BTT Mathematics / Primary Mathematics Learning Hub / Unitary Method
The unitary method reduces a known quantity to one equal unit, then rebuilds the quantity required. It is useful when a problem describes several equal groups, ratio units, a fraction of a whole, a repeated rate or a price for several identical items.
If five notebooks cost fifteen dollars, one notebook costs fifteen divided by five, or three dollars. Eight notebooks at the same unit price cost eight times three, or twenty-four dollars. The first step finds one unit. The second scales that unit.
This method is powerful because it makes the reference unit explicit. But it only works when the units being compared are genuinely equal under the stated conditions. A packet of six apples and a packet of eight apples are not interchangeable units. A constant speed or constant price per item must be stated or reasonably built into the mathematical model.
This guide develops equal groups, ratio units, fractions of quantities, rates, prices, measurement and reverse unitary reasoning. It complements Ratio, Rate and Percentage and Fractions of a Quantity, Equivalence and Operations by concentrating on the “find one, then scale” structure.
Use the learner’s current school scope and the MOE Primary curriculum and syllabus page as the official reference point. The explanations and questions below are original teaching material, not official examination items.
Find one unit · Ratio units · Fraction units · Rates and prices · Reverse unitary reasoning · When the method does not apply · 24 questions · Worked solutions
1. One unit is a quantity with a name
Suppose seven identical boxes contain fifty-six counters altogether. One box contains 56 ÷ 7 = 8 counters. Three such boxes contain 3 × 8 = 24 counters.
The number eight is not just “the answer after division.” It means eight counters per box. Keeping that label attached prevents the next multiplication from becoming mechanical.
Worked example A: Equal groups
Nine trays hold seventy-two buns equally. One tray holds 72 ÷ 9 = 8 buns. Five trays hold 5 × 8 = 40 buns.
Worked example B: Same group size, different group count
Four packets contain twenty-eight stickers equally. One packet contains seven. Eleven packets of the same size contain 77 stickers.
Worked example C: Find the number of units instead
One carton holds six bottles. Forty-two bottles fill 42 ÷ 6 = 7 cartons. Here the one-unit quantity is already known, so the final operation counts how many equal units fit into the total.
Do not confuse unit value and unit count
If five bags contain forty marbles, eight is the number of marbles in one bag. It is not the number of bags. If the question later asks how many bags are needed for eighty marbles, calculate 80 ÷ 8 = ten bags.
Write “8 marbles per bag” beside the one-unit step. Units act as a check on whether multiplication or division is producing the right kind of quantity.
2. A ratio divides quantities into equal comparison units
In the ratio 3:5, the first quantity contains three equal ratio units and the second contains five equal ratio units. The unit is not automatically one object. Its actual size depends on the known quantity in the problem.
Worked example D: One part is known
Red to blue counters are in the ratio 3:5. There are twenty-five blue counters. Five units correspond to twenty-five, so one unit is five counters. Red contains three units, giving 15 red counters.
Worked example E: Total is known
Two quantities are in ratio 2:7 and total eighty-one. The total contains nine ratio units. One unit is 81 ÷ 9 = 9. The quantities are 18 and 63.
Worked example F: Difference is known
A:B = 7:4 and A exceeds B by twenty-one. The difference is three ratio units, so one unit is 21 ÷ 3 = 7. A=49 and B=28.
The same printed ratio can lead to different one-unit calculations depending on whether the known amount represents one side, the total or the difference. Identify what the known quantity labels before dividing.
Equivalent ratios preserve the comparison
3:5 and 6:10 describe the same proportion. Their printed “one units” are not meant to be equal across the two forms; the second representation simply uses twice as many smaller-labelled parts. Simplify or scale ratios as needed, but keep the actual quantities fixed.
3. A fraction of a quantity is also an equal-unit problem
To find three-sevenths of fifty-six, divide the whole into seven equal parts. One seventh is 56 ÷ 7 = 8. Three sevenths is 3 × 8 = 24.
