📚 8C Unit 5: Ratio, Proportion and Rates | 8C 第五单元：比、比例与速率
In Unit 5 of the 8C mathematics course, ratio, proportion and rates are introduced as essential tools for comparing quantities, scaling real-life situations and solving fair-sharing problems. This article reviews the core methods and applications you need to master.
在 8C 数学课程第五单元中，比、比例和速率是用于比较数量、处理实际缩放问题以及解决公平分配问题的重要工具。本文回顾你需要掌握的核心方法和应用。
1. What is a ratio? | 什么是比？
A ratio compares two or more quantities of the same kind. We write a ratio like 3 : 2, and we say ‘3 to 2’. The order matters because 3 : 2 is not the same as 2 : 3.
比用来比较两个或多个同类量的大小。我们把它写成 3 : 2，读作“3 比 2”。顺序很重要，因为 3 : 2 与 2 : 3 不同。
Ratios have no units, and they can compare parts to parts, parts to a whole, or whole quantities in a given situation.
比没有单位，它可以表示部分与部分、部分与整体，或特定情境下的整体数量之间的比较。
2. Simplifying ratios | 化简比
To simplify a ratio, divide every part by the highest common factor, or HCF. For example, 12 : 18 can be simplified by dividing both numbers by 6.
化简比时，将每一项都除以最大公因数。例如，12 : 18 可以同时除以 6 来化简。
12 : 18 = 2 : 3
If the ratio contains decimals or fractions, first multiply every part by the same power of 10 or by the lowest common multiple of the denominators. This clears the decimals or fractions before simplifying.
如果比中含有小数或分数，先给所有项乘以相同的 10 的幂或分母的最小公倍数。这样可以先消去小数或分数，再继续化简。
3. Equivalent ratios | 等价比
Equivalent ratios are obtained by multiplying or dividing both parts by the same non-zero number. They are used to scale recipes, models or mixtures without changing the proportion.
等价比通过将比的各项同时乘以或除以同一个非零数得到。它们用于缩放食谱、模型或混合物，而不会改变比例关系。
3 : 4 = 6 : 8 = 9 : 12
You can check equivalence by writing each ratio as a fraction and seeing whether the fractions are equal.
你可以把每个比写成分数，检查分数是否相等，来判断它们是否为等价比。
4. Sharing a quantity in a given ratio | 按给定比例分配
To share a total in the ratio a : b, first add the parts to find the total number of shares, then divide the total by this number, and finally multiply by each part.
按 a : b 分配一个总量时，先把两部分相加得到总份数，再用总量除以总份数，最后分别乘以每一份。
Share £120 in the ratio 2 : 3 → 2 + 3 = 5 parts → 1 part = £24 → £48 : £72
Always label the final shares in the context of the question, for example £48 for the first person and £72 for the second person.
务必在题目情境中标明最终份额，例如第一个人 £48，第二个人 £72。
5. Ratio and fractions | 比与分数
A ratio a : b can be expressed as fractions of the whole. In a 2 : 3 mixture, the first component is 2/5 of the total and the second component is 3/5 of the total.
比 a : b 可以表示成整体的分数。在 2 : 3 的混合物中，第一种成分占总量的 2/5，第二种成分占总量的 3/5。
2 : 3 → 2/5 and 3/5 of the whole
Do not confuse a ratio part with the whole. If the ratio of boys to girls is 2 : 3, then boys are 2/5 of the class, not 2/3.
不要把一个比的部分与整体混淆。如果男生与女生的比是 2 : 3，那么男生占全班的 2/5，而不是 2/3。
6. Direct proportion | 正比例
Two quantities are directly proportional if their ratio stays constant. As one doubles, the other doubles; as one trebles, the other trebles. We can write y = kx, where k is the constant of proportionality.
如果两个量的比保持恒定，它们就成正比例。一个翻倍，另一个也翻倍；一个变为三倍，另一个也变为三倍。可以写成 y = kx，其中 k 是比例常数。
If 5 pens cost £3, then 15 pens cost £9.
To solve, find the cost of one pen first: £3 ÷ 5 = £0.60 per pen, then multiply by 15.
