative renamings
2 : 3 = 4 : 6 = 6 : 9.
Why?
Both parts are multiplied by the same scale factor.
2:3 multiplied by 2 becomes 4:6.
2:3 multiplied by 3 becomes 6:9.
The relationship stays the same because the two quantities scale together.
Simplifying a ratio
12 : 18.
Both terms share the factor 6.
Divide both by 6:
12 : 18 = 2 : 3.
This is similar to simplifying a fraction:
12/18 = 2/3.
The factor structure is the same.
Bar models make ratio units visible
Suppose boys : girls = 2 : 3.
Draw:
- boys = 2 equal units;
- girls = 3 equal units.
Total = 5 equal units.
If total students = 35:
5 units = 35.
1 unit = 7.
Boys = 2×7 = 14.
Girls = 3×7 = 21.
The model converts the symbolic ratio into equal multiplicative units.
Finding one quantity from another
Red : blue = 3 : 5.
There are 18 red counters.
Three ratio units = 18.
One unit = 6.
Blue = 5×6 = 30.
This is unitary reasoning applied to ratio.
The learner finds one ratio unit before rebuilding the required amount.
Unit ratio
Sometimes it is useful to express one side as 1.
For example:
6 : 15.
Divide both parts by 6:
1 : 2.5.
This says the second quantity is 2.5 times the first.
At upper-primary level, whole-number ratio forms are often preferred when convenient, but unit ratios become especially important later in rates, scale drawings and proportional reasoning.
Ratio answers “how many times as much?”
Suppose A = 12 and B = 30.
A : B = 12 : 30 = 2 : 5.
Then:
B/A = 30/12 = 2.5.
B is 2.5 times A.
This is very different from saying B is 18 more than A.
Additive and multiplicative comparison answer different questions about the same pair.
Ratio in recipes
A drink uses syrup : water = 1 : 4.
That means every 1 equal unit of syrup corresponds to 4 equal units of water.
If syrup = 250 mL:
1 unit = 250 mL.
Water = 4×250 = 1,000 mL.
Total drink = 1,250 mL.
The part-to-whole fraction of syrup is:
1/(1+4) = 1/5 = 20%.
Ratio, fraction and percentage now describe the same mixture from different perspectives.
Ratio in scale drawings
A map scale of 1 : 50,000 compares map distance with actual distance in the same unit.
1 cm on the map corresponds to 50,000 cm in reality.
50,000 cm = 500 m.
So 1 cm represents 500 m.
The ratio is multiplicative: every map length scales by the same factor to become real-world length.
Ratio can compare unlike units later
Strict ratio work often compares quantities in the same unit.
Later, related multiplicative comparisons become rates:
- 60 km in 1 hour;
- $12 for 3 kg;
- 80 words per minute.
The key transition is that rate compares quantities with different units, while a pure ratio often compares like quantities or dimensionless scale relationships.
This boundary matters because ratio and rate are connected but not identical.
Worked example: total given
Red : blue = 4 : 7.
Total counters = 55.
Total ratio units:
4 + 7 = 11 units.
One unit:
55 ÷ 11 = 5.
Red:
4×5 = 20.
Blue:
7×5 = 35.
Check:
20 + 35 = 55.
Worked example: difference given
A : B = 3 : 5.
B is 18 more than A.
Difference in ratio units:
5 − 3 = 2 units.
2 units = 18.
1 unit = 9.
A = 3×9 = 27.
B = 5×9 = 45.
Check:
45 − 27 = 18.
This is where ratio and additive difference meet in one model.
Worked example: one part given
Cats : dogs = 2 : 7.
There are 16 cats.
2 units = 16.
1 unit = 8.
Dogs = 7×8 = 56.
Common misconception 1: 2:3 means difference 1
The difference between ratio terms is not the actual difference unless the size of one ratio unit is known.
4:6 and 20:30 both simplify to 2:3 but have differences 2 and 10.
Repair: distinguish ratio units from actual quantity units.
Common misconception 2: part-to-part ratio is automatically the fraction of the whole
If red:blue = 2:3, red is not 2/3 of the total.
Total units = 5.
Red fraction of whole = 2/5.
Common misconception 3: add the same number to both sides to make an equivalent ratio
2:3 is not equivalent to 4:5.
Equivalent ratios require multiplying or dividing both terms by the same non-zero factor.
Adding the same amount preserves difference, not ratio.
Common misconception 4: ratio order does not matter
2:3 and 3:2 describe inverse comparisons.
The order must match the named quantities.
Common misconception 5: ratio units are physical objects
A ratio unit is a relational scale unit.
If 2 ratio units correspond to 16 cats, one ratio unit represents 8 cats in that problem.
In another problem, one ratio unit may represent 5 litres or $12.
A diagnostic ladder for ratio
- Can the learner distinguish additive difference from multiplicative ratio?
- Can the learner read A:B in the correct order?
- Can the learner simplify a ratio using common factors?
- Can the learner generate equivalent ratios by scaling both terms?
- Can the learner distinguish part-to-part from part-to-whole ratio?
- Can the learner connect ratio to fractions without confusing the reference whole?
- Can the learner represent a ratio with equal bar units?
- Can the learner find one actual unit from a known part, total or difference?
- Can the learner solve for an unknown quantity?
- Can the learner distinguish ratio from rate in a context with unlike units?
A five-minute home investigation
Use counters in two colours.
Build 2 red and 3 blue.
Then double both:
4 red and 6 blue.
Ask:
- What changed?
- What stayed the same?
- What is the red:blue ratio?
- What fraction of the total is red?
- What is the difference?
Then add one counter to each colour.
Ask whether the ratio stayed the same.
This exposes the contrast between multiplicative equivalence and additive change.
What parents should listen for
- “The ratio tells me how the quantities scale, not how many more there are.”
- “2:3 and 4:6 are equivalent because both parts doubled.”
- “If red:blue is 2:3, red is 2/5 of the total, not 2/3.”
- “I found one ratio unit before finding the unknown amount.”
- “Adding the same number preserves difference, not ratio.”
How this fits Singapore Primary 5 Mathematics
The updated October 2025 MOE Primary Mathematics syllabus places ratio in Primary 5 alongside percentage, fractions and decimals. The official syllabus remains the authoritative source for exact level-specific problem forms.
The mathematical dependency is clear: ratio requires multiplicative thinking, factors, fractions and unitary reasoning, and later becomes a foundation for rate, scale, proportion and algebra.
The deeper lesson: ratio measures relationship, not size
2:3 can describe tiny quantities or enormous ones.
The absolute numbers may change.
The multiplicative relationship stays fixed when both quantities scale together.
Difference tells us how far apart two quantities are. Ratio tells us how they are scaled relative to each other.
Final thought
Ratio is the moment comparison becomes multiplicative.
That shift changes how learners think about mixtures, maps, rates, percentages, scale factors and later algebra.
Once one ratio unit has meaning, the whole relationship can be rebuilt.