Unit 8

Rates and Ratios

About this unit

Solve ratio, proportion, distance-speed-time, and work-rate problems. Handle meeting, chasing, and combined-rate scenarios with a systematic approach.

What types of questions will you face?

  • 1Scale a ratio to a new total — divide the given total by the matching ratio part to find the number of groups, then multiply the other ratio part by that number
  • 2Compare two travel times for the same distance at different speeds — apply T = D ÷ S independently to each leg and subtract to find the time difference
  • 3Calculate distance or speed using D = S × T with time conversions — convert minutes to a fraction of an hour before applying the formula
  • 4Work rate problems — two people or machines each complete a task in different times; add their rates (1÷time) to find the combined rate, then find the finish time
  • 5Before-and-after ratio problems — quantities change and the ratio changes; track the total or an unchanged quantity across both states to find the new amounts

Skills you will build

  • Applying the unit-groups method for ratio scaling: groups = total ÷ ratio part; answer = groups × other ratio part
  • Using T = D ÷ S and D = S × T fluently, including converting 24 minutes to 2/5 of an hour before multiplying
  • Computing two journey times separately and comparing them — never subtracting speeds or averaging them
  • Adding worker/pipe rates as fractions (1/time_A + 1/time_B), simplifying to a combined rate, and taking the reciprocal for the total time
  • Identifying the unchanged quantity in a before-after ratio problem and using it to link the two ratio states

By the end of this unit, you will be able to

  • Scale any ratio to a new total in two arithmetic steps without setting up a proportion equation
  • Find distance, speed, or time for any journey including those with unit conversions between minutes and hours
  • Calculate how much longer one leg of a trip takes than another when the speeds differ
  • Find the time for two workers or pipes to complete a task together using the rate-addition method
  • Solve before-and-after ratio problems by anchoring on the unchanged quantity

Difficulty profile

Questions in this unit range from Easy to Difficult. Simple ratio scaling and basic D=S×T calculations are Easy. Same-distance time comparisons and work rate problems are Medium. Meeting, chasing, and multi-step before-after ratio problems reach Difficult.

Exam tip: Rates and Ratios

For any ratio scaling question, avoid cross-multiplication — the unit-groups method (divide total by ratio part, multiply by other part) is faster and less error-prone. For speed questions, always compute each time separately using T = D ÷ S — never subtract or average the speeds. These are the two most common method errors in this unit.

Sample Questions

Lesson 1 of 3Rates and RatiosEasy

A car park charges $2.25/hr before 5 pm and $1.00/hr after. Coen parks from 3 pm to 9 pm. The price changes in the middle of his visit — so split the stay at 1700, calculate each chunk separately, then add.

Tiered pricing questions (different rates for different time bands) appear regularly on NSW Selective MR. The key skill is identifying where the price boundary falls within the visit and splitting the duration accordingly.

The examiner tests whether students convert 12-hour to 24-hour time, identify that 1700 falls within the 1500–2100 stay, correctly split into 2 hours at $2.25 and 4 hours at $1.00, and avoid the common trap of applying the $2.25 rate to the full 6 hours (giving $13.50).

A pricing table lists different hourly rates for different time windows. A person's start and end times are given in 12-hour format. The visit spans two (or more) price windows. Students must split the visit at the boundary and calculate each segment's cost.

Best approach: Step 1: Convert all times to 24-hour format. Step 2: Find which price boundaries fall within the visit. Step 3: Split the visit at each boundary. Step 4: Multiply hours × rate for each segment. Step 5: Add the segment costs.

Question

Here are the costs for a car park:

  • Between 0800 and 1700: $2.25 per hour
  • Between 1700 and 2300: $1.00 per hour

Coen arrives at the car park at 3 pm and leaves at 9 pm on the same day.

How much does he pay for parking?

  1. A$6.50
  2. B$8.50
  3. C$11.00
  4. D$13.00
  5. E$13.50

Decided on your answer? Check how you went below.

Lesson 2 of 3Rates and RatiosIntermediate

Ratio scaling always starts with total parts: add the ratio numbers, divide the given amount by that total, then multiply each part you need.

Mixture and sharing ratios appear throughout Selective MR — students who find “how many groups” first avoid cross-multiplying under pressure.

The examiner checks whether you can scale a part-to-part ratio to a real total (e.g. litres of drink) using unit groups, not guesswork.

Two ingredients are mixed in a fixed ratio. A total amount of mixture is made. You find how much of one ingredient is required.

Best approach: Add ratio parts (5 + 2 = 7). One group size = total ÷ 7. Multiply the required part (concentrate = 2 groups) by that size.

Question

A sports drink is made by mixing water and concentrate in the ratio 5 : 2.

How much concentrate is needed to make 3.5 litres of drink?

  1. A750 mL
  2. B800 mL
  3. C900 mL
  4. D1000 mL
  5. E1200 mL

Decided on your answer? Check how you went below.

Lesson 3 of 3Rates and RatiosIntermediate

Average speed is total distance divided by total time — never average the two speeds unless the times are equal.

Multi-leg journey items appear in the medium band of Selective MR — add kilometres and hours separately before dividing.

The examiner tests whether you compute distance ÷ time for each leg, sum both distances and both times, then find overall km/h.

A trip has two segments with different distances and times. You find the average speed for the whole journey.

Best approach: Distance = 160 + 210 km. Time = 2 + 3 hours. Average speed = total distance ÷ total time. Do not use (speed1 + speed2) ÷ 2.

Question

A vehicle travelled from Place A to Place B. It travelled 160 kilometres in the first 2 hours and 210 kilometres in the next 3 hours. How many kilometres did the vehicle travel per hour on average?

  1. A72
  2. B73
  3. C74
  4. D75
  5. E76

Decided on your answer? Check how you went below.

Up next: Practice Set 4

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