Reference · module 5
Ratio and rates
Everything this module explains, on one page.
Ratios
Simplify ratios and share an amount in a given ratio.
- A ratio compares amounts: a recipe with 2 cups of flour for every 3 cups of milk has flour and milk in the ratio 2 : 3. Order matters, since 3 : 2 is a different recipe.
- Ratios simplify like fractions: divide both sides by a common factor. 12 : 18 becomes 2 : 3 by dividing by 6. The ratio stays the same, only the numbers get smaller.
- To share 50 euros in the ratio 2 : 3, count the parts: 2 + 3 is 5. One part is 50 ÷ 5, which is 10, so the shares are 20 and 30.
The mistake you are about to make
50 in 2 : 3 = 25 and 25
50 in 2 : 3 = 20 and 30
Sharing usually means splitting in half, and a ratio seems like a detail. In 2 : 3 there are five equal parts, and one side gets three of them.
Proportion
Scale quantities up and down, and see when more means less.
- Two quantities are in direct proportion when doubling one doubles the other. If 3 notebooks cost 6 euros, 6 notebooks cost 12 and 9 cost 18.
- The unit rate makes every question easy: find the value of one. If 5 kg of apples cost 12 euros, 1 kg costs 2.40, so 3 kg cost 7.20. It also tells which pack is cheaper.
- In inverse proportion, more of one means less of the other. If 4 workers paint a fence in 6 hours, 8 workers need 3 hours: the total work, 24 worker-hours, stays the same.
The mistake you are about to make
4 workers: 6 h → 8 workers: 12 h
4 workers: 6 h → 8 workers: 3 h
Doubling one number makes the other double in most proportion problems, so the habit takes over. More workers finish faster: the hours are halved, not doubled.
Speed, distance and time
Solve travel problems and handle minutes and hours correctly.
- Distance is speed times time. A car at 60 km/h drives 180 km in 3 hours. Rearranged, speed is distance ÷ time and time is distance ÷ speed.
- Keep the units matching. With km/h, time must be in hours: 30 minutes is 0.5 hours, 45 minutes is 0.75, and 20 minutes is one third of an hour.
- Average speed is the whole distance divided by the whole time, not the average of the speeds. 60 km in 1 hour and 60 km in 2 hours is 120 km in 3 hours: 40 km/h.
The mistake you are about to make
1 h 30 min = 1.3 h
1 h 30 min = 1.5 h
Minutes are written after the hours like decimals, so 1 h 30 min looks like 1.3. An hour has 60 minutes, not 100: half an hour is 0.5.