PSLE Mathematics Learning Guide · Guide 58 · Editorial Review Companion to Guide 20
Return to the PSLE Learning Guide · Parent guide: Unitary Method and Rate
A rate is not simply “another number in the question”. It tells how much of one quantity belongs to one unit of another quantity.
$4 per notebook. 6 litres per container. 18 pages per minute. 250 grams per packet. Every example has the same deep structure: one quantity is attached to one unit of something else.
The current Primary 5 Standard Mathematics syllabus explicitly defines rate as the amount of a quantity per unit of another quantity and includes finding the rate, total amount or number of units given the other two quantities.
This page isolates that three-state system. Public practice sites often give mixed rate worksheets; eduKate’s missing learning job is to teach the learner to identify which of the three quantities is unknown before choosing multiplication or division.
Total amount = rate × number of units.
From that one relationship:
Rate = total amount ÷ number of units.
Number of units = total amount ÷ rate.
The formulas are not three unrelated facts. They are three views of the same relationship.
Rate language contains the word “per” even when “per” is not written
A machine fills 8 bottles every minute.
Rate = 8 bottles per minute.
A packet contains 250 g of rice.
Rate-like structure = 250 g per packet.
A ticket costs $6.
Cost rate = $6 per ticket.
The second unit can be implied by the context. Ask: “How much for one what?”
The unitary method is the bridge to rate
Five identical notebooks cost $20.
Cost per notebook = $20 ÷ 5 = $4 per notebook.
Once the one-unit value is known, any number of notebooks can be priced.
This is exactly the unitary method developed in Guide 20. The present guide extends that idea into a complete three-quantity control system.
Case 1: rate and number of units are known
A printer produces 24 pages per minute for 7 minutes.
Total pages = 24 × 7 = 168 pages.
Unit check:
(pages/minute) × minutes = pages.
The time unit cancels conceptually, leaving the requested total quantity.
Case 2: total and number of units are known
Six boxes contain 2700 g altogether. Each box contains the same mass.
Rate = 2700 ÷ 6 = 450 g per box.
Do not report 450 g as a total. The phrase “per box” belongs to the answer.
Case 3: total and rate are known
A tank dispenses 12 L per container. There are 96 L available.
Number of full containers = 96 ÷ 12 = 8 containers.
Unit check:
litres ÷ (litres/container) = containers.
Use a state table before choosing the operation
| Rate | Number of units | Total amount |
|---|---|---|
| $5 per book | 8 books | $40 |
| 300 g per packet | 7 packets | 2100 g |
| 15 L per container | ? | 120 L |
In the last row, the missing state is number of units. Divide total by rate:
120 ÷ 15 = 8 containers.
The table prevents the learner from choosing multiplication merely because two numbers appear.
Do not confuse the one-unit amount with the total
“A tray holds 18 cups. There are 6 trays.”
18 is not the total number of cups. It is the amount per tray.
Total = 18×6 = 108 cups.
A learner who reports 18 has read the rate correctly but stopped before scaling it to all six units.
Do not divide when the given number is already a rate
“A machine packs 40 items per minute for 9 minutes.”
40 already describes one minute. Dividing 40 by 9 would answer no meaningful quantity in the story.
The 9 tells how many rate-units occur. Multiply.
A constant rate means the per-unit amount stays fixed
If a printer works at 25 pages per minute for 3 minutes, it prints 75 pages.
If the same rate continues for 8 minutes, it prints 200 pages.
The total changes because the number of time-units changes. The rate remains 25 pages/min.
Do not assume constancy unless the problem states or clearly implies a uniform rate.
If the rate changes, split the problem into states
A machine produces 20 items/min for 4 minutes, then 30 items/min for 3 minutes.
First stage: 20×4 = 80 items.
Second stage: 30×3 = 90 items.
Total = 170 items.
There is no single constant rate across the whole process unless an average rate is separately requested.
Average rate is total amount ÷ total number of units
Using the previous example:
Total output = 170 items.
Total time = 7 minutes.
Average output rate = 170 ÷ 7 = 24 2/7 items per minute.
This is an extension of the rate relationship. It is not the simple mean of 20 and 30 because the two rates operated for different durations.
Make the rate unit match the number-of-units quantity
A tap fills 6 L per minute. It runs for 2 hours.
Do not multiply 6 by 2 immediately because the rate is per minute while the duration is in hours.
2 hours = 120 minutes.
Total = 6×120 = 720 L.
Guide 2’s rule applies: units must agree before calculation.
The number of units can be fractional
A material costs $8 per metre. A customer buys 2.5 m.
Total cost = 8×2.5 = $20.
The unit count does not have to be a whole number when the measured quantity can be divided continuously.
The rate itself can also be fractional or decimal
A dispenser releases 1.5 L per cycle. After 8 cycles:
Total = 1.5×8 = 12 L.
The three-state relationship does not depend on rate being a whole number.
Unit price is a rate
Three pens cost $7.50 altogether.
Unit price = 7.50÷3 = $2.50 per pen.
Nine pens at the same unit price cost:
2.50×9 = $22.50.
Unit price problems are rate problems wearing money language.
Mass per packet is a rate
Eight equal packets have total mass 3.2 kg.
Mass per packet = 3.2÷8 = 0.4 kg per packet = 400 g per packet.
