Ratio, proportion and rates
| English | Chinese | Pinyin |
|---|---|---|
| ratio | 比 | bǐ |
| direct proportion | 正比例 | zhèng bǐ lì |
| rate | 比率 | bǐ lǜ |
| average speed | 平均速度 | píng jūn sù dù |
Splitting a pizza fairly
- You and a friend share a pizza in ratio 比 $3:5$. You get $\dfrac{3}{8}$, your friend gets $\dfrac{5}{8}$.
- Ratios are about fair sharing — splitting a total into parts that keep a fixed relationship.
What is a ratio?
- A ratio $a:b$ compares two quantities — "for every $a$ of one thing, there are $b$ of the other".
- Simplify by dividing both parts by their HCF: $20:30 = 2:3$.
- Ratios have no units — $3\text{ cm}:5\text{ cm}$ is just $3:5$.
Ratios ≠ fractions. The ratio $3:5$ does NOT mean "$\dfrac{3}{5}$ of the total". It means the total is split into $3+5=8$ parts: one gets $\dfrac{3}{8}$, the other $\dfrac{5}{8}$.
Direct proportion
y = ax
Direct proportion is a straight line through the origin — double x and you double y.
What is the ratio 24:36 in its simplest form?
HCF of 24 and 36 is 12. 24÷12 = 2, 36÷12 = 3. Simplest form: 2:3.
The ratio 3:5 means one person gets 3/5 of the total.
3:5 means 3+5 = 8 parts total. One gets 3/8, the other gets 5/8 — not 3/5.
Sharing in a ratio
- Method: add the parts, divide the total, multiply back.
- Share 48 in the ratio $3:5$:
- Total parts $= 3 + 5 = 8$. One part $= 48 \div 8 = 6$.
- Shares: $3 \times 6 = 18$ and $5 \times 6 = 30$. Check: $18 + 30 = 48$. ✓
Bar model: split the total into equal parts, then group them by the ratio.
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.
Share 120 in the ratio 1:3. What is the smaller share?
4 parts, one part = 120/4 = 30. Smaller share = 1 × 30 = 30.
Direct proportion 正比例
- Two quantities are in direct proportion if they increase at the same rate 比率.
- If 3 pens cost 1.80, then 1 pen costs $1.80 \div 3 = 0.60$, so 7 pens cost $7 \times 0.60 = 4.20$.
- The key idea: find the cost of one, then scale up.
Recipe scaling. A cake for 4 people needs 200 g flour. For 10 people: flour per person $= 50$ g, so $10 \times 50 = 500$ g.
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 quantities: price per kg, km per hour, words per minute.
- Average speed 平均速度 $= \dfrac{\text{total distance}}{\text{total time}}$.
- 45 km in 3 hours 45 minutes $= 45 \div 3.75 = 12$ km/h.
A cyclist travels 45 km in 3 hours 45 minutes. What is the average speed in km/h?
3 h 45 min = 3.75 hours. Average speed = 45 ÷ 3.75 = 12 km/h.
You've got it
- simplify a ratio by the HCF; share by counting parts (add, divide, multiply back)
- proportion: find the cost of one, then scale up
- average speed $= \text{distance} \div \text{time}$ (convert time to decimal hours first)
- ratio $3:5$ means $\dfrac{3}{8}$ and $\dfrac{5}{8}$ of the total — not $\dfrac{3}{5}$