Ratio, proportion and rates
Introduced| English | Meaning |
|---|---|
| ratio/ˈreɪʃɪəʊ/ | A comparison of quantities expressed through division or the relative number of parts. |
| direct proportion/daɪˈrekt prəˈpɔːʃn/ | A relationship y = kx, a straight line through the origin with constant ratio y/x. |
| rate/reɪt/ | A quantity measured or calculated per unit of another quantity, often per unit time. |
| average speed/ˈævrɪdʒ spiːd/ | Total distance divided by total time, including any stops. |
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?
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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.
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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?
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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?
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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?
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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?
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3 h 45 min = 3.75 hours. Average speed = 45 ÷ 3.75 = 12 km/h.
Rates beyond speed
- Compare equal quantities for best value: 3 kg for 18 yuan costs $c=18/3=6$ yuan per kg; 5 kg for 28 yuan costs $c=28/5=5.6$ yuan per kg, so the larger pack is cheaper per kg.
- A tap at 12 litres per minute fills 90 litres in $t=V/r=90/12=7.5$ min. If density is given as $\rho=m/V$, then 90 g in $30\text{ cm}^3$ gives $\rho=90/30=3\text{ g/cm}^3$. For average speed include stops in the time.
At 12 litres per minute, how many minutes are needed for 90 litres?
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Time = volume / rate = 90 / 12 = 7.5 minutes.
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 $\dfrac{3}{5}$