Mastering Ratio and Proportion | 掌握比与比例

📚 Mastering Ratio and Proportion | 掌握比与比例

Ratio and proportion are powerful tools used to compare quantities and solve real-life problems. This revision guide covers the key concepts, methods, and common pitfalls you need for your IGCSE Mathematics exam.

比与比例是比较数量、解决实际问题的有力工具。本复习指南涵盖了IGCSE数学考试所需的核心概念、方法和常见易错点。

1. What is a Ratio? | 什么是比？

A ratio compares two or more quantities of the same kind. For example, if a fruit basket has 3 apples and 2 oranges, the ratio of apples to oranges is written as 3:2.

比用于比较同类型的两个或多个数量。例如，如果一个水果篮里有3个苹果和2个橙子，那么苹果与橙子的比写作3:2。

The order of a ratio is important. 3:2 is not the same as 2:3. The ratio 3:2 means for every 3 apples, there are 2 oranges.

比的顺序很重要。3:2与2:3不同。3:2表示每有3个苹果，就有2个橙子。

2. Equivalent Ratios | 等价比

Equivalent ratios are obtained by multiplying or dividing both parts of a ratio by the same non-zero number. For example, 3:2 is equivalent to 6:4 and 15:10.

等价比是通过对比的两部分同时乘以或除以同一个非零数得到的。例如，3:2等价于6:4和15:10。

To simplify a ratio, divide both parts by their highest common factor (HCF). If 12:18 is simplified, divide both by 6 to get 2:3.

要化简比，将比的两部分同时除以它们的最大公因数（HCF）。例如12:18，两边同时除以6得到2:3。

Original Ratio	Simplified Ratio
20:30	2:3
45:60	3:4
16:24	2:3

3. Dividing a Quantity in a Given Ratio | 按给定比分配数量

To divide a quantity in a given ratio, first add the parts of the ratio to find the total number of parts. Then divide the quantity by the total parts and multiply by each part.

按给定比分配数量时，先将比的各项相加得到总份数，然后用数量除以总份数，再乘以每一项。

For example, divide $500 in the ratio 2:3. Total parts = 2 + 3 = 5. One part = 500 ÷ 5 = 100. The two shares are 2×100 = 200 and 3×100 = 300.

例如，将500美元按2:3分配。总份数=2+3=5。每份=500÷5=100。两份分别为2×100=200和3×100=300。

Total parts = a + b (for ratio a:b)

4. Direct Proportion | 正比例

Two quantities are in direct proportion if they increase or decrease in the same ratio. If y is directly proportional to x, then y = kx, where k is a constant called the constant of proportionality.

两个量如果按相同的比增加或减少，则它们成正比例。如果y与x成正比，则y = kx，其中k是常数，称为比例常数。

For example, if y ∝ x and y = 10 when x = 2, then k = 10 ÷ 2 = 5, so y = 5x. When x = 7, y = 35.

例如，若y ∝ x，且x=2时y=10，则k=10÷2=5，所以y=5x。当x=7时，y=35。

In direct proportion, the graph of y against x is a straight line that passes through the origin (0,0).

在正比例中，y关于x的图像是一条经过原点的直线。

5. Inverse Proportion | 反比例

Two quantities are in inverse proportion if one increases while the other decreases proportionally. If y is inversely proportional to x, then y = k/x, where k is a constant.

如果一个量按比例增加而另一个量按比例减少，则这两个量成反比例。如果y与x成反比，则y = k/x，其中k是常数。

For example, speed and time are inversely proportional for a fixed distance. If speed is 60 km/h and time is 2 hours, then k = 60×2 = 120. If speed becomes 80 km/h, time = 120 ÷ 80 = 1.5 hours.

例如，在固定距离下，速度和时间成反比。若速度为60公里/小时，时间为2小时，则k=60×2=120。若速度变为80公里/小时，时间=120÷80=1.5小时。

xy = k (for inverse proportion)

6. Using Proportion to Solve Word Problems | 用比例解决文字题

Word problems often involve proportional reasoning. Set up a proportion statement and solve for the unknown value.

