Use ratio, proportion and rates in context
Simplify and share ratios, scale recipes and maps, and calculate rates such as average speed and unit cost.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a rate of 4 litres per minute compare?
What you will learn
- Simplify a ratio and divide a quantity in a given ratio
- Use proportional reasoning in context
- Calculate common rates and average speed with consistent units
Make ratio parts comparable
To simplify a ratio, divide every part by the same common factor. To share a total, add the ratio parts, divide the total by that sum, then multiply by each part.
Proportional reasoning scales every relevant quantity by the same factor. Check that the order of the compared quantities matches the order in the ratio.
A rate compares different units
A rate has a unit per another unit, such as kilometres per hour, dollars per item or litres per minute. Average speed is total distance divided by total time, not the average of two speeds unless durations match.
Convert time before calculating: 45 minutes is 0.75 hours, so 3 hours 45 minutes is 3.75 hours. Keep units on each line to catch conversions that run in the wrong direction.
Pause and explain
A car travels 150 km in 2.5 hours. What is its average speed?
Worked example
Simplify 18:30 and share $64 in that ratio. Then find the average speed for 45 km travelled in 3 h 45 min.
Show the worked solution
- Divide 18:30 by 6 to get 3:5. There are 8 parts altogether.
- Each part is $64 ÷ 8 = $8, so the shares are $24 and $40.
- 3 h 45 min = 3.75 h; average speed = 45 ÷ 3.75.
Answer Ratio 3:5; shares $24 and $40; average speed 12 km/h.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A drink uses concentrate and water in the ratio 2:7. How much of each is needed for 1.8 litres altogether?
Give me a hint
Add the parts, find one part, then multiply by 2 and 7.
Compare my reasoning
- There are 2 + 7 = 9 parts.
- One part is 1.8 ÷ 9 = 0.2 litres.
- Concentrate is 0.4 litres and water is 1.4 litres.
0.4 L concentrate and 1.4 L water.
Look for these in your work
- Total parts counted
- One part found
- Shares total 1.8 L
A runner covers 5 km in 28 minutes. Find the average speed in km/h, to 3 significant figures.
Give me a hint
Convert minutes to hours before dividing distance by time.
Compare my reasoning
- 28 minutes = 28/60 hours.
- Average speed = 5 ÷ (28/60) = 10.714… km/h.
- Round only the final value to 3 significant figures.
10.7 km/h.
Look for these in your work
- Time converted
- Distance divided by elapsed time
- Final value rounded once
A map scale is 1:50 000. Two towns are 7.6 cm apart on the map. Find the real distance in kilometres. A second route is 13.5 cm and takes 18 minutes; compare its average speed.
Give me a hint
Convert map centimetres to real centimetres, then kilometres; convert the travel time to hours.
Compare my reasoning
- 7.6 × 50 000 = 380 000 cm = 3.8 km.
- 13.5 cm represents 6.75 km.
- 18 minutes = 0.3 hours, so the second route's average speed is 6.75 ÷ 0.3 = 22.5 km/h.
First route distance 3.8 km; second route average speed 22.5 km/h.
Look for these in your work
- Scale applied in the correct direction
- Centimetres converted to kilometres
- Time rate calculated with matching units
Common mistakes
- Dividing the total by only one ratio part
- Averaging speeds without accounting for travel time
- Mixing minutes with hours
- Reversing a currency or unit rate
What can you explain now?
How many dollars are in the smaller share when $56 is divided in the ratio 3:5?
Compare with the explanation
$21
There are 8 parts, so each is $7 and the smaller share is 3 × $7.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you share a total using a ratio, and what must be converted before calculating average speed?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.