Mathematics Standard • Year 11 • Module 2 • Lesson 11
Rates — Past-Paper Style
HSC Mathematics Standard 2-style writing on rates and fuel — short answers and one structured extended response with explicit marking criteria.
1. Short-answer questions
1.1 A car uses 10.5 L of petrol for every 100 km travelled. Petrol costs $1.98 per litre. Calculate the fuel cost for a 600 km trip, to the nearest cent. 3 marks Band 3
1.2 A cyclist completes a 75 km charity ride in 3 hours 45 minutes.
(a) Find the cyclist's average speed in km/h, to 2 decimal places.
(b) If she maintained the same average speed for an additional 20 km, how long (in minutes, to the nearest minute) would those extra 20 km take? 3 marks Band 3-4
1.3 A 1.25 L bottle of orange juice costs $5.20. A 2 L bottle of the same brand costs $7.80.
(a) Find the unit cost of each bottle, in $/L, to the nearest cent.
(b) Which bottle is better value, and by how many cents per litre is it cheaper? Show your conclusion clearly. 4 marks Band 4
2. Extended response
2.1 Marcus plans a one-day driving trip from Newcastle to Tamworth. The total driving distance is 282 km. He plans to drive at an average speed of 95 km/h and stop for 25 minutes for lunch.
His car has fuel consumption 7.2 L/100 km. The tank holds 50 L and is currently full.
Unleaded petrol along the route costs $2.05 per litre.
(a) Calculate the driving time (in hours and minutes, to the nearest minute) and the total trip time including the lunch stop.
(b) Calculate the number of litres of fuel used for the trip, to 1 d.p., and confirm Marcus does NOT need to refuel during the trip.
(c) Calculate the total fuel cost for the round trip (Newcastle → Tamworth → Newcastle). State the cost to the nearest cent with a clear conclusion sentence. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — correct driving time using T = D ÷ S, expressed in h and min.
• 1 mark — correct total time = driving time + 25 minutes.
Part (b) — 2 marks
• 1 mark — correct litres for the one-way trip using L = c × d ÷ 100.
• 1 mark — explicit comparison of litres used vs 50 L tank, with conclusion.
Part (c) — 3 marks
• 1 mark — doubles the one-way distance OR the one-way litres for the round trip.
• 1 mark — correct cost = round-trip litres × $2.05.
• 1 mark — explicit conclusion sentence stating the total fuel cost to the nearest cent.
Your response:
Stuck on (b)? Compute litres for one way and compare to the 50 L tank. For (c), the round trip is twice the one-way litres at the same price per litre.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Fuel cost for 600 km (3 marks)
Sample response.
Litres used = 10.5 × 600 ÷ 100 = 10.5 × 6 = 63 L.
Cost = 63 × $1.98 = $124.74.
Fuel cost for 600 km is $124.74.
Marking notes. 1 mark — correct setup for litres (10.5 × 600 ÷ 100 or equivalent). 1 mark — correct litres value (63 L). 1 mark — correct multiplication × $1.98 giving $124.74 with units/sentence. A bare "$124.74" with no working scores 1/3.
1.2 — Cyclist 75 km in 3 h 45 min (3 marks)
(a) Sample. T = 3 + 45 ÷ 60 = 3.75 h. S = 75 ÷ 3.75 = 20.00 km/h.
(b) Sample. T = D ÷ S = 20 ÷ 20 = 1 h = 60 minutes.
Marking notes. (a) 1 mark — correct conversion 3 h 45 min → 3.75 h. 1 mark — correct S = 20.00 km/h with units. (b) 1 mark — correct application T = 20 ÷ 20 = 1 h = 60 min. Common error: forgetting to convert minutes to hours before dividing.
1.3 — Orange juice best value (4 marks)
(a) Sample. 1.25 L bottle: $5.20 ÷ 1.25 = $4.16/L. 2 L bottle: $7.80 ÷ 2 = $3.90/L.
(b) Sample. The 2 L bottle is better value. It is cheaper by $4.16 − $3.90 = $0.26/L, i.e. 26 cents per litre cheaper.
Marking notes. (a) 1 mark — correct $/L for 1.25 L bottle. 1 mark — correct $/L for 2 L bottle. (b) 1 mark — identifies the 2 L bottle as cheaper per litre. 1 mark — correct difference (26 c/L) with units/sentence. Common error: comparing total prices ($5.20 vs $7.80) instead of unit prices.
2.1 — Newcastle to Tamworth (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Driving time and total trip time.
T_drive = D ÷ S = 282 ÷ 95 = 2.9684... h.
Hours = 2; minutes = 0.9684 × 60 ≈ 58.1 → 58 min. Driving time ≈ 2 h 58 min. [1 mark — driving time.]
Total trip time = 2 h 58 min + 25 min = 2 h 83 min = 3 h 23 min. [1 mark — total including lunch.]
(b) Litres used one-way + refuel check.
Litres = 7.2 × 282 ÷ 100 = 7.2 × 2.82 = 20.3 L. [1 mark — litres for one way.]
20.3 L < 50 L tank, so Marcus does not need to refuel during the one-way trip. [1 mark — explicit comparison and conclusion.]
(c) Total fuel cost (round trip).
Round-trip litres = 2 × 20.3 = 40.6 L. [1 mark — doubled distance/litres for round trip.]
Cost = 40.6 × $2.05 = $83.23. [1 mark — multiplication × $2.05.]
Conclusion: the total fuel cost for the round trip Newcastle → Tamworth → Newcastle is $83.23. [1 mark — explicit conclusion to the nearest cent.]
Total: 7/7.
Band descriptors for marker.
Band 3: Driving time computed (perhaps as decimal hours only) and one of the litres or cost calculations attempted. No round-trip handling. ≈ 3 marks.
Band 4: Driving time + lunch added correctly; litres for one way correct; misses the round-trip doubling or omits the conclusion. ≈ 4-5 marks.
Band 5: Full numerical solution including round trip but conclusion sentence missing — e.g. just "$83.23" with no naming of "total round-trip fuel cost". ≈ 6 marks.
Band 6: Complete with correct round-trip handling, refuel comparison stated explicitly, and a clear final conclusion sentence. 7/7.