📚 IGCSE Maths Chapter 7: Ratio, Proportion and Percentage | IGCSE 数学第7章：比、比例与百分数
Ratio, proportion and percentage are three connected tools that allow you to compare quantities, share amounts fairly, describe change and solve real-life problems in IGCSE Mathematics. This chapter brings together the essential methods you need for both the Core and Extended tiers, from simplifying a ratio to applying compound interest and inverse proportion.
比、比例与百分数是三个相互关联的数学工具，用来比较数量、公平分配、描述变化并解决 IGCSE 数学中的实际问题。本章涵盖 Core 与 Extended 两个层级所需的核心方法，从化简比到复利和反比例的应用。
1. Understanding Ratio Notation | 理解比的表示法
A ratio is a way of comparing two or more quantities of the same kind. For example, if a fruit basket has 3 apples and 5 oranges, the ratio of apples to oranges is written as 3 : 5. You must keep the order the same as the question states.
比是比较两个或多个同类数量的方式。例如，一个水果篮里有 3 个苹果和 5 个橙子，苹果与橙子的比写作 3 : 5。比的顺序必须与题目给出的顺序一致。
Ratios do not have units. The ratio 3 cm : 5 cm is the same as 3 : 5 because both quantities are measured in the same unit. You can also compare three or more quantities, such as 2 : 3 : 5 for red, blue and green counters.
比没有单位。3 cm : 5 cm 与 3 : 5 相同，因为两个量使用相同单位。你也可以比较三个或更多数量，例如红、蓝、绿计数器的比为 2 : 3 : 5。
2. Simplifying Ratios | 化简比
To simplify a ratio, divide every part of the ratio by the highest common factor of all the numbers. For instance, the ratio 12 : 18 can be simplified by dividing both parts by 6, giving 2 : 3. The simplified ratio has the same meaning as the original ratio.
要化简比，将比的每一项除以所有数的最大公因数。例如，12 : 18 可以除以 6，得到 2 : 3。化简后的比与原比含义相同。
If the ratio contains fractions or decimals, multiply every part by the same number to produce whole numbers first. The ratio 0.2 : 0.5 can be multiplied by 10 to give 2 : 5. The ratio 1/2 : 3/4 can be multiplied by 4 to give 2 : 3.
如果比包含分数或小数，先将每一项乘以同一个数化为整数。0.2 : 0.5 乘以 10 得到 2 : 5。1/2 : 3/4 乘以 4 得到 2 : 3。
You should always leave a ratio in its simplest form unless a question asks otherwise. This makes the comparison clearer and helps in later calculations.
除非题目另有要求，你应始终把比化为最简形式。这样比较更清晰，也有助于后续计算。
3. Dividing Quantities in a Given Ratio | 按给定比分配量
a quantity in a given ratio, first find the total number of parts. For example, to divide £60 in the ratio 2 : 3, the total number of parts is 2 + 3 = 5. One part is worth £60 ÷ 5 = £12, so the shares are 2 × £12 = £24 and 3 × £12 = £36.
要按给定比分配一个量，先求出总份数。例如，按 2 : 3 分配 £60，总份数为 2 + 3 = 5。一份为 £60 ÷ 5 = £12，因此份额为 2 × £12 = £24 和 3 × £12 = £36。
For three or more parts, the same rule applies. Divide £120 in the ratio 3 : 4 : 5. The total parts are 3 + 4 + 5 = 12, so one part is £120 ÷ 12 = £10. The shares are £30, £40 and £50.
对于三个或更多部分，规则相同。按 3 : 4 : 5 分配 £120，总份数为 3 + 4 + 5 = 12，一份为 £120 ÷ 12 = £10。份额分别为 £30、£40 和 £50。
This method is often tested with money, lengths, angles in a triangle, or ingredients in a recipe. Always check that the parts add up to the original total.
这个方法常出现在金钱、长度、三角形角度或食谱配料问题中。始终检查各部分之和是否等于原来的总量。
4. Direct Proportion | 正比例
y proportional if one quantity increases at the same rate as the other. We write this as y ∝ means y = kx, where k is the constant of proportionality. If you double x, y also doubles.
