Understanding Percentages | 理解百分比

📚 Understanding Percentages | 理解百分比

Percentages are one of the most useful mathematical tools in everyday life. From shopping discounts to exam scores, percentages help us compare quantities and make sense of the world. In this Year 7 topic series, we will build a confident understanding of percentages step by step.

百分比是日常生活中最实用的数学工具之一。从购物折扣到考试分数，百分比帮助我们比较数量、理解世界。在本次七年级主题系列中，我们将一步一步建立对百分比的扎实理解。

1. What is a Percentage? | 什么是百分比？

A percentage is a number or ratio expressed as a fraction of 100. The word ‘percent’ comes from the Latin phrase ‘per centum’, meaning ‘by the hundred’. The symbol % is used to represent percent.

百分比是表示为100之几的数或比率。单词“percent”源自拉丁语“per centum”，意为“每一百”。符号 % 用来表示百分比。

For example, 35% means 35 out of every 100, or the fraction 35⁄100. If you see 50%, it means half of the whole.

例如，35% 表示每100个中有35个，即分数 35⁄100。如果你看到 50%，它表示整体的一半。

Percentages are always compared to 100.
100% represents the whole.

百分比总是以100为基准进行比较。
100% 表示整体。

2. Converting between Percentages and Fractions | 百分比与分数的互转

To convert a percentage to a fraction, write the percentage as the numerator over a denominator of 100, then simplify if possible.

将百分比转换为分数时，把百分比作为分子，分母写为100，然后尽可能化简。

45% = 45⁄100 = 9⁄20

To convert a fraction to a percentage, multiply the fraction by 100 out of 1, or find an equivalent fraction with denominator 100.

将分数转换为百分比时，用分数乘以100，或者找到一个分母为100的等价分数。

3⁄4 = (3 × 25)⁄(4 × 25) = 75⁄100 = 75%

Common conversions you should memorise include 1⁄2 = 50%, 1⁄4 = 25%, 1⁄5 = 20%, 1⁄10 = 10%, and 1⁄3 ≈ 33.3%.

你应该记住的常见转换包括 1⁄2 = 50%、1⁄4 = 25%、1⁄5 = 20%、1⁄10 = 10% 和 1⁄3 ≈ 33.3%。

3. Converting between Percentages and Decimals | 百分比与小数的互转

Converting between percentages and decimals is very simple: divide by 100 to change a percentage into a decimal, and multiply by 100 to change a decimal into a percentage.

百分比与小数之间的转换非常简单：百分比除以100变成小数，小数乘以100变成百分比。

32% = 32 ÷ 100 = 0.32

0.07 = 0.07 × 100 = 7%

Remember that the decimal point moves two places to the left when converting from a percentage to a decimal, and two places to the right when converting the other way.

记住：从百分比转换为小数时，小数点向左移动两位；反向转换时向右移动两位。

4. Finding a 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) and multiply by the quantity.

要求一个量的百分比，先把百分比转换为小数（或分数），再乘以该量。

Find 20% of £60 → 0.20 × 60 = £12

For more complex percentages like 15%, you can find 10% first and then add 5%, or directly multiply by 0.15.

对于像15%这样更复杂的百分比，你可以先求10%，再加上5%，或者直接乘以0.15。

10% of 60 = 6, so 20% of 60 = 12.
1% of 60 = 0.6, so 15% = 10% + 5% = 6 + 3 = 9.

60的10%是6，所以60的20%是12。
60的1%是0.6，所以15% = 10% + 5% = 6 + 3 = 9。

5. Percentage Increase and Decrease | 百分比增加与减少

When a quantity increases or decreases by a percentage, you can calculate the new value by finding the percentage change and adding or subtracting it from the original.

当一个量增加或减少某个百分比时，你可以计算变化量，然后从原量上加上或减去它，得到新值。

Increase 40 by 15% → 40 × 0.15 = 6 → New = 40 + 6 = 46

Decrease 80 by 25% → 80 × 0.25 = 20 → New = 80 − 20 = 60

A faster method is to use a multiplier. For an increase of 15%, use the multiplier 1.15; for a decrease of 25%, use 0.75.

更快捷的方法是使用乘数。增加15%用乘数1.15；减少25%用乘数0.75。

6. Percentage Change and Percentage Difference | 百分比变化与百分比差

Percentage change measures how much a value has increased or decreased relative to the original value. It is calculated using the formula:

百分比变化衡量一个值相对原值增加或减少了多少，计算公式如下：

Percentage Change = (Change ÷ Original) × 100%

For example, if a price goes from £50 to £65, the change is £15, and the percentage change is (15 ÷ 50) × 100% = 30%.

