Five notebooks cost $17.50.
How much do eight notebooks cost at the same rate?
One route is to find the cost of one notebook first.
$17.50 ÷ 5 = $3.50.
Then rebuild the required quantity:
$3.50 × 8 = $28.
The unitary method reduces a proportional relationship to one unit, then scales from that unit to the quantity required.
This method appears throughout Primary Mathematics even when it is not named explicitly.
It is the logic behind:
- finding one ratio unit;
- finding price per item;
- finding distance per hour;
- finding one percent;
- finding one fractional unit;
- using scale drawings;
- reconstructing totals from one part.
The quick answer: divide to one, multiply to many
The classic unitary pattern is:
- Use division to find the value of one unit.
- Use multiplication to find the required number of units.
But the word “unit” changes by context.
- one notebook;
- one ratio unit;
- one kilogram;
- one hour;
- one percent;
- one fifth;
- one centimetre on a map.
The power of the unitary method comes from naming the unit correctly before dividing.
Unitary Method Type 1: cost per item
Six pens cost $15.
Cost of one pen:
$15 ÷ 6 = $2.50.
Cost of ten pens:
$2.50 × 10 = $25.
The method works because the cost per pen is constant.
If a bulk discount changes the price after six pens, the proportional assumption no longer holds.
This is an important model boundary.
Unitary Method Type 2: ratio units
Red : blue = 3 : 5.
Total = 96.
Total ratio units:
3 + 5 = 8.
One unit:
96 ÷ 8 = 12.
Red:
3×12 = 36.
Blue:
5×12 = 60.
The equal ratio unit is the bridge between symbolic ratio and actual quantity.
Unitary Method Type 3: percentage through 1%
25% of a quantity is 45.
Find the whole.
25% corresponds to 45.
1%:
45 ÷ 25 = 1.8.
100%:
1.8 × 100 = 180.
This method is transparent but not always the fastest.
Since 25% = 1/4, multiplying 45 by 4 gives 180 immediately.
A mature learner should compare methods rather than treat “find 1%” as mandatory.
Unitary Method Type 4: fractions
3/5 of a quantity is 42.
Find the whole.
3 fractional units = 42.
1 fifth = 42 ÷ 3 = 14.
5 fifths = 14 × 5 = 70.
The unit is one fifth of the same whole.
Unitary Method Type 5: rate
A car travels 210 km in 3 hours at constant average rate over the interval.
Distance in one hour:
210 ÷ 3 = 70 km.
Distance in 5 hours at the same rate:
70 × 5 = 350 km.
Here the unitary method is identical to finding a unit rate.
Unitary Method Type 6: recipe scaling
A recipe uses 750 g flour for 6 batches.
Flour per batch:
750 ÷ 6 = 125 g.
For 10 batches:
125×10 = 1,250 g.
The method assumes the recipe scales proportionally.
Unitary Method Type 7: map scale
3 cm on a map represents 12 km in reality.
1 cm represents:
12 ÷ 3 = 4 km.
7.5 cm represents:
4×7.5 = 30 km.
The one-centimetre unit creates the conversion rate.
Unitary Method Type 8: difference in a ratio
A:B = 4:7.
B is 33 more than A.
Difference in ratio units:
7 − 4 = 3 units.
3 units = 33.
1 unit = 11.
A = 44.
B = 77.
The method begins from a known block of units rather than a total.
The unitary method is not always “divide first”
Suppose 1/4 of a quantity is 18.
One unit is already given:
1/4 = 18.
Whole:
18×4 = 72.
No division is needed because the one-unit value is already known.
The unitary method means reason through one unit. It does not mean mechanically divide on the first line of every problem.
Choose the unit carefully
Suppose 4 boxes contain 72 bottles.
One box contains 18 bottles.
But if the question asks about bottles per packet and each box contains 3 packets, “one box” is not the final useful unit.
You may need:
72 bottles ÷ 12 packets = 6 bottles per packet.
Unit selection should match the relationship required by the question.
Direct proportion is the hidden assumption
The unitary method works cleanly when the relationship scales proportionally.
If 5 identical notebooks cost $17.50, ten identical notebooks cost twice as much.
