Ratio, proportion and rates
Ratio and proportion
- A ratio compares quantities ($a:b$); simplify by dividing by the HCF ($20:30 = 2:3$).
- Dividing in a ratio: share 48 in 3:5 → 8 parts, one part = 6, shares 18 and 30.
- Proportion: 3 pens cost 1.80 → one pen 0.60 → 7 pens 4.20.
Practice
Share 48 in the ratio 3:5. What is the larger share?
8 parts, one part = 48/8 = 6, so the larger share is 5 × 6 = 30.
Practice
If 3 pens cost 1.80, what do 7 pens cost?
One pen = 1.80/3 = 0.60, so 7 pens = 7 × 0.60 = 4.20.
Rates and speed
- A rate compares two different units (price per kg, km per hour).
- Average speed $= \dfrac{\text{total distance}}{\text{total time}}$. e.g. 45 km in 3.75 h = 12 km/h.
- Density $= \dfrac{\text{mass}}{\text{volume}}$.
Practice
A cyclist travels 45 km in 3.75 hours. What is the average speed (km/h)?
average speed = distance ÷ time = 45 ÷ 3.75 = 12 km/h.
You've got it
Key idea
- simplify a ratio by the HCF; share by counting parts
- proportion: find the cost of one, then scale up
- average speed = distance ÷ time (convert time to hours first)