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.