Mathematics Standard • Year 11 • Module 2 • Lesson 11
Rates — Problem Set
Apply rates, speed-distance-time and fuel consumption to realistic Australian driving, supermarket and workplace scenarios.
Problem 1 — Sydney to Dubbo road trip
The Larsen family drives from Sydney to Dubbo, a total distance of 405 km. They drive at an average speed of 100 km/h on the highway for the first 240 km, then stop for 30 minutes for lunch, and then drive at an average speed of 90 km/h for the remainder.
Set up: What are we solving for?
(i) Find the time taken for the first 240 km, in hours and minutes. 1 mark
(ii) Find the time taken for the remainder of the trip, in hours and minutes. 2 marks
(iii) Find the total travel time including the lunch stop. If they left Sydney at 8:00 am, at what time do they arrive in Dubbo? 2 marks
Stuck? Revisit lesson § Speed, Distance, Time — work each leg separately, then add the rest stop as minutes.Problem 2 — Olive oil at the supermarket (best value)
A supermarket sells olive oil in three sizes.
Bottle A: 375 mL for $6.45
Bottle B: 750 mL for $11.40
Bottle C: 1.5 L for $24.00
Set up: What are we solving for?
(i) Find the unit cost in $/100 mL for each bottle, to the nearest cent. 2 marks
(ii) State which bottle is the best value, and the price per 100 mL. 1 mark
(iii) A shopper says "the biggest bottle is always the best value". Use the prices above to explain in one sentence why this is not true. 2 marks
Stuck? Revisit lesson § Worked Example 1 — Best Value. Convert all sizes to the same unit (100 mL or 1 L) before comparing.Problem 3 — Comparing two cars on fuel cost
A family is choosing between two cars.
Car X: Fuel consumption 6.4 L/100 km (uses 91 unleaded at $1.95/L).
Car Y: Fuel consumption 8.8 L/100 km (uses 91 unleaded at $1.95/L).
Set up: What are we solving for?
(i) They drive 15 000 km per year. Calculate the litres of fuel used by each car per year. 2 marks
(ii) Find the annual fuel cost for each car. 1 mark
(iii) Car X costs $4,200 more to buy than Car Y. After how many years does the lower fuel cost of Car X "pay back" the extra purchase price? Round up to the nearest whole year. 2 marks
Stuck on (iii)? Find the annual fuel cost saving by using Car X, then divide $4,200 by that saving.Problem 4 — Filling a backyard pool (flow rates)
A backyard pool holds 24 000 L of water. The garden tap fills the pool at 18 L/min. A second tap on the side of the house fills at 12 L/min and can be run at the same time.
Set up: What are we solving for?
(i) Using only the garden tap, how long does it take to fill the pool? Give your answer in hours and minutes. 2 marks
(ii) Using both taps together, find the combined flow rate and the time to fill the pool, in hours and minutes. 2 marks
(iii) Sydney Water charges $2.85 per kL (kilolitre, 1000 L). Find the cost of filling the pool. 1 mark
Stuck? Revisit lesson § Practice Q2 (flow rate). Combined flow rate = sum of individual flow rates.Problem 5 — Tradie's weekly fuel and travel costs
A plumber drives a van with fuel consumption 11.2 L/100 km. Diesel costs $2.05/L. In a typical work week she drives 680 km on call-outs across western Sydney.
Set up: What are we solving for?
(i) Calculate the litres of diesel used in a typical week. 1 mark
(ii) Calculate the weekly fuel cost. 1 mark
(iii) If she works 48 weeks per year, calculate her annual fuel bill. 1 mark
(iv) She is considering replacing the van with one that uses 9.0 L/100 km. Calculate the annual fuel saving she would make (same kms, same diesel price, same number of weeks). State your answer in dollars per year with a one-sentence conclusion. 2 marks
Stuck? Revisit lesson § Worked Example 3 — Fuel Consumption. Find the litres-difference per week first, then scale to a year.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Sydney to Dubbo road trip
Set up. Decompose the trip into two driving legs and the lunch stop. Find time for each leg, sum, then add the rest.
(i) T₁ = 240 ÷ 100 = 2.4 h = 2 h 24 min.
(ii) Distance remaining = 405 − 240 = 165 km. T₂ = 165 ÷ 90 = 1.8333... h = 1 h + (0.8333 × 60) min = 1 h 50 min = 1 h 50 min.
(iii) Total = 2 h 24 min + 30 min + 1 h 50 min = 3 h 104 min = 4 h 44 min. Departure 8:00 am + 4 h 44 min = arrives at 12:44 pm.
Problem 2 — Olive oil best value
Set up. Convert each price to a common unit ($/100 mL), then compare.
(i) Bottle A: $6.45 ÷ 3.75 ≈ $1.72/100 mL. Bottle B: $11.40 ÷ 7.5 = $1.52/100 mL. Bottle C: $24.00 ÷ 15 = $1.60/100 mL.
(ii) Bottle B (750 mL) is best value at $1.52 per 100 mL.
(iii) Sample: Bottle C is the biggest but costs $1.60/100 mL — more than Bottle B at $1.52/100 mL — so the biggest is not always best; you must compare unit prices each time.
Problem 3 — Car X vs Car Y
Set up. Find annual litres for each car, then annual cost, then how many years the fuel saving offsets the $4,200 extra purchase price.
(i) Car X: 6.4 × 15 000 ÷ 100 = 960 L/year. Car Y: 8.8 × 15 000 ÷ 100 = 1,320 L/year.
(ii) Car X: 960 × $1.95 = $1,872.00/year. Car Y: 1,320 × $1.95 = $2,574.00/year.
(iii) Annual saving with Car X = $2,574 − $1,872 = $702.00. Payback years = $4,200 ÷ $702 ≈ 5.98 → 6 years (round up, because at 5 years the saving has not yet covered the extra purchase price).
Problem 4 — Pool filling
Set up. Pool volume ÷ flow rate gives time. With two taps, add the flow rates first.
(i) T = 24 000 ÷ 18 = 1333.33... min = 22 h 13.33 min ≈ 22 h 13 min.
(ii) Combined flow = 18 + 12 = 30 L/min. T = 24 000 ÷ 30 = 800 min = 13 h 20 min.
(iii) Volume in kL = 24 000 ÷ 1000 = 24 kL. Cost = 24 × $2.85 = $68.40.
Problem 5 — Plumber's fuel costs
Set up. Litres per week × $/L gives weekly cost; scale to year; repeat for new van; subtract to get saving.
(i) Litres/week = 11.2 × 680 ÷ 100 = 76.16 L.
(ii) Weekly cost = 76.16 × $2.05 = $156.13.
(iii) Annual = $156.13 × 48 = $7,494.14.
(iv) New van litres/week = 9.0 × 680 ÷ 100 = 61.2 L. New annual = 61.2 × 48 × $2.05 = $6,022.08. Saving = $7,494.14 − $6,022.08 = $1,472.06 per year. Conclusion: switching to the more efficient van would save about $1,472 per year in diesel.