例如，如果价格从50英镑涨到65英镑，变化量为15英镑，百分比变化为 (15 ÷ 50) × 100% = 30%。

Percentage difference is different: it compares the absolute difference to the average of the two values, but at Year 7 level we usually focus on percentage change.

百分比差与之不同：它把绝对差与两个值的平均值进行比较，但在七年级我们通常只关注百分比变化。

7. Reverse Percentage Problems | 反百分比问题

Sometimes you know the final amount after a percentage change, and you need to find the original amount. This is called a reverse percentage problem.

有时你已知百分比变化后的最终数量，而需要求原来的数量。这被称为反百分比问题。

After a 20% increase, a price is £96. Find the original price.

Since the final price is 120% of the original, you divide by 1.20: 96 ÷ 1.20 = £80.

因为最终价格是原价的120%，所以除以1.20：96 ÷ 1.20 = 80英镑。

For a 20% decrease, the final amount is 80% of the original, so divide by 0.80 to reverse the change.

对于减少20%的情况，最终数量是原价的80%，因此除以0.80来还原变化。

8. Simple Interest and Percentages | 简单利率与百分比

Simple interest is calculated on the original amount (principal) only. The formula is:

简单利息只按原始金额（本金）计算，公式为：

Simple Interest = Principal × Rate × Time

If you invest £200 at an annual rate of 4% for 3 years, the interest is 200 × 0.04 × 3 = £24. The total amount becomes £224.

如果你将200英镑以每年4%的利率存3年，利息为 200 × 0.04 × 3 = 24英镑，总金额变为224英镑。

Notice that the rate is written as a decimal in calculations. Always check whether the time is in years or months.

注意在计算中利率要写成小数。同时要检查时间是以年还是月为单位。

9. Percentage in Real-Life Contexts | 百分比在实际生活中的应用

Percentages appear everywhere: sales discounts, tax, tips, exam results, sports statistics, and population data.

百分比无处不在：打折、税收、小费、考试成绩、体育统计和人口数据。

Context	Example
Shopping	30% off a £45 jacket → £45 × 0.70 = £31.50
Test Score	28 out of 40 = (28 ÷ 40) × 100 = 70%
VAT	20% VAT on £50 = £10 tax, total £60

Understanding percentages helps you make smart decisions, such as which discount is better or how much tip to leave.

理解百分比帮助你做出明智决策，比如哪个折扣更划算，或者该留多少小费。

10. Common Percentage Mistakes | 常见百分比错误

Many students make the same errors when working with percentages. Here are some pitfalls to avoid.

许多学生在处理百分比时会犯同样的错误。以下是一些需要避免的陷阱。

Adding percentages to different bases directly: 10% of 50 + 10% of 100 is not 10% of 150.
Forgetting that a 20% decrease then a 20% increase does not return to the original value.
Writing 30% as 30 instead of 0.30 in calculations.
Confusing fraction conversion with decimal conversion.

直接把不同基数的百分比相加：50的10%加100的10%并不等于150的10%。
忘记先减少20%再增加20%并不会回到原值。
在计算中把30%写成30而不是0.30。
混淆分数转换和小数转换。

11. Practice Questions | 练习题目

Try these questions to test your understanding. Work step by step and check your answers.

尝试以下问题来检验你的理解。一步步计算并检查答案。

Convert 0.64 into a percentage.
Find 35% of 200.
Decrease 90 by 15%.
A shirt was £25 and is now £19. Find the percentage decrease.
After a 10% increase, a toy costs £33. What was the original price?

将0.64转换为百分比。
求200的35%。
把90减少15%。
一件衬衫原价25英镑，现价19英镑，求百分比减少。
一个玩具涨价10%后售价为33英镑，原价是多少？

Answers: 1) 64% 2) 70 3) 76.5 4) 24% 5) £30

答案：1) 64% 2) 70 3) 76.5 4) 24% 5) 30英镑

12. Summary | 总结

You have now learned the key ideas about percentages: what they mean, how to convert them, how to find a percentage of a quantity, and how to solve increase, decrease, and reverse problems. Percentages are not just a school topic; they are a life skill.

现在你已经掌握了百分比的核心概念：百分比的含义、如何转换、如何求一个量的百分比，以及如何解决增加、减少和反问题。百分比不只是学校里的一个知识点，它是一种生活技能。

Keep practising with real-world examples, and soon you will handle percentages with confidence.

坚持用现实中的例子练习，很快你就能自信地处理百分比问题了。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Understanding Percentages | 理解百分比