如果两个量以相同速率增大，它们就成正比例。写作 y ∝ x，即 y = kx，其中 k 是比例常数。如果 x 加倍，y 也加倍。
n 12 = k × 4, so k = 3. The formula is y = 3x. You can then use it to find unknown values.
求 k 时，代入一组已知值。如果 y = 12 且 x = 4，则 12 = k × 4，所以 k = 3。公式为 y = 3x。然后可以用它求未知值。
y = kx
Common examples include cost and number of items, distance and time at constant speed, and mass and volume of a substance.
常见例子包括商品数量与总价、匀速下的距离与时间、物质的质量与体积。
5. Inverse Proportion | 反比例
portional if one quantity increases while the other decreases at the same rate. We write y ∝ 1/x, which means y = k/x. If you double x, y becomes half of its previous value.
如果两个量一个增大而另一个以相同速率减小，它们就成反比例。写作 y ∝ 1/x，即 y = k/x。如果 x 加倍，y 变成原来的一半。
k/2, so k = 16. The formula is y = 16/x. Use this formula to calculate y for any other x.
求 k 时，代入一组已知值。如果 y = 8 且 x = 2，则 8 = k/2，所以 k = 16。公式为 y = 16/x。用这个公式可以计算任意 x 对应的 y。
y = k/x
Typical inverse proportion problems include the number of workers and time to complete a job, speed and travel time for a fixed distance, and number of taps and time to fill a tank.
典型反比例问题包括工人数量与完成工作的时间、固定路程下的速度与行驶时间、水龙头数量与注满水箱的时间。
6. Percentage Increase and Decrease | 百分数增减
A percentage increase or decrease compares the change in a quantity with its original value. The percentage change is calculated as change divided by original value, multiplied by 100.
百分数增减将数量的变化与原值进行比较。百分数变化等于变化量除以原值，再乘以 100。
Percentage change = (change ÷ original value) × 100%
To increase a quantity by r%, multiply the original amount by (1 + r/100). For example, increasing £80 by 15% gives £80 × 1.15 = £92.
要将一个量增加 r%，将原量乘以 (1 + r/100)。例如，£80 增加 15% 得到 £80 × 1.15 = £92。
To decrease a quantity by r%, multiply the original amount by (1 − r/100). Decreasing £80 by 15% gives £80 × 0.85 = £68.
要将一个量减少 r%，将原量乘以 (1 − r/100)。£80 减少 15% 得到 £80 × 0.85 = £68。
Be careful with repeated percentage changes: a 10% increase followed by a 10% o the original value. The total multiplier is 1.10 × 0.90 = 0.99, so the final value is 99% of the original.
注意连续百分数变化：先增加 10% 再减少 10% 不会回到原值。总乘数为 1.10 × 0.90 = 0.99，因此最终值是原值的 99%。
7. Reverse Percentages | 逆推百分数
Reverse percentage problems ask you to find the original amount before a percentage increase or decrease. If a price after a 20% increase is £96, then the final amount is 120% of the original.
逆推百分数问题要求你求百分数增减之前的原值。如果价格增加 20% 后是 £96，那么最终金额是原值的 120%。
Write 120% as 1.20 and divide the final value by this multiplier. The original price is £96 ÷ 1.20 = £80. This is faster and safer than guessing and checking.
将 120% 写作 1.20，用最终值除以这个乘数。原价为 £96 ÷ 1.20 = £80。这比猜测和检查更快更可靠。
For a decrease, use a multiplier less than 1. If a price after a 25% discount is £60, then £60 represents 75% of the original. The original price is £60 ÷ 0.75 = £80.
对于减少，使用小于 1 的乘数。如果折扣 25% 后的价格是 £60，那么 £60 代表原值的 75%。原价为 £60 ÷ 0.75 = £80。
Original value = final value ÷ (1 ± r/100)
Always identify whether the final value is more than or less than 100% of the original before choosing the multiplier.
在选择乘数之前，始终判断最终值是原值的 100% 以上还是以下。
8. Simple and Compound Interest | 单利与复利
al, R is the annual rate of interest, and T is the time in years. The final amount is P + I.
