Mathematics Standard • Year 11 • Module 2 • Lesson 11

Rates — Skill Drill

Build fluency in the core Rates calculations: unitary method, speed-distance-time, fuel consumption and cost — one mechanic at a time.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the rearrangements of D = S × T.

S = ____________ T = ____________

Q1.2 A car has fuel consumption of c L/100 km and travels d km. Write the formula for litres of fuel used.

Litres used = ____________

Q1.3 Convert 1 h 30 min into hours as a decimal. ____________ h

Stuck? Revisit lesson § Key Rate Formulas — the D = S × T panel and the L/100 km panel.

2. Worked example — average speed in km/h

Follow each line of working. Every step has a reason on the right.

Problem. A car travels 270 km in 3 hours and 15 minutes. Find the average speed in km/h, correct to 1 decimal place.

Step 1 — Convert mixed time to hours.

T = 3 + 15 ÷ 60 = 3.25 h

Reason: 15 minutes ÷ 60 = 0.25 h. Units must match (km/h needs hours).

Step 2 — Choose the correct rearrangement.

S = D ÷ T = 270 ÷ 3.25

Reason: D and T are known, S is unknown — cover S in the DST triangle, D sits over T.

Step 3 — Evaluate and round.

S = 83.0769... ≈ 83.1 km/h

Reason: 1 d.p. rounding; include units in the final answer.

Conclusion. The average speed is 83.1 km/h.

3. Faded example — fill in the missing steps

A runner completes a 10 km race in 52 minutes 30 seconds. Find the average speed in km/h, correct to 2 decimal places. Fill in each blank line. 4 marks

Step 1 — Convert time to hours:

T = 52 min + 30 s = 52.5 min = 52.5 ÷ ____ h = ____________ h

Step 2 — Rearrangement: S = D ÷ T = ____________ ÷ ____________

Step 3 — Evaluate: S = ____________ km/h (unrounded)

Step 4 — Round to 2 d.p.: S ≈ ____________ km/h

Conclusion. The runner's average speed is ____________ km/h.

Stuck? Revisit lesson § Worked Example 2 — Speed–Distance–Time. The runner's time must be converted to hours before dividing.

4. Graduated practice — Rates calculations

Show your working in the space below each part. Always include units in the final answer.

Foundation — single-step rates and conversions (4 questions)

Q	Problem	Answer
4.1 1	A printer produces 420 pages in 7 minutes. Find the rate in pages per minute.
4.2 1	A tap fills a 180 L bath in 12 minutes. Find the flow rate in L/min.
4.3 1	A cyclist travels at 24 km/h for 2.5 hours. Find the distance covered.
4.4 1	Convert 1 h 45 min into hours as a decimal.

Standard — typical HSC difficulty (6 questions)

Show at least one line of substitution and clearly label your final answer with units.

4.5 Pack A: 400 g of coffee for $7.60. Pack B: 600 g for $11.10. Which is better value (per gram)? 2 marks

4.6 A train travels 480 km at an average speed of 120 km/h. Find the time taken in hours. 2 marks

4.7 A car has fuel consumption 9.2 L/100 km. How much fuel is needed for a 350 km journey? 2 marks

4.8 Using your answer to 4.7, petrol costs $2.15/L. Find the total fuel cost for the 350 km journey. 2 marks

4.9 A car travels 140 km in 1 h 45 min. Find the average speed in km/h. 2 marks

4.10 A tap drips at 0.3 L/min. How many litres are wasted in 24 hours? 2 marks

Extension — combine two ideas (2 questions)

4.11 A family drives from Sydney to Melbourne, a distance of 880 km. They drive at 100 km/h for the first 400 km, stop for 45 min, then drive at 90 km/h for the rest. Find the total travel time including the rest stop, in hours and minutes. 3 marks

4.12 A car has consumption 7.8 L/100 km. Its tank holds 60 L. Starting on a full tank, find (i) how many km it can travel on a full tank, to the nearest km, and (ii) the cost of one full tank of petrol at $2.10/L. 3 marks

Stuck on 4.11? Find time for each leg separately, add the rest stop in the same units (minutes), then convert the total back to h and min.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I divided pages by minutes (420 ÷ 7), not the other way around.

For 4.3 my units check out: km/h × h gives km — so multiplying speed by time gives distance.

For 4.4 I divided the 45 minutes by 60 (not 100) to convert minutes to hours.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Rearrangements of D = S × T

S = D ÷ T. T = D ÷ S.

Q1.2 — Fuel-used formula

Litres used = c × d ÷ 100 (consumption multiplied by the number of hundreds of km).

Q1.3 — Decimal hours

1 h 30 min = 1 + 30 ÷ 60 = 1.5 h.

Q3 — Faded example (10 km race in 52 min 30 s)

Step 1: T = 52.5 ÷ 60 = 0.875 h.
Step 2: S = 10 ÷ 0.875.
Step 3: S = 11.42857... km/h.
Step 4: S ≈ 11.43 km/h.
Conclusion: The runner's average speed is 11.43 km/h.

Q4.1 — Pages per minute

420 ÷ 7 = 60 pages/min.

Q4.2 — Flow rate

180 ÷ 12 = 15 L/min.

Q4.3 — Distance from speed and time

D = S × T = 24 × 2.5 = 60 km.

Q4.4 — 1 h 45 min as decimal hours

1 + 45 ÷ 60 = 1 + 0.75 = 1.75 h.

Q4.5 — Coffee: better value per gram

A: $7.60 ÷ 400 = $0.0190/g. B: $11.10 ÷ 600 = $0.0185/g. Pack B is better value (cheaper per gram).

Q4.6 — Time from distance and speed

T = D ÷ S = 480 ÷ 120 = 4 hours.

Q4.7 — Fuel for 350 km

Litres = 9.2 × 350 ÷ 100 = 9.2 × 3.5 = 32.2 L.

Q4.8 — Fuel cost

Cost = 32.2 × $2.15 = $69.23.

Q4.9 — Speed from D and T

T = 1.75 h. S = 140 ÷ 1.75 = 80 km/h.

Q4.10 — Dripping tap over 24 hours

Minutes in 24 h = 60 × 24 = 1440 min. Wasted = 0.3 × 1440 = 432 L.

Q4.11 — Sydney → Melbourne total time

Leg 1: T₁ = 400 ÷ 100 = 4 h.
Leg 2: distance left = 880 − 400 = 480 km; T₂ = 480 ÷ 90 = 5.333... h = 5 h 20 min.
Total = 4 h + 5 h 20 min + 45 min = 9 h 65 min = 10 h 5 min.

Q4.12 — Full-tank range and cost

(i) Litres used per km = 7.8 ÷ 100 = 0.078 L/km. D = 60 ÷ 0.078 = 769.23... ≈ 769 km.
(ii) Cost = 60 × $2.10 = $126.00.