How they connect
A ratio compares like quantities in the same unit (no unit), e.g. 2 : 3. A rate compares unlike quantities in different units (has a unit), e.g. 80 km/h. A proportion says two ratios or two rates are equal, e.g. 2/3 = 4/6. Together they solve many real-life problems.
Key idea / Key idea
Use a ratio to compare or share, a rate to find "per one unit", and a proportion (cross-multiply) to scale up or down.
A combined problem
Worked example / Worked example
Fruit juice mixes water and syrup in the ratio 5 : 2. To make 3.5 litres of syrup, how much water is needed?
Set up a proportion: water/syrup = 5/2 = x/3.5.
Cross-multiply: 2x = 5 × 3.5 = 17.5, so x = 17.5 ÷ 2 = 8.75 litres of water.
Total drink = 8.75 + 3.5 = 12.25 litres.
Rates help you plan too. If a machine fills 6 bottles per minute (a rate), then in 15 minutes it fills 6 × 15 = 90 bottles, and to fill 300 bottles it needs 300 ÷ 6 = 50 minutes. Notice the same three tools appear again and again: compare with a ratio, find a unit value with a rate, then scale with a proportion. Choosing the right tool for a problem is the real skill, and with practice you will spot which one to reach for straight away.
Remember / Remember
- Ratio: same units, no unit in the answer.
- Rate: different units, keep the unit.
- Proportion: two equal ratios or rates.
- Cross-multiply to find any missing value.