Worked example G: Fraction of a set
Five-eighths of forty-eight pupils choose an activity. One eighth is six pupils, so five eighths is 30 pupils.
Worked example H: Reverse fraction problem
Three-fifths of a collection is twenty-four. One fifth is 24 ÷ 3 = 8. Five fifths is 40.
Notice that the first division uses the number of known fractional units, not the denominator automatically. Twenty-four represents three units, so divide by three.
Worked example I: The remainder is known
Two-sevenths of a collection is removed and thirty-five remain. The remainder is five-sevenths. Five units correspond to thirty-five, so one unit is seven and the original whole is 49.
Before using a unitary step, convert the wording into the fraction that the known amount represents. “Removed” and “left” refer to different parts of the whole.
Mixed numbers can still be units
If four identical lengths total ten metres, one length is 10 ÷ 4 = 2.5 metres. The method does not require every unit value to be a whole number. However, a count of people, chairs or packets normally must be whole unless the context explicitly allows fractions.
4. Rates compare unlike quantities through a common unit
A rate such as dollars per kilogram or kilometres per hour gives the amount of one quantity for each unit of another. The unitary method can recover that rate from a known pair, then scale it.
Worked example J: Unit price
Six identical pens cost eighteen dollars. One pen costs three dollars. Ten pens at the same price cost $30.
Worked example K: Price per kilogram
Four kilograms of rice cost twenty-eight dollars. One kilogram costs seven dollars. Seven kilograms cost $49, assuming the stated constant unit price.
Worked example L: Constant speed
A vehicle covers 180 km in three hours at a constant average rate for the model. One hour corresponds to sixty kilometres. Five hours at the same rate corresponds to 300 km.
This does not claim a real vehicle always travels at exactly the same speed. The calculation follows the constant-rate condition of the mathematical problem.
Worked example M: Production rate
Eight identical machines produce four hundred parts in one hour, with equal output under the stated model. One machine produces fifty parts per hour. Twelve such machines would produce 600 parts per hour.
If machines have different speeds or interfere with one another, the equal-unit model would need to be changed. The word identical is doing mathematical work.
Worked example N: Recipe scaling
A recipe for six equal portions uses 450 g of flour. One portion corresponds to 75 g. Ten equal portions require 750 g, assuming the recipe scales proportionally.
Real cooking can involve practical non-linearities, but the school-style mathematical model treats the stated quantities proportionally.
5. Reverse unitary reasoning reconstructs the number or size of units
Worked example O: Find how many items
Five identical notebooks cost twenty dollars. One notebook costs four dollars. A budget of thirty-six dollars buys 9 notebooks at that same exact unit price, because 36 ÷ 4 = 9.
Worked example P: Find a missing total
Seven ninths of a quantity is sixty-three. One ninth is 63 ÷ 7 = 9. The whole is nine ninths, so 9 × 9 = 81.
Worked example Q: Find the number of groups from a total
Four boxes contain twenty-six kilograms equally, so one box contains 6.5 kg. A total of fifty-two kilograms fills 8 boxes of the same size.
Worked example R: Two-stage scale
Three metres of fabric cost twenty-four dollars. One metre costs eight dollars. A roll costing seventy-two dollars contains 9 metres at the same unit price.
The same one-unit value supports both directions: quantity to cost by multiplication, and cost to quantity by division.
Check with a ratio or product
If five items cost twenty and nine cost thirty-six, the costs per item are both four. Alternatively, 5:9 should match 20:36, which simplifies to 5:9. A second representation can expose a scale error.
6. The unitary method fails when the units are not equal or the relationship is not proportional
Suppose a taxi journey has a fixed booking charge plus a distance charge. If five kilometres costs fourteen dollars and ten kilometres costs twenty-two dollars, doubling the distance does not double the total fare. Dividing fourteen by five and multiplying by ten would wrongly spread the fixed fee across distance.
