Ratio, proportion and rates

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 $3:5$.

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.

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.

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.

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.

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