Mini Mathematics
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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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A rate connects two kinds of quantity

A printer produces 84 pages in 3 minutes at a steady rate. In one minute it produces 84÷3 = 28 pages. The rate is 28 pages per minute. Writing only “28” loses what is being compared.

Time in minutesPages
128
256
384
5140

For this constant-rate model, total pages = pages per minute × minutes. For 140 pages, the time is 140÷28 = 5 minutes. These relationships assume the rate stays steady; starting delays or a jam would need extra information.

Find the unit before comparing

One pack costs $9 for 6 identical notebooks: $1.50 per notebook. Another costs $8 for 4: $2 per notebook. The lower pack price does not give the lower unit price. Compare amounts for the same unit and check that the products are comparable.

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 for 27 ℓ.

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The rate is 18÷4 = 4.5 ℓ per minute. The time is 27÷4.5 = 6 minutes. Reversing the first division gives minutes per litre, a different rate.

A rate is a model

A worker packs 48 boxes in 6 hours. Using that average rate predicts 80 boxes in 10 hours. Explain what assumption this prediction needs.

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The rate is 8 boxes per hour, and 8 × 10 = 80. The prediction assumes the same rate throughout all 10 hours. Breaks, different box sizes or fatigue could make the prediction inaccurate.

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