Fixed charges break simple proportionality
When a total contains a one-off fee, remove or model that fee separately before applying equal-unit reasoning to the proportional part. This is similar to the fixed-charge examples in the Assumption Method guide.
Bulk discounts can change the unit price
If one shop charges three dollars per item for the first five and two dollars per item thereafter, the unit price is not constant across all items. A one-unit value found from the first five cannot simply be scaled to twenty items.
Average rate is not always a constant rate
A journey covering sixty kilometres in one hour and another sixty in two hours has a whole-journey average speed of forty kilometres per hour. It does not mean every hour contains forty kilometres. Use the Speed, Distance and Time guide when stages have different rates.
Equal units must refer to the same whole or base
Twenty per cent of eighty and twenty per cent of one hundred are both “twenty per cent,” but their actual values are sixteen and twenty. The percentage unit only becomes an actual amount after the base is known.
The unitary method is therefore not “always divide then multiply.” It is a way to reason about equal units after the structure and reference quantity have been identified correctly.
7. Practice: 24 questions
For each question, write what one unit represents before scaling. Keep the worked solutions covered until the first attempt is complete.
Questions 1–8: Equal groups and ratio units
1. Seven identical bags contain fifty-six marbles. How many marbles are in four bags?
2. Nine trays hold seventy-two buns equally. How many buns are in six trays?
3. Red:Blue = 3:5. There are forty blue counters. Find the number of red counters.
4. A:B = 2:7 and A+B=72. Find A and B.
5. P:Q = 8:3 and P−Q=35. Find P and Q.
6. Twelve identical boxes contain one hundred eight books. How many boxes contain sixty-three books?
7. Five packets contain thirty-five cards equally. How many cards are in thirteen such packets?
8. A:B = 4:9 and B=63. Find A and the total A+B.
Questions 9–16: Fractions, rates and prices
9. Find 5/8 of 64.
10. Three-sevenths of a quantity is thirty. Find the whole quantity.
11. Two-fifths of a collection is removed and forty-two remain. Find the original collection.
12. Six pens cost $21 at a constant unit price. Find the cost of fourteen pens.
13. Four kilograms of fruit cost $18 at a constant unit price. Find the cost of ten kilograms.
14. A cyclist covers ninety-six kilometres in four hours at a constant rate. How far is covered in seven hours at the same rate?
15. Eight equal portions use 600 g of rice. How much rice is used for twelve equal portions under proportional scaling?
16. Five identical machines make 350 parts in one hour. How many parts would eight such machines make in one hour under equal output?
Questions 17–24: Reverse reasoning and method limits
17. Five notebooks cost $20. How many notebooks cost $52 at the same unit price?
18. Seven ninths of a number is seventy. Find the number.
19. Six identical containers hold 27 L altogether. How many such containers hold 45 L?
20. Three metres of fabric cost $25.50 at a constant unit price. How much do eight metres cost?
21. A service charges a fixed $6 fee plus $2 per unit used. Explain why dividing the total cost for four units by four does not give the variable cost per unit.
22. A store charges $4 per item for the first five items and $3 per item after that. Can the cost of ten items be found by taking the cost of one of the first five items and multiplying by ten? Explain.
23. Twenty per cent of one quantity is eighteen. Find the quantity using equal percentage units.
24. A quantity is in ratio 5:8 with another. The second quantity is increased, but the first stays unchanged and the new ratio is 5:10. If the increase is fourteen, find both original quantities.
8. Worked solutions
Solutions 1–8
1. 32 marbles. One bag contains 56 ÷ 7 = 8 marbles. Four bags contain 4 × 8 = 32.
2. 48 buns. One tray contains 72 ÷ 9 = 8. Six trays contain 48.
3. 24 red counters. Five ratio units are forty, so one unit is eight. Three units are twenty-four.