But consider a taxi fare with:
- $4 fixed booking fee;
- $2 per kilometre.
The total fare is not directly proportional to distance because of the fixed charge.
Finding “cost per kilometre” by dividing total fare by distance and scaling may give the wrong result.
This is why the method needs a constant-ratio relationship.
Unitary method versus equivalent-ratio scaling
Suppose 3 items cost $12 and 9 items are required.
Unitary route:
1 item = $4, then 9 items = $36.
Direct scaling route:
9 is 3 times 3, so cost is 3×$12 = $36.
The direct scale-factor method is shorter.
Both are correct.
The unitary route is especially useful when the target is not an obvious whole-number multiple of the known quantity.
Unitary method versus bar model
In ratio problems, a bar model can reveal the equal units.
The unitary method then operates on those units.
Example:
A:B = 3:5, total=96.
Bar model reveals 8 equal units.
Unitary method finds:
1 unit=12.
These methods are complementary rather than competing.
Unitary method versus algebra
Five notebooks cost $17.50.
If c is cost per notebook:
5c = 17.50.
c = 3.50.
Then cost for n notebooks:
3.50n.
The unitary method is the arithmetic form of isolating the per-one multiplier.
Algebra generalises the same relationship.
Worked example: best buy
Pack A: 6 kg for $27.
Pack B: 8 kg for $34.40.
Find unit price.
Pack A:
$27÷6 = $4.50/kg.
Pack B:
$34.40÷8 = $4.30/kg.
Pack B has the lower unit rate.
The unitary method creates a common basis for comparison.
Worked example: worker output
A machine makes 360 parts in 6 hours at a constant rate.
One hour:
360÷6 = 60 parts.
Eight hours:
60×8 = 480 parts.
If the machine slows after six hours, the constant-rate assumption would fail and the problem would need multiple stages instead.
Worked example: percentage reverse problem
18% of a number is 63.
1%:
63÷18 = 3.5.
100%:
3.5×100 = 350.
Check:
18% of 350 = 63.
Worked example: map scale with unit conversion
4 cm represents 10 km.
1 cm represents 2.5 km.
6.8 cm represents:
2.5×6.8 = 17 km.
The unitary method handles the non-integer scaling factor naturally.
Common misconception 1: divide the larger number by the smaller
The division direction is determined by the unit sought, not by number size.
Repair: write “5 notebooks = $17.50; 1 notebook = ?” before dividing.
Common misconception 2: one unit always means one physical object
One unit may mean one ratio unit, one percent, one hour or one fractional part.
Common misconception 3: unitary method works for every changing relationship
Fixed charges, bulk discounts and changing speeds can break direct proportionality.
Repair: test whether the ratio between the quantities stays constant.
Common misconception 4: always reduce to one even when direct scaling is easier
If the target is an obvious multiple, equivalent-ratio scaling may be faster.
Common misconception 5: one-unit value needs no unit label
3.5 could mean dollars per notebook, kilometres per hour or litres per container.
Keep the unit attached.
A unitary-method diagnostic ladder
- Can the learner identify what “one unit” should mean?
- Can the learner divide correctly to find one unit?
- Can the learner scale from one unit to the target?
- Can the learner use the method in ratio problems?
- Can the learner use it with fractions and percentages?
- Can the learner use it to find a unit rate?
- Can the learner preserve units throughout?
- Can the learner identify a non-proportional relationship where the method fails?
- Can the learner choose direct scaling when it is more efficient?
- Can the learner connect the unitary method to bar models and algebra?
How this fits Singapore Primary Mathematics
The updated October 2025 MOE Primary Mathematics syllabus places problem solving, representation, applying and modelling at the centre of the curriculum. The unitary method supports many upper-primary relationships involving ratio, percentage, fractions, rates and scale.
It is best treated as a general problem-solving structure rather than a keyword-triggered trick.
The deeper lesson: one unit exposes the multiplier
Many proportional problems look different because the nouns change.
Underneath, the same relationship survives:
known quantity → value per one unit → required quantity.
The unitary method is powerful because it turns an unfamiliar scale into a familiar question: what does one unit represent?