Ratios and Percentages | 比例与百分比

📚 Ratios and Percentages | 比例与百分比

Ratios and percentages are fundamental tools in mathematics that help us compare quantities, share amounts fairly, and understand changes in real-world situations. In this article, we will explore the key concepts, methods, and common pitfalls for both topics, with clear examples and practice questions.

比例和百分比是数学中的基础工具，帮助我们比较数量、公平分配以及理解现实世界中的变化。本文将深入讲解这两个主题的关键概念、方法和常见错误，并配有清晰的例子和练习题。

1. What is a Ratio? | 什么是比？

A ratio is a way of comparing two or more quantities of the same kind. For example, if a fruit basket contains 3 apples and 2 oranges, we say the ratio of apples to oranges is 3:2. The order of the numbers matters, so 2:3 would mean oranges to apples.

比是一种比较两个或多个同类型数量的方式。例如，如果一个水果篮里有3个苹果和2个橙子，我们说苹果与橙子的比是3:2。数字的顺序很重要，因此2:3表示橙子与苹果的比。

The ratio is written as a:b or a/b (but usually a:b).

比通常写成a:b或a/b的形式，但更常用a:b。
A ratio has no units, as the units cancel out.

比没有单位，因为单位会被约去。
Ratios can be simplified by dividing all parts by a common factor.

比可以通过将所有部分除以公因数来化简。

2. Simplifying Ratios | 化简比

To simplify a ratio, divide every part by the same number (the highest common factor) until the numbers are as small as possible. For example, 12:8 simplifies to 3:2 by dividing both parts by 4.

要化简比，需将所有部分同时除以同一个数（最大公因数），直到数字尽可能小。例如，12:8同时除以4，化简为3:2。

12 : 8 = 3 : 2

If the ratio involves decimals or fractions, multiply to make whole numbers first. For instance, 0.5:1.5 can be multiplied by 2 to get 1:3.

如果比中含有小数或分数，先乘一个数使其成为整数。例如，0.5:1.5可以同时乘以2得到1:3。

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

When we share a total amount in a given ratio, we first add the parts of the ratio to find the total number of shares. Then we divide the amount by this total, and multiply each part by the share value.

当按给定比分配总量时，我们先将比的所有部分相加得到总份数，然后用总量除以总份数，再乘以每一部分。

Example: Share $60 in the ratio 2:3.

示例：将60美元按2:3分配。

Total shares = 2 + 3 = 5

总份数 = 2 + 3 = 5
One share = 60 ÷ 5 = 12

每份 = 60 ÷ 5 = 12
First part = 2 × 12 = 24; second part = 3 × 12 = 36

第一部分 = 2 × 12 = 24；第二部分 = 3 × 12 = 36

Always check that the sum of the parts equals the original total: 24 + 36 = 60.

务必检查各部分之和等于原总量：24 + 36 = 60。

4. Ratio and Proportion | 比与比例

Proportion describes the equality of two ratios. If two ratios are equal, they are in proportion. For example, 2:3 and 4:6 are in proportion because 2/3 = 4/6.

比例描述两个比的相等关系。如果两个比相等，则它们成比例。例如，2:3和4:6成比例，因为2/3 = 4/6。

To solve a proportion problem, use cross-multiplication. For example, if x/5 = 6/10, then 10x = 30, so x = 3.

解决比例问题可使用交叉相乘。例如，若x/5 = 6/10，则10x = 30，因此x = 3。

a/b = c/d → ad = bc

Proportion is useful for scaling recipes, maps, and models.

比例在食谱调整、地图缩放和模型中非常有用。

5. Introduction to Percentages | 百分比入门

A percentage is a fraction with a denominator of 100. The symbol ‘%’ means ‘out of 100’. For instance, 25% means 25 out of 100, which is equivalent to the fraction 25/100 or the decimal 0.25.

百分比是分母为100的分数。符号’%’表示’每一百’。例如，25%表示每一百中的25，相当于分数25/100或小数0.25。

50% = 1/2 = 0.5

50% = 1/2 = 0.5
10% = 1/10 = 0.1

10% = 1/10 = 0.1
100% = 1 (the whole amount)

100% = 1（整体量）

Percentages are used in discounts, interest rates, statistics, and many daily contexts.

百分比广泛应用于折扣、利率、统计以及许多日常场景。

6. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分比的互化

To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100. For example, 0.35 = 35%, and 8% = 0.08.

将百分比转换为小数，需除以100；将小数转换为百分比，需乘以100。例如，0.35 = 35%，8% = 0.08。

To convert a fraction to a percentage, multiply the fraction by 100. For example, 3/4 = 75% because (3/4) × 100 = 75.

将分数转换为百分比，需将分数乘以100。例如，3/4 = 75%，因为(3/4) × 100 = 75。

Here is a quick conversion table:

以下为快速换算表：

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%

7. Percentage of a Quantity | 求一个量的百分比

To find a percentage of a quantity, convert the percentage to a decimal or fraction, then multiply by the quantity. For example, to find 15% of 80, write 0.15 × 80 = 12.