文字题通常涉及比例推理。先建立比例关系式，再解出未知量。

Example: 5 pens cost $6.50. What is the cost of 8 pens? Since cost is directly proportional to number of pens, 5 : 6.50 = 8 : x. So x = (8 × 6.50) ÷ 5 = 52 ÷ 5 = $10.40.

例：5支笔花费6.50美元。8支笔多少钱？因为总价与支数成正比，所以5:6.50=8:x。因此x=(8×6.50)÷5=52÷5=10.40美元。

Check your answer: is it reasonable? More pens should cost more, so $10.40 is sensible.

检查答案：是否合理？更多的笔应该花费更多，所以10.40美元是合理的。

7. Proportion and Percentages | 百分比与比例

Percentages are a special form of ratio with a denominator of 100. 50% means the ratio 50:100 or 1:2.

百分比是一种特殊形式的比，其分母为100。50%表示比50:100，即1:2。

To find a percentage of a quantity, multiply by the percentage and divide by 100. For example, 15% of $200 = (15 ÷ 100) × 200 = $30.

求一个数的百分比，用该数乘以百分数再除以100。例如，200美元的15% = (15 ÷ 100) × 200 = 30美元。

To express a quantity as a percentage of another, divide the first by the second and multiply by 100. For example, 30 out of 50 is (30 ÷ 50) × 100 = 60%.

将一个量表示为另一个量的百分比，用第一个量除以第二个量再乘以100。例如，50中的30是(30÷50)×100=60%。

8. Rates and Unit Rates | 比率与单位比率

A rate is a ratio that compares quantities with different units, such as speed (km/h) or density (kg/m³). A unit rate has a denominator of 1.

比率是比较不同单位数量的比，如速度（公里/小时）或密度（公斤/立方米）。单位比率的分母为1。

For example, if a car travels 240 km in 3 hours, the unit rate is 240 ÷ 3 = 80 km per hour.

例如，一辆汽车3小时行驶240公里，其单位比率为240÷3=80公里/小时。

To find a unit rate, divide the first quantity by the second. This helps compare values.

要找到单位比率，用第一个数量除以第二个数量。这有助于比较数值。

9. Common Mistakes and Tips | 常见错误与提示

One common mistake is writing ratios in the wrong order. Always read the question carefully to see which quantity comes first.

一个常见错误是把比的前后顺序写反。务必仔细阅读题目，确认哪个数量在前。

Always simplify ratios to their lowest terms.
Check whether the proportion is direct or inverse before solving.
Use units in rate problems to avoid confusion.
When dividing in a ratio, make sure the sum of the parts matches the total.

始终将比化为最简形式。
先判断是正比例还是反比例，再求解。
在比率问题中使用单位以避免混淆。
按比分配时，确保各部分之和等于总量。

10. Practice Questions | 练习题

Try these questions and check your answers at the end.

尝试以下练习，并在最后核对答案。

Share £84 in the ratio 3:4.
If 4 kg of apples cost $9, how much does 7 kg cost?
y is inversely proportional to x. If y = 12 when x = 5, find y when x = 10.
Express 18 as a percentage of 45.

将84英镑按3:4分配。
如果4公斤苹果花费9美元，7公斤需要多少？
y与x成反比例。若x=5时y=12，求x=10时y的值。
将18表示为45的百分比。

Answers: 1) £36 and £48; 2) $15.75; 3) y = 6; 4) 40%

答案：1) 36英镑和48英镑；2) 15.75美元；3) y=6；4) 40%

11. Summary | 总结

In this guide, we covered the meaning of ratios, equivalent ratios, dividing quantities in a ratio, direct and inverse proportion, proportional reasoning, percentages, and rates. These skills are essential for many exam topics.

在本指南中，我们学习了比的含义、等价比、按比分配数量、正比例与反比例、比例推理、百分比和比率。这些技能在许多考试题目中都至关重要。

Remember: read carefully, identify the type of proportion, and always check your answer for reasonableness.

记住：仔细审题，判断比例类型，并始终检查答案是否合理。

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Mastering Ratio and Proportion | 掌握比与比例