Guide 44 helps if the main difficulty is kg/g conversion.
Capacity per container is a rate structure
72 L is poured equally into 9 containers.
72÷9 = 8 L per container.
If 15 such containers are filled at the same rate, total volume = 8×15 = 120 L.
A changed total can reveal a missing rate
A machine works for 12 identical cycles and produces 540 components.
Rate = 540÷12 = 45 components per cycle.
Check: 45×12=540.
A changed total can reveal number of units
Tickets cost $7.50 each and the total paid is $60.
Number of tickets = 60÷7.50 = 8.
Check: 8×7.50=$60.
Compare rates only in compatible units
Machine A: 30 items/minute.
Machine B: 150 items/5 minutes.
Convert B to a one-minute rate:
150÷5 = 30 items/minute.
The rates are equal.
Comparing 30 and 150 directly would compare quantities measured over different unit counts.
Rate and ratio are related but not identical labels
A rate compares quantities with different units, such as dollars per book or litres per minute. A ratio can compare quantities of the same kind or different kinds depending on context.
Both rely on proportionality when the relationship stays constant.
Guide 3 develops ratio units; the current guide keeps the “per one unit” structure explicit.
Main worked workshop: twenty original rate problems
1.
7 bags at 12 apples per bag.
Answer: 84 apples.
2.
360 g in 8 equal packets.
Answer: 45 g per packet.
3.
$54 total at $6 per ticket.
Answer: 9 tickets.
4.
18 pages/min for 15 min.
Answer: 270 pages.
5.
420 pages in 14 min at constant rate.
Answer: 30 pages/min.
6.
480 L at 16 L/container.
Answer: 30 containers.
7.
$2.80 per item for 15 items.
Answer: $42.
8.
$63 for 9 identical items.
Answer: $7 per item.
9.
4.5 kg across 6 packets.
Answer: 0.75 kg per packet.
10.
0.75 kg/packet for 20 packets.
Answer: 15 kg.
11.
A machine produces 25 units/min for 2 h.
Answer: 25×120=3000 units.
12.
A learner multiplies $48 total by 6 items to find unit price.
Repair: rate = total÷units = $8/item.
13.
A learner divides 15 L/container by 8 containers.
Repair: total = rate×units = 120 L.
14.
900 g total at 150 g/packet.
Answer: 6 packets.
15.
32 books in 4 shelves, equal number per shelf.
Answer: 8 books/shelf.
16.
Rate 3.5 L/min for 12 min.
Answer: 42 L.
17.
42 L at 3.5 L/min.
Answer: 12 min.
18.
42 L in 12 min.
Answer: 3.5 L/min.
19.
20 units/min for 5 min, then 25 units/min for 4 min.
Answer: 100+100=200 units.
20.
200 units over total 9 min.
Average rate: 200/9 units/min, if requested.
Diagnose the first weak link
The learner cannot identify “per one”. Return to unitary method.
The learner knows the rate but chooses division for total. Use the three-state table.
The operation is correct but units mismatch. Convert units before arithmetic.
The answer is numerically correct but missing “per”. Restore the rate unit.
The learner averages two rates directly despite unequal durations. Rebuild total amount and total unit count.
Independent transfer check
- 18 kg is packed equally into 24 bags. Find kg per bag.
- A copier prints 35 pages/min for 18 min. Find total.
- 560 mL is poured at 70 mL per cup. Find cups.
- $72 buys 12 identical books. Find unit price.
- Each book costs that unit price. Find cost of 25 books.
- A machine makes 12 items/min for 30 min, then 18 items/min for 20 min. Find total items.
- Find the average items/min across the full 50 min for Question 6.
- Explain why 5 L/min × 2 hours cannot be calculated as 5×2 litres.
- Explain why unit price is a rate.
- Give the inverse check for 96 L ÷ 12 L/container = 8 containers.
Independent-check answers
1. 0.75 kg/bag.
2. 630 pages.
3. 8 cups.
4. $6/book.
5. $150.
6. 360+360=720 items.
7. 720÷50=14.4 items/min.
8. The rate unit is minutes, so 2 h must first become 120 min.
9. It states an amount of money for one unit of product.
10. 8 containers×12 L/container=96 L.
Editorial Review return test
Change the surface story but keep the same three states. A pupil who can move from $/item to g/packet to L/container to pages/minute without changing the relationship has learned rate as structure rather than memorised a worksheet category.
Then move the unknown: first total, then rate, then number of units. A robust learner should choose the operation from the missing state, not from the order in which the numbers appear.
The learner’s final card
What amount belongs to one unit? What is the rate unit? How many units are involved? Am I finding rate, total amount or number of units? Are the units compatible? Can the inverse operation rebuild the given quantities?
Continue through Batch 15
Use Guide 57: Reverse Percentage. Continue with Guide 59: Reverse Cuboid Dimensions and Guide 60: Semicircle and Quarter-Circle Boundaries.
Return to the PSLE Learning Guide Mathematics route.
Sources and boundaries
MOE Primary Mathematics syllabus, updated October 2025; SEAB 2026 PSLE formats.
All rates, contexts and solutions are original eduKate teaching material. This page is subordinate to the existing unitary-method owner and isolates the rate–total–number-of-units state system.