求一个量的百分比，先将百分比转换为小数或分数，再乘以该量。例如，求80的15%，计算0.15 × 80 = 12。

Percentage of quantity = (percentage ÷ 100) × quantity

Another method: find 10% first, then adjust. 10% of 80 is 8, 5% is 4, so 15% is 8 + 4 = 12.

另一种方法：先求10%，再调整。80的10%是8，5%是4，所以15%是8 + 4 = 12。

8. Percentage Increase and Decrease | 百分比的增加和减少

To increase a quantity by a percentage, calculate the percentage of the quantity and add it. To decrease, subtract it. For example, increase 40 by 25%: 25% of 40 is 10, so the new value is 40 + 10 = 50.

增加一个量的百分比，先算出该百分比对应的值再加上；减少则减去。例如，将40增加25%：25%的40是10，所以新值为40 + 10 = 50。

A quicker method uses multipliers. An increase of 25% means the multiplier is 1.25; a decrease of 25% means 0.75.

更快捷的方法是使用倍数。增加25%意味着倍数为1.25；减少25%意味着倍数为0.75。

New value = original × (1 ± percentage/100)

Careful: a 10% increase followed by a 10% decrease does not return to the original value.

注意：先增加10%再减少10%，不会回到原值。

9. Reverse Percentage Problems | 反向百分比问题

Sometimes we know the final value after a percentage change and need to find the original amount. For example, after a 20% increase, the price is $120. What was the original price?

有时我们知道变化后的最终值，需要求原始值。例如，价格上涨20%后为120美元，原价是多少？

If there was a 20% increase, the final value is 120% of the original.

如果增加了20%，最终值是原值的120%。
So original = 120 ÷ 1.20 = 100.

因此原值 = 120 ÷ 1.20 = 100。

Use multiplication: for an increase of 20%, multiplier = 1.20. For a decrease of 20%, multiplier = 0.80. Divide the final value by the multiplier to reverse the change.

使用倍数：增加20%，倍数为1.20；减少20%，倍数为0.80。将最终值除以倍数即可还原。

10. Ratios and Percentages in Real Life | 比例和百分比在生活中的应用

Ratios are commonly used in cooking, mixing paints, and dividing bills. For example, a concrete mix might require cement, sand, and gravel in the ratio 1:2:4. Percentages are used in sales discounts, bank interest, and exam scores.

比例常用于烹饪、调配油漆和分摊账单。例如，混凝土可能需要水泥、沙子和碎石按1:2:4的比例混合。百分比用于销售折扣、银行利息和考试成绩。

Example: A shirt costs $40 and is on a 30% discount. The sale price is 40 × 0.70 = $28.

示例：一件衬衫售价40美元，打7折（30%折扣）。销售价为40 × 0.70 = 28美元。

Understanding these concepts helps in making informed financial decisions.

理解这些概念有助于做出明智的财务决策。

11. Common Mistakes and Tips | 常见错误与技巧

Mistake: Adding percentages directly in multi-step changes. A 10% increase then a 10% decrease is not the same as no change.

错误：在多步变化中直接加减百分比。先增加10%再减少10%并不等于不变。
Mistake: Forgetting to divide by the total number of shares in a ratio.

错误：在按比分配时忘记除以总份数。
Mistake: Writing the order of a ratio incorrectly; 3:2 is not the same as 2:3.

错误：写错比的顺序；3:2与2:3不同。
Tip: To find 5% of a number, find 10% and halve it.

技巧：求一个数的5%，先求10%再除以2。
Tip: Always ask ‘percentage of what?’ to avoid confusion.

技巧：始终明确’是哪个量的百分比’，避免混淆。

12. Practice Questions | 练习题

Try these problems and check your answers by reading the explanations below.

尝试以下题目，并通过下方解析检验你的答案。

Simplify the ratio 18:24.

化简比18:24。
Divide $80 in the ratio 3:5.

将80美元按3:5分配。
Write 7/8 as a percentage.

将7/8写成百分比。
Find 40% of 90.

求90的40%。
Increase 60 by 15%.

将60增加15%。
After a 10% decrease, a book costs $27. Find the original price.

一本书降价10%后售价27美元，求原价。

Answers: 1) 3:4, 2) 30 and 50, 3) 87.5%, 4) 36, 5) 69, 6) 30.

答案：1) 3:4，2) 30和50，3) 87.5%，4) 36，5) 69，6) 30。

Practice these skills regularly to build confidence and speed. Mastery of ratios and percentages will support your progress in algebra, geometry, and statistics.

定期练习这些技能，以增强信心和速度。掌握比例和百分比将为你在代数、几何和统计中的学习进步提供支持。

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Ratios and Percentages | 比例与百分比