What you'll learn
Ratio, rates and proportion are everyday number skills that form an important part of the CSEC Mathematics syllabus under Computation and Number. A ratio compares quantities of the same kind, a rate compares quantities of different kinds (such as distance per hour), and proportion describes how ratios stay equal. In this guide you will learn to simplify ratios, share a quantity in a given ratio, work with direct and inverse proportion, and handle rates including speed, currency exchange and best-buy comparisons. These skills appear throughout Paper 1 and Paper 2 and connect to percentages, variation and measurement. The Caribbean context — currency conversion, recipes, fuel consumption — makes them especially practical.
Key terms and definitions
Ratio — a comparison of two or more quantities of the same kind, written a : b.
Rate — a comparison of two quantities of different kinds, e.g. km/h or $/kg.
Proportion — a statement that two ratios are equal, e.g. a : b = c : d.
Direct proportion — as one quantity increases, the other increases in the same ratio.
Inverse proportion — as one quantity increases, the other decreases in the same ratio.
Unitary method — finding the value of one unit first, then scaling to the amount needed.
Exchange rate — the rate at which one currency converts to another.
Core concepts
Simplifying ratios
A ratio is simplified by dividing every part by their highest common factor, just like a fraction. So 12 : 18 divides by 6 to give 2 : 3. Quantities must be in the same units before forming a ratio: 50 cm to 2 m becomes 50 cm to 200 cm, i.e. 1 : 4. Ratios can also be written with more than two parts, such as 2 : 3 : 5.
Sharing in a given ratio
To divide a quantity in a ratio, add the parts to find the total number of shares, find the value of one share, then multiply for each portion. To share $240 in the ratio 3 : 5, the total is 8 shares, one share is $240 ÷ 8 = $30, so the portions are 3 × 30 = $90 and 5 × 30 = $150. Always check the portions add back to the original total.
Direct proportion and the unitary method
In direct proportion, doubling one quantity doubles the other. The unitary method is the most reliable approach: find the value of one unit, then scale. If 5 exercise books cost $30, one book costs $6, so 8 books cost $48. This method works for recipes, fuel, wages and many CSEC problems.
Inverse proportion
In inverse proportion, increasing one quantity decreases the other, so their product stays constant. If 4 workers take 12 days to finish a job, the total work is 4 × 12 = 48 worker-days; with 6 workers the time is 48 ÷ 6 = 8 days. Recognising that the product is fixed is the key.
Rates: speed, density and exchange
A rate links different units. Average speed = distance ÷ time (km/h); be sure the units match before dividing. Currency exchange is a rate: to convert, multiply or divide by the exchange rate consistently — for example, if US$1 = J$155, then US$40 = J$6 200. Best-buy problems compare rates such as price per gram, choosing the lower unit price.
Worked examples
Example 1: Sharing in a ratio (Paper 2 style)
Three siblings share a $4 500 gift in the ratio 2 : 3 : 4. How much does each receive?
Total shares = 2 + 3 + 4 = 9. One share = $4 500 ÷ 9 = $500. So they receive 2 × 500 = $1 000, 3 × 500 = $1 500, and 4 × 500 = $2 000. Check: 1 000 + 1 500 + 2 000 = $4 500. ✓
Example 2: Direct proportion (Paper 1/2 style)
A car uses 9 litres of fuel to travel 108 km. How much fuel is needed for 156 km?
By the unitary method, fuel per km = 9 ÷ 108 = 0.0833 litres. For 156 km, fuel = 0.0833 × 156 ≈ 13 litres. (Equivalently, 156 ÷ 108 × 9 = 13.)
Example 3: Inverse proportion (Paper 2 style)
It takes 8 pumps 6 hours to empty a tank. How long would 12 pumps take, working at the same rate?
The product is constant: 8 × 6 = 48 pump-hours. With 12 pumps, time = 48 ÷ 12 = 4 hours.
Common mistakes and how to avoid them
Comparing different units. Convert to the same unit before forming a ratio (e.g. metres and centimetres, hours and minutes).
Treating inverse proportion as direct. More workers means less time. If the answer should get smaller but your method makes it bigger, you have used the wrong type.
Forgetting to add the ratio parts. When sharing, the number of shares is the sum of the parts, not just one part.
Dividing currency the wrong way. Decide clearly which currency you are converting to and apply the rate consistently; estimate to check the answer is sensible.
Rounding mid-calculation in rate problems. Keep accuracy until the end, especially with fuel and speed.
Exam technique for Ratio, Rates and Proportion
Decide direct or inverse first. Ask: as one goes up, does the other go up (direct) or down (inverse)? This chooses your method.
Use the unitary method. Finding the value of one unit is the safest route through most proportion problems.
Check totals when sharing. The parts must sum back to the original amount.
Match units before dividing rates. Convert minutes to hours, grams to kilograms, etc., so the rate is meaningful.
Estimate to sense-check. A rough mental estimate catches errors in conversions and best-buy comparisons.
Quick revision summary
A ratio compares like quantities (a : b) and is simplified by dividing by the highest common factor, after converting to the same units. To share a quantity in a ratio, add the parts to get the total shares, find one share, then multiply for each portion and check the total. A rate compares different quantities, such as speed (distance ÷ time), density, or an exchange rate; ensure units match before dividing. In direct proportion quantities rise and fall together — use the unitary method: find the value of one unit, then scale. In inverse proportion one rises as the other falls, so their product stays constant (e.g. worker-days). Decide whether a problem is direct or inverse before choosing your method, keep full accuracy until the final step, estimate to sense-check conversions and best-buys, and always confirm that shared portions add back to the original total.