Two trays contain red and blue counters in the ratio 2 : 3.
One tray has 4 red and 6 blue counters.
Another has 20 red and 30 blue counters.
The numbers are different.
The relationship is the same.
Equivalent ratios are different numerical descriptions of one unchanged multiplicative relationship.
That idea is more important than the rule “multiply or divide both sides by the same number”.
The rule works because both quantities are being scaled by the same factor.
When the scale changes equally, the relationship survives.
Under Singapore’s current Primary Mathematics syllabus, equivalent ratios are explicit Primary 6 content. The same syllabus also includes simplifying ratios, finding missing terms in pairs of equivalent ratios, dividing quantities in a given ratio and connecting ratio with fraction. Those are not separate tricks. They are all consequences of the same scaling structure.
The quick answer: equivalent ratios keep the multiplier between quantities unchanged
Take:
2 : 3.
This says that whenever the first quantity contains 2 equal ratio units, the second contains 3 equal ratio units.
Multiply both terms by 4:
8 : 12.
Multiply both by 10:
20 : 30.
Every pair says the same thing:
the second quantity is 3/2 times the first.
Or:
the first is 2/3 of the second.
The ratio notation changes size.
The multiplicative comparison does not.
Scaling is not the same as adding
A common mistake is to think:
2 : 3 → 4 : 5
because 2 was added to both terms.
But 2:3 and 4:5 are not equivalent.
Compare their fractions:
2/3 ≈ 0.667.
4/5 = 0.8.
The relationship changed.
Adding the same amount preserves difference.
Multiplying both quantities by the same factor preserves ratio.
Equal addition is an additive invariant. Equal scaling is a multiplicative invariant.
This distinction becomes increasingly important in upper-primary mathematics because before-and-after problems often contain both additive changes and multiplicative relationships.
A ratio table makes equivalent scaling visible
Suppose flour : water = 3 : 2.
A ratio table can be read like this:
- 3 : 2
- 6 : 4
- 9 : 6
- 12 : 8
- 15 : 10
Each row is generated by multiplying both terms by the same whole-number scale factor.
The table is useful because it separates two jobs:
- the ratio unit structure, which stays fixed;
- the scale factor, which changes the actual quantities.
This is the same logic used later in direct proportion, scale drawings, rates and similar figures.
Bar models show why scaling both terms works
For 2 : 3, draw:
- first quantity = 2 equal units;
- second quantity = 3 equal units.
If one ratio unit is worth 5 counters:
first quantity = 2×5 = 10.
second quantity = 3×5 = 15.
So 2:3 and 10:15 are equivalent.
If one unit becomes 20 counters:
the quantities become 40 and 60.
The ratio is still 2:3.
The bar model reveals an important idea:
the ratio terms are counts of equal relational units, not necessarily the actual quantities themselves.
Simplifying a ratio is reverse scaling
Take:
18 : 24.
Both terms share a factor of 6.
Divide both by 6:
18 : 24 = 3 : 4.
This is not changing the relationship.
It is expressing the relationship using smaller ratio units.
In effect:
18 : 24
means 6 groups of 3 compared with 6 groups of 4.
Remove the common scale factor 6 and the core relationship is 3:4.
Why common factors matter
To simplify 42 : 63:
the highest common factor is 21.
42 ÷ 21 = 2.
63 ÷ 21 = 3.
So:
42 : 63 = 2 : 3.
Using the highest common factor is efficient because it removes the entire common scale in one step.
But repeated division by smaller common factors also works because each step preserves the ratio.
The mathematics is not “find HCF because ratios say so”.
The mathematics is:
remove a common multiplicative scale without changing the comparison.
Equivalent ratios and equivalent fractions share a mechanism
Consider:
2 : 3.
Written as a quotient:
2/3.
Scale both numerator and denominator by 4:
8/12.
The fraction is equivalent because:
2/3 = 8/12.
The ratio is equivalent because:
2:3 = 8:12.
Both ideas rely on one multiplicative transformation applied to both parts of a relationship.
This connection is explicitly useful in Primary 6 because the current syllabus includes the relationship between fraction and ratio.
But ratio and fraction do not always answer the same question
If boys : girls = 2 : 3, then:
boys/girls = 2/3.
But boys as a fraction of the whole class is:
2/(2+3) = 2/5.
The ratio compares part with part.
The fraction 2/5 compares one part with the whole.
Equivalent-ratio work must therefore keep the comparison target visible.
Finding a missing term in equivalent ratios
Suppose:
3 : 5 = 12 : ?
Find the scale factor:
3 × 4 = 12.
Apply the same factor to the corresponding second term:
5 × 4 = 20.
So:
3 : 5 = 12 : 20.
The critical word is corresponding.
A learner must match first term with first term and second with second.
When the scale factor is not obvious
Suppose:
8 : 12 = 14 : ?
The scale factor from 8 to 14 is 14/8 = 7/4.
Apply it to 12:
12 × 7/4 = 21.
So the missing term is 21.
At Primary 6, whole-number ratio terms are central, but the underlying scaling factor can still be understood through division even if it is not a whole number.
Another efficient route is to simplify first:
8 : 12 = 2 : 3.
If 2 units correspond to 14, one unit is 7.
Then 3 units = 21.
Unitary reasoning and equivalent ratios
Suppose 5 notebooks cost $20.
Quantity : cost = 5 : 20.
Divide both by 5:
1 : 4.
One notebook costs $4.
For 8 notebooks:
8 : 32.
