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.
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