单利只按原始本金计算。公式为 I = PRT/100，其中 P 为本金，R 为年利率，T 为时间（年）。最终金额为 P + I。
I = PRT/100
For example, £500 invested at 4% per year for 3 years earns simple interest of £500 × 4 × 3 ÷ 100 = £60. The total amount is £560.
例如，£500 以 4% 年利率投资 3 年，单利为 £500 × 4 × 3 ÷ 100 = £60。总金额为 £560。
rincipal plus any interest already earned. The final amount A is given by A = P(1 + r/100)n, where n is the number of compounding periods.
复利按本金加上已赚取利息计算。最终金额 A 由 A = P(1 + r/100)n 给出，其中 n 是复利计息期数。
A = P(1 + r/100)n
The same £500 invested at 4% per year for 3 years with compound interest gives £500 × (1.04)3 = £ e difference from simple interest is small at first but grows quickly over longer periods.
同样 £500 以 4% 年利率复利投资 3 年，得到 £500 × (1.04)3 = £562.43。起初与单利的差别很小，但在较长时间内会迅速扩大。
9. Ratio and Proportion in Real Life: Scale and Maps | 生活中的比与比例：比例尺与地图
Maps, scale drawings and models use ratios to represent real distances. A scale of 1 : 50 means that 1 cm on the drawing represents 50 cm in real life. A map scale of 1 : 25 000 means 1 cm on the map represents 25 000 cm, or 250 m, on the ground.
地图、比例图和模型使用比来表示实际距离。比例尺 1 : 50 表示图上 1 cm 代表实际 50 cm。地图比例尺 1 : 25 000 表示地图上 1 cm 代表地面 25 000 cm，即 250 m。
A 3 cm line on a 1 : 50 scale drawing represents 3 cm × 50 = 150 cm = 1.5 m in real life.
求实际长度时，将图上长度乘以比例系数。比例尺为 1 : 50 的图上 3 cm 线段代表实际 3 cm × 50 = 150 cm = 1.5 m。
the scale factor. A real wall of 6 m on a scale of 1 : 40 is drawn as 600 cm ÷ 40 = 15 cm.
求图上长度时，将实际长度除以比例系数。比例尺为 1 : 40 时，实际 6 m 的墙画作 600 cm ÷ 40 = 15 cm。
- Always convert to the same unit before multiplying or dividing.
- Scale 1 : n means multiplying by n to go from drawing to real life.
- Scale n : 1 is used for enlargements, such as a large diagram of a small insect.
- 在乘除之前，始终换算为相同单位。
- 比例尺 1 : n 表示从图上到实际乘以 n。
- 放大比例尺 n : 1 用于放大图，例如小昆虫的放大图。
10. Exam Technique and Common Mistakes | 考试技巧与常见错误
When answering ratio and proportion questions, always write down the ratio in the order given and simplify it fully. If the question asks for a share of money, give the answer in the correct unit and check that the parts sum to the total.
回答比和比例问题时，始终按给定顺序写出比，并完全化简。如果题目要求金钱份额，答案要使用正确单位，并检查各部分之和等于总量。
For percentage questions, decide whether you are finding the change, the final amount, or the original amount. Use multipliers rather than repeated addition or subtraction, because they work for both increase and decrease and support clean working.
对于百分数问题，先判断是求变化量、最终量还是原量。使用乘数而不是反复加减，因为乘数对增加和减少都适用，书写也更清晰。
| Skill | Common mistake | Correct approach |
| Simplifying ratio | Dividing only one part | Divide all parts by the HCF |
| Reverse percentage | Subtracting 20% from a price after a 20% increase | Divide by the correct multiplier, such as 1.20 |
| Map scale | Mixing cm and m | Convert to the same unit first |
Before the exam, practise identifying direct and inverse proportion from a word problem: if increasing one quantity makes the other increase at the same rate, use y = kx; if increasing one quantity makes the other decrease, use y = k/x.
考前练习从文字题中识别正比例与反比例：如果一个量增大使另一个量以相同速率增大，使用 y = kx；如果一个量增大使另一个量减小，使用 y = k/x。
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