The unitary method and equivalent-ratio method are the same proportional structure viewed at different scales.
Equivalent ratios can be checked by cross-products—but understand what the check means
To test whether 4:6 and 10:15 are equivalent, compare:
4×15 = 60.
6×10 = 60.
The cross-products are equal.
Why does this work?
Because equivalent ratios satisfy:
4/6 = 10/15.
Multiplying both sides by 6×15 produces:
4×15 = 10×6.
Cross multiplication is therefore a consequence of equal fractions, not a mysterious test.
For primary learners, scaling and unitary reasoning are usually more transparent starting points.
Worked example: mixture scaling
A drink uses syrup : water = 2 : 7.
If 8 cups of syrup are used, how much water is needed?
2 ratio units = 8 cups.
1 unit = 4 cups.
7 units = 28 cups.
Equivalent ratio:
2 : 7 = 8 : 28.
Answer: 28 cups of water.
Check the multiplier:
8/2 = 4.
28/7 = 4.
Both quantities used the same scale factor.
Worked example: total given
Red : blue = 3 : 5.
Total = 64 counters.
Total ratio units:
3 + 5 = 8.
One unit:
64 ÷ 8 = 8.
Red:
3×8 = 24.
Blue:
5×8 = 40.
The actual pair 24:40 is equivalent to 3:5.
Worked example: difference given
A : B = 4 : 7.
B exceeds A by 18.
Difference in ratio units:
7 − 4 = 3 units.
3 units = 18.
1 unit = 6.
A = 24.
B = 42.
24:42 simplifies to 4:7.
The difference information identifies the scale factor.
Three-term ratios use the same logic
For:
2 : 3 : 5,
multiply all three terms by the same scale factor.
×4 gives:
8 : 12 : 20.
It is not enough to scale only two terms.
The shared relationship includes all three quantities.
Common misconception 1: equivalent means the terms are equal
2:3 is not made of equal terms.
Equivalent means two ratios represent the same relationship.
Repair: compare the scale factor or quotient rather than looking for matching terms.
Common misconception 2: add the same number to both terms
2:3 → 4:5 preserves a difference of 1 but changes the multiplicative relationship.
Repair: ask whether both terms were multiplied or divided by the same factor.
Common misconception 3: simplify each term independently
Both terms must be divided by the same common factor.
18:24 cannot become 9:8 by halving 18 and dividing 24 by 3.
Different scale factors destroy equivalence.
Common misconception 4: reverse the order while simplifying
2:3 and 3:2 are not equivalent.
They compare the same quantities in opposite directions.
Ratio order carries meaning.
Common misconception 5: the ratio terms are always the actual quantities
In a 2:3 ratio, actual quantities may be 10 and 15, 40 and 60, or many other equivalent pairs.
The ratio terms describe relative units, not one fixed scale.
A diagnostic ladder for equivalent ratios
- Can the learner explain ratio as multiplicative comparison?
- Can the learner generate a simple equivalent ratio by doubling or tripling both terms?
- Can the learner explain why adding the same amount does not preserve ratio?
- Can the learner simplify a ratio using common factors?
- Can the learner find a missing term using a scale factor?
- Can the learner use a bar model or ratio table to make scaling visible?
- Can the learner connect equivalent ratios with equivalent fractions?
- Can the learner solve total-given and difference-given ratio problems?
- Can the learner preserve a three-term ratio under scaling?
- Can the learner identify the actual comparison before writing a ratio?
A five-minute home investigation
Use two colours of counters.
Build 2 red and 3 blue.
Then build:
- 4 red and 6 blue;
- 6 red and 9 blue;
- 8 red and 12 blue.
Ask what changed.
Then add one counter to each colour and ask whether the ratio remains equivalent.
This small experiment reveals the difference between additive and multiplicative change.
What parents and teachers should listen for
- “Both quantities were multiplied by the same scale factor.”
- “The actual numbers changed but the multiplicative relationship stayed the same.”
- “Adding the same number keeps the difference, not the ratio.”
- “I simplified by removing a common factor from both terms.”
- “I know these ratios are equivalent because corresponding terms use the same multiplier.”
- “The ratio is part-to-part, while the fraction of the total uses all the ratio units.”
How this fits Singapore Primary 6 Mathematics
Singapore’s current Primary Mathematics syllabus, applicable to Primary 6 from 2026, explicitly includes equivalent ratios, dividing a quantity in a given ratio, expressing ratios in simplest form, finding ratios of given quantities, finding a missing term in a pair of equivalent ratios, and the relationship between fraction and ratio.
The 2026 PSLE Mathematics syllabus assesses recall and computation, application in varied contexts, and mathematical reasoning and strategy selection. Equivalent-ratio work therefore matters not only as a procedure but as a relationship students must recognise when a problem changes surface.
This article stays within that current conceptual frame while also making the underlying scaling logic explicit so the same idea can transfer later into proportion, scale, rate and algebra.
The deeper lesson: scale can change while structure stays fixed
Equivalent ratios are one of the clearest examples of an invariant in school mathematics.
Quantities can become larger.
They can become smaller.
The ratio can still remain unchanged.
What survives is the multiplier between the quantities.
Equivalent ratios teach a powerful mathematical habit: separate the size of a system from the relationship that organises it.
Final thought
When a learner sees 2:3 and 20:30, the important observation is not that a zero appeared.
It is that both quantities were scaled together.
Once that becomes automatic, equivalent ratios stop being a special chapter and become a general way to recognise preserved multiplicative structure.