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.