解题时，先求出一支笔的价格：£3 ÷ 5 = £0.60/支，再乘以 15。
7. Unitary method and rates | 单位法与速率
The unitary method finds the value of one unit first, then scales up or down. Rates compare two different units, such as km/h, £/kg or words per minute.
单位法先求出一个单位的量，再进行放大或缩小。速率比较两个不同的单位，例如 km/h、£/kg 或每分钟字数。
If a car travels 180 km in 2 h, its speed is 180 ÷ 2 = 90 km/h.
When using rates, keep the units in the calculation so you can check that the final unit makes sense.
使用速率时，计算过程中保留单位，这样可以检查最终单位是否合理。
8. Scale drawings, maps and models | 比例图、地图与模型
A scale ratio such as 1 : 50 000 means that 1 cm on the map represents 50 000 cm, or 0.5 km, in real life. Convert units carefully before multiplying.
比例尺如 1 : 50 000 表示地图上的 1 cm 代表实际 50 000 cm，即 0.5 km。计算前要小心换算单位。
Map distance 4 cm at scale 1 : 25 000 gives 4 × 25 000 = 100 000 cm = 1 km.
Remember that 100 000 cm = 1 000 m = 1 km. Write down each conversion step to avoid place-value errors.
记住 100 000 cm = 1 000 m = 1 km。写出每一步换算，避免数位错误。
9. Percentages as proportions | 百分比作为比例
A percentage is a ratio compared with 100. To convert a ratio to percentages, find the total number of parts, divide each part by the total and multiply by 100%.
百分比是与 100 相比的比例。要把比转换为百分比，先求出总份数，再用每份除以总数并乘以 100%。
3 : 2 → total = 5 → 3/5 = 60% and 2/5 = 40%
This is useful when comparing groups of different sizes or when a question asks for a percentage breakdown.
这在比较不同规模的组或题目要求给出百分比构成时非常有用。
10. Percentage increase and decrease | 百分数增减
To increase an amount by p%, multiply by (1 + p/100). To decrease by p%, multiply by (1 – p/100). Never add or subtract the percentage directly without identifying the correct multiplier.
将某量增加 p%，乘以 (1 + p/100)；减少 p%，乘以 (1 – p/100)。不要在没有确定正确乘数的情况下直接加减百分数。
| Change | Multiplier |
|---|---|
| Increase by 15% | ×1.15 |
| Decrease by 15% | ×0.85 |
| Increase by 5% | ×1.05 |
| Decrease by 47% | ×0.53 |
Using multipliers keeps percentage calculations consistent and makes it easier to combine several changes.
使用乘数可以让百分比计算保持一致，也更容易组合多个变化。
11. Reverse percentages and combining changes | 逆运算百分比与复合变化
If a price after a 20% increase is £96, the original price is found by dividing by 1.20, not by subtracting 20%.
如果某价格增加 20% 后为 £96，原价应除以 1.20，而不是减去 20%。
Original price = £96 ÷ 1.20 = £80
For successive changes, apply the multipliers in order. For example, +10% then -10% gives ×1.10 ×0.90 = ×0.99, so there is a net 1% decrease.
对于连续变化，按顺序乘以相应的乘数。例如 +10% 再 -10% 得到 ×1.10 ×0.90 = ×0.99，即净减少 1%。
This shows that percentage increases and decreases do not cancel out simply by addition or subtraction.
这说明百分数增加和减少不能简单地通过加法或减法相互抵消。
12. Exam-style tips and common mistakes | 考试技巧与常见错误
Always check the order of a ratio and write the final shares in the correct context.
务必检查比的顺序，并在正确的情境中写出最终份额。
Find the value of one part before sharing a total in a given ratio.
在按给定比例分配总量之前，先求出一份的量。
Convert units carefully in scale and rate questions, and show each step.
在比例尺和速率问题中仔细换算单位，并写出每一步。
Use multipliers for percentage change, especially in reverse percentage and successive-change problems.
对于百分比变化，尤其是逆运算和连续变化问题，使用乘数。
Finally, simplify fully and double-check that ratio parts add to the correct total when needed.
最后要彻底化简，并在需要时再次检查比的各部分加起来是否等于正确的总量。
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