Modelling practical rate problems and checking assumptions | Year 8

01 / LEARN

The idea, step by step

Start with the situation

A mathematical model simplifies a real situation. Identify the quantities and units, decide which rate can reasonably stay constant, then write a rule. A useful answer includes its units and an explanation of the assumptions.

A model can fit a calculation but fail in practice. A printer rate ignores warm-up and paper jams; a walking model may ignore crossings and rests. Name the limitation that could actually matter rather than writing a vague warning.

Check and improve the result

Substitute a known input to see whether your model reproduces the data. Check the size of the answer and whether it can be used in the context. Some quantities must be rounded up: 2.6 buses cannot carry a group when whole buses must be booked.

If observations show changing rates, do not keep claiming exact direct proportion. You may need separate stages or a fixed-fee-plus-rate model. Explain what extra measurement would improve a prediction.

A common trap

An answer from a proportional calculation must be valid in the real situation.

Instead: Check the assumptions, units and allowable quantities. Whole packs, buses or people cannot always use the unrounded model answer.

02 / WORKED EXAMPLES

See the thinking

Example 1

Whole buses

A bus has 48 seats. How many buses are needed for 125 passengers?

Assume all 48 seats can be used for these passengers.
$125 \div 48 \approx 2.60$ .
Two buses provide only 96 seats; three provide 144. Round up, not to the nearest whole number.

3 buses.

Example 2

A filling model

A 240 L tank already contains 60 L. A tap adds 6 L/min steadily. How long until the tank is full?

The amount still needed is $240 - 60 = 180$ L.
At 6 L/min, $t = 180 \div 6 = 30$ min.
Check the model $V = 60 + 6 t$ : at $t = 30$ , $V = 240$ .

30 minutes, assuming the rate stays constant.

Check your understanding

A van holds 20 boxes. How many trips are needed for 53 boxes?

Show answer

3 trips.

Two trips carry only 40 boxes, so the third trip is necessary even though it is not full.

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A printer produces 18 pages/min. Predict its output in 7 min and state one assumption.
2 marks
Five volunteers pack 120 boxes in 2 h. At the same rate per volunteer, how many boxes can 8 volunteers pack in 3 h?
3 marks
One litre of paint covers 8 square metres. A wall is 38 square metres. Paint is sold in whole 1 L tins. How many tins are needed?
3 marks
A walker averages 4.8 km/h while moving. Predict the time for 3 km. State why the whole journey might take longer.
3 marks
A service costs AUD 30 plus AUD 50 per hour. Is the total cost directly proportional to hours? Find the cost of 3 h.
3 marks
A tap supplies 4 L after 1 min, 8 L after 2 min and 11 L after 3 min. Does one exact constant-rate model fit all the data? Explain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A printer produces 18 pages/min. Predict its output in 7 min and state one assumption.
2 marks
$18 \times 7 = 126$ .
The prediction excludes warm-up, paper changes and jams.
126 pages, assuming it prints at a steady rate without stops.
Five volunteers pack 120 boxes in 2 h. At the same rate per volunteer, how many boxes can 8 volunteers pack in 3 h?
3 marks
Rate per volunteer per hour: $120 \div (5 \times 2) = 12$ .
Prediction: $12 \times 8 \times 3 = 288$ .
288 boxes.
One litre of paint covers 8 square metres. A wall is 38 square metres. Paint is sold in whole 1 L tins. How many tins are needed?
3 marks
The model requires $38 \div 8 = 4.75$ L.
Four tins cover only 32 square metres, so buy 5.
5 tins.
A walker averages 4.8 km/h while moving. Predict the time for 3 km. State why the whole journey might take longer.
3 marks
$t = 3 \div 4.8 = 0.625$ h.
$0.625 \times 60 = 37.5$ min.
37.5 minutes; rests or crossings may add time.
A service costs AUD 30 plus AUD 50 per hour. Is the total cost directly proportional to hours? Find the cost of 3 h.
3 marks
The fixed AUD 30 means zero hours does not give zero cost.
$C = 30 + 50 t$ ; $C (3) = 30 + 150 = 180$ .
No; the cost is AUD 180.
A tap supplies 4 L after 1 min, 8 L after 2 min and 11 L after 3 min. Does one exact constant-rate model fit all the data? Explain.
3 marks
The first two records suggest 4 L/min.
That model predicts 12 L at 3 min, but 11 L was measured.
The flow may have changed, or measurement error may be present.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Ratio scaling and division review; unit rates; comparing rates and best buys; speed, distance and time; converting compatible units; direct-proportion tables and graphs; modelling practical rate problems and checking assumptions.

Official curriculum source ↗

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