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Rates and unit amounts

Use a per-unit amount to connect two different quantities and check the units.

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Find the amount for one minute

84 pages in 3 minutes. How many pages in one minute?

A printer produces 84 pages in 3 minutes at a steady rate. The rate tells us the number of pages produced per minute: in one minute. Divide the total output into three equal one-minute amounts.

Three equal times share the total output

Each unknown card counts pages produced during one minute. The three equal cards together account for 84 pages during three minutes.

Together, the three cards contain 84 pages. Each card shows one minute's output. The equal unknown amounts model a steady printing rate.
Figure data

This table is included in print while it is open.

Values shown on the cards
SetValues
Three equal minutes? pages; ? pages; ? pages

For this lesson’s value fields, use a whole number or decimal and choose the unit separately. A slash in a unit means per: page/min means pages per minute.

Printing rate: unit
Show workingHide answer

84÷3 = 28, so the rate is 28 pages per minute. The unit page/min keeps both quantities and their order visible. Writing only “28” leaves its meaning incomplete.

This model assumes a steady rate. A starting delay or a jam would need extra information.

How much does the printer make in all five minutes?

Find a total from the rate

The printer continues at 28 pages per minute. How many pages does it produce in 5 minutes? Combine five equal one-minute outputs.

Time in minutesPages produced
128
256
384
5?
Total output: unit
Show workingHide answer
total pages = pages per minute × minutes.

28 × 5 = 140 pages. The time has already been included in the calculation, so the answer’s unit is pages, not pages per minute.

How many one-minute amounts make the total?

Find how many minutes you need

The printer produces 28 pages per minute. How many minutes are needed for 140 pages, assuming that rate stays steady? This time, the total output is known and the number of one-minute units is missing.

Time needed: unit
Show workingHide answer

140÷28 = 5 minutes. Check 28 × 5 = 140 pages.

The same relationship answers three questions:

  • Rate: divide the total amount by the number of units.
  • Total amount: multiply the rate by the number of units.
  • Number of units: divide the total amount by the rate. The rate cannot be zero.

Decide which quantity is missing before choosing an operation.

How much does one notebook cost in each pack?

Find the unit before comparing

One pack costs $9 for 6 identical notebooks. Another costs $8 for 4 of the same notebooks. Which has the lower cost per notebook? SGD means Singapore dollars. Enter the numerical prices without a dollar sign.

First unit price: unit
Second unit price: unit
Lower cost per notebook
Show workingHide answer

9÷6 = 1.50 dollars per notebook; 8÷4 = 2 dollars per notebook. The first pack has the lower unit price.

A lower pack price does not necessarily give a lower unit price. Use the same unit and check that the products being compared are equivalent.

Are you finding pages per minute or minutes per page?

Check which way you divide

For 84 pages in 3 minutes, Nora calculates 3÷84 = 1/28. She calls her result “pages per minute”. Identify the unit her division actually produces, then repair the printing rate.

Unit produced by minutes ÷ pages
Show workingHide answer

3÷84 = 1/28 minute per page. That is a meaningful rate, but it answers a different question.

The rate asked for is 84÷3 = 28 pages per minute. Check 28 × 3 = 84 pages. Changing only the label cannot turn one rate into the other.

Which amount belongs to one minute?

Practise: Primary 5 Mathematics Rates, total amounts and numbers of units practice · Primary 5 MathematicsPractise: Primary 5 Foundation Mathematics Rates, total amounts and numbers of units practice · Primary 5 Foundation Mathematics

Check your understanding 1

Choose what to divide

A tap fills 18 ℓ in 4 minutes at a constant rate. Find the rate in litres per minute, then the time needed for 27 ℓ. Keep the rate and the time as different quantities.

Filling rate: unit
Time for 27 ℓ: unit
Which calculation checks the filling time?
Show answerHide answer

The rate is 18÷4 = 4.5 ℓ/min. The filling time for 27 ℓ is 27÷4.5 = 6 minutes. Check 4.5 × 6 = 27 ℓ.

An equal-groups route gives the same time: 18 ℓ in 4 minutes → 9 ℓ in 2 minutes → 27 ℓ in 6 minutes. Reversing the first division would give minutes per litre, a different rate.

What must stay the same for your prediction to work?

Use a rate to predict

A worker packs 48 boxes in 6 hours. Use that average rate to predict the output in 10 hours. Explain what assumption makes the prediction useful.

Average packing rate: unit
Predicted output: unit
Assumption needed for the prediction
Show answerHide answer

The observed average rate is 48÷6 = 8 boxes per hour. At that same average rate, 8 × 10 = 80 boxes would be packed in 10 hours.

The prediction assumes the same average rate applies over the new interval. Different breaks, box sizes or fatigue could change that average. Your calculation predicts what happens if the rate stays the same. The real rate could change.

Does the question ask for paint or the area it covers?

Use one rate for two questions

At a uniform coverage rate, 3 ℓ of paint covers 18 m². Ignore paint losses and keep the same coverage rate. Find the area covered by 4.5 ℓ, then the paint needed for 30 m². A rate can connect two quantities without either quantity being time.

Covered area: unit
Paint needed: unit
At 6 m²/L, which operations find area, then paint volume?
Show answerHide answer

The coverage rate is 18÷3 = 6 m²/L: six square metres per litre. For 4.5 ℓ, the covered area is 6 × 4.5 = 27 m². For 30 m², the required volume is 30÷6 = 5 ℓ.

The first answer is an area and the second is a volume. The rate connects them; its unit is not automatically the unit of every answer.

For a grown-up

Pip’s synthetic voice uses the University of Edinburgh’s Alba speech dataset, via Piper’s Alba voice, under CC BY 4.0. Audio is optional.

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