Modelling practical rate problems and checking assumptions | Year 8

01 / LEARN

Understand the idea.

Start with the situation

A model connects the quantities in a situation using a simplified rule. Identify what is known, what is needed and the units. Decide whether a rate can reasonably stay constant and whether there is already an initial amount.

Make the assumption specific: a tap model may assume steady inflow and no leak, while a bus model assumes every listed passenger seat is available. These conditions explain when the calculation can be trusted and what could change its answer.

Check and improve the result

Check that your rule reproduces a known value and gives an answer of a sensible size. Then ask whether the result is usable. A fractional number of buses may be a correct quotient but cannot be a booking; enough whole buses are needed.

A starting amount needs to be included separately from a rate. If the rate later changes, split the process into stages instead of extending one rule indefinitely. Better measurements of delays or changing rates can improve a prediction.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Whole buses

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

Worked solution

1
Use the stated capacity for each bus.
$48 seats per bus$
Assume all 48 seats are available to these passengers, as the model requires.
2
Calculate the capacity quotient.
$\frac{125}{48} \approx 2.604$
The quotient lies between two and three buses, so a whole-bus decision is needed.
3
Test the neighbouring whole numbers.
$\begin{aligned} 2 \times 48 & = 96 < 125 \\ 3 \times 48 & = 144 \geq 125 \end{aligned}$
Two buses leave passengers without seats; three are sufficient and are the smallest possible number.

Answer

3 buses.

EXAMPLE 02

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?

Worked solution

1
Find how much more water is needed.
$240 - 60 = 180 L$
The sixty litres already in the tank count towards its capacity.
2
Divide the remaining volume by the rate.
$t = \frac{180}{6} = 30 min$
A steady addition of six litres per minute supplies 180 litres in thirty minutes.
3
Check the model at that time.
$V = 60 + 6 (30) = 240 L$
The result reaches capacity if the inflow stays steady and no water leaves the tank.

Answer

30 minutes, assuming the rate stays constant.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

3 trips.

Two trips carry at most $2 \times 20 = 40$ boxes, short of $53$ . Three trips can carry $3 \times 20 = 60$ boxes, so $3$ trips are needed even though the last is not full.

Keep practising

6 worksheet questions · 17 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Start with the situation

Check and improve the result

See the method in action.

Whole buses

Use the stated capacity for each bus.

Calculate the capacity quotient.

Test the neighbouring whole numbers.

A filling model

Find how much more water is needed.

Divide the remaining volume by the rate.

Check the model at that time.

Try it yourself.

Check your understanding

Keep practising