4. A=16, B=56. Nine total ratio units represent seventy-two, so one unit is eight.
5. P=56, Q=21. The difference is five units and equals thirty-five, so one unit is seven.
6. 7 boxes. One box contains 108 ÷ 12 = 9 books. Sixty-three books require 63 ÷ 9 = 7 boxes.
7. 91 cards. One packet contains seven cards. Thirteen packets contain 91.
8. A=28; total 91. Nine units represent sixty-three, so one unit is seven. Four units give twenty-eight.
Solutions 9–16
9. 40. One eighth of sixty-four is eight; five eighths is forty.
10. 70. Three units represent thirty, so one seventh-unit is ten. Seven units make seventy.
11. 70. Three-fifths remains and equals forty-two. One fifth is fourteen; five fifths are seventy.
12. $49. One pen costs $21 ÷ 6 = $3.50. Fourteen cost $49.
13. $45. One kilogram costs $4.50. Ten kilograms cost $45.
14. 168 km. One hour corresponds to 24 km. Seven hours correspond to 168 km at the same stated rate.
15. 900 g. One portion corresponds to 75 g. Twelve portions require 900 g.
16. 560 parts. One machine makes seventy parts per hour. Eight make 560 under the equal-output assumption.
Solutions 17–24
17. 13 notebooks. One notebook costs four dollars. Fifty-two dollars buys thirteen at that unit price.
18. 90. Seven ninth-units total seventy, so one unit is ten and nine units are ninety.
19. 10 containers. One container holds 4.5 L. Forty-five litres require ten containers.
20. $68. One metre costs $25.50 ÷ 3 = $8.50. Eight metres cost $68.
21. The total for four units is $6 + 4×$2 = $14. Dividing fourteen by four gives an average cost of $3.50 per unit that includes a share of the fixed fee. The variable charge remains $2 per unit.
22. No. The pricing rule changes after five items. Ten items cost five at four dollars plus five at three dollars, or $35. Multiplying the first-tier unit price by ten would give $40 and ignore the discount.
23. 90. Twenty per cent contains two ten-per-cent units. If two units equal eighteen, one ten-per-cent unit equals nine. Ten such units make ninety.
24. Original quantities 35 and 56. The first quantity is unchanged and occupies five units in both ratios, so align the unit scale. The second rises from eight to ten units, an increase of two units representing fourteen. One unit is seven. Original amounts are thirty-five and fifty-six.
9. Teaching and diagnosis
If a learner divides correctly but then scales the wrong quantity, ask them to state what the one-unit result measures. “Seven” is too vague; “seven dollars per kilogram” or “seven counters per ratio unit” is usable information.
When the wrong divisor is used
Return to the known amount and ask how many equal units it represents. If three-fifths equals twenty-four, the number twenty-four represents three units, so divide by three. The denominator tells the number of units in the whole, not automatically the number represented by the given part.
When proportionality is assumed without evidence
Present one problem with a constant unit price and another with a fixed fee. Ask whether doubling the quantity must double the total. The contrast helps the learner identify the structural condition rather than applying a memorised divide-then-multiply routine.
When a ratio unit changes across stages
Use the Invariants and Before-and-After guide. A simplified ratio unit before a change need not have the same actual size afterwards unless an unchanged quantity justifies the alignment.
Use an independent return check
After finding a unit value, rebuild the known quantity as well as the requested quantity. If one box is eight counters, seven boxes must reconstruct fifty-six. If that return fails, the one-unit step or its label is wrong.
Continue through the structural-reasoning collection
For two-type replacement problems, use The Assumption Method and Guess-and-Check. For cases described by too much or too little under two plans, continue to Excess and Shortage Problems. For several conditions acting together, use Simultaneous Conditions and Multiple Constraints.
Return to the BTT Primary Mathematics Learning Hub.
Original explanations and 24 original practice questions with separate worked solutions. Prices, rates and production examples are hypothetical mathematical models. No examination prediction or performance guarantee is implied.