Percentages: Conversions and Calculations | 百分比：转换与计算

📚 Percentages: Conversions and Calculations | 百分比：转换与计算

Percentages appear everywhere – from shop discounts to test scores and interest rates. Understanding how to convert between fractions, decimals and percentages, and how to solve percentage problems, is a fundamental skill in Key Stage 3 mathematics. This article breaks down the essential concepts with clear examples and British English terminology aligned with the Cambridge KS3 curriculum.

百分比无处不在——从商店折扣到考试成绩和利率。掌握分数、小数和百分比之间的转换以及如何解决百分比问题，是KS3数学的基本技能。本文用清晰的例子和符合剑桥KS3课程要求的英式术语，逐步解析核心概念。

1. Understanding Percentages | 理解百分比

The word ‘percent’ comes from the Latin per centum, meaning ‘out of one hundred’. So 30% means 30 out of 100, or 30/100. Percentages are an efficient way to compare proportions, especially when the total number of items is not 100. We can always scale a percentage back to a fraction with denominator 100 and then simplify.

“百分之”（percent）一词源自拉丁语 per centum，意为”每一百”。因此 30% 表示 100 份中的 30 份，即 30/100。百分比是比较比例的便捷方式，尤其当总数不是 100 时。我们总可以将百分比还原为分母为 100 的分数，然后进行约分。

For example, 65% = 65/100 = 13/20 as a simplified fraction. Similarly, 8% = 8/100 = 2/25. This basic equivalence is the foundation for all conversion work.

例如，65% = 65/100 = 13/20（化简后）。同样地，8% = 8/100 = 2/25。这个基本等价关系是所有转换工作的基础。

2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分比的相互转换

The three forms – fraction, decimal and percentage – are just different ways of expressing the same proportion. The table below shows common conversions that every KS3 student should memorise:

分数、小数和百分比这三种形式只不过是表示同一比例的不同方式。下表展示了每位KS3学生都应记住的常见转换：

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%
1/3	0.333… (recurring)	33⅓%

Remember that the decimal form of 1/3 is a recurring decimal, so its percentage equivalent is often written as 33.3% or 33⅓% exactly. Being fluent in these core equivalents speeds up mental calculation enormously.

请记住，1/3 的小数形式是循环小数，因此其百分比等价形式通常精确写作 33.3% 或 33⅓%。熟练运用这些核心等价关系可以极大地加快心算速度。

3. Converting Fractions to Percentages | 分数转化为百分比

To convert any fraction to a percentage, either find an equivalent fraction with denominator 100, or multiply the fraction by 100%. For example, 3/8: calculate 3 ÷ 8 = 0.375, then multiply by 100 to get 37.5%. The general rule is:

要将任何分数转换为百分比，可以找到一个分母为 100 的等值分数，或者将该分数乘以 100%。例如 3/8：先计算 3 ÷ 8 = 0.375，再乘以 100 得到 37.5%。一般规则如下：

Percentage = (Fraction) × 100%

If the fraction is already out of a factor of 100, the job is simpler: 17/20 = (17 × 5)/(20 × 5) = 85/100 = 85%. For fractions with denominator 3, 6 or 7, long division may be needed, and you should express the percentage to one decimal place unless told otherwise.

如果分数本身的分母已是 100 的因数，就更简单了：17/20 = (17 × 5)/(20 × 5) = 85/100 = 85%。对于分母为 3、6 或 7 的分数，可能需要作长除法，此时除非另有说明，应将百分比保留一位小数。

4. Converting Decimals to Percentages and Vice Versa | 小数与百分比的相互转换

Since percent means ‘per hundred’, converting a decimal to a percentage is done by multiplying by 100. This moves the decimal point two places to the right. 0.47 becomes 47%; 0.03 becomes 3%; 0.6 becomes 60% (because 0.6 = 0.60). Going from a percentage to a decimal, you divide by 100, moving the decimal point two places to the left. 85% = 0.85; 7% = 0.07; 120% = 1.20.

由于百分数意为”每一百”，将小数转换为百分比只需乘以 100，即将小数点向右移动两位。0.47 变成 47%；0.03 变成 3%；0.6 变成 60%（因为 0.6 = 0.60）。从百分比转换为小数时，则除以 100，即将小数点向左移动两位。85% = 0.85；7% = 0.07；120% = 1.20。

A useful method is to remember the ‘times 100 or divide by 100’ switch. Many mistakes occur when students forget to move the decimal point the correct number of places. Always check: is your answer sensible? For instance, 0.45 as a percentage should be less than 100%, and indeed it is 45%.

一个有用的方法是牢记”乘以 100 或除以 100″的转换开关。当学生忘记正确移动小数点位数时，常常会出错。始终检查一下：你的答案合理吗？例如，0.45 作为百分比应小于 100%，实际上它正是 45%。

5. Finding a Percentage of an Amount | 求一个数量的百分之几

To find a percentage of a quantity, convert the percentage to a decimal and multiply by the quantity. For example, to find 15% of £80: 15% = 0.15, so 0.15 × 80 = 12. Hence 15% of £80 is £12. An alternative method uses fractions: 15% = 15/100 = 3/20, then multiply 80 × 3/20 = 12.

要计算一个数量的百分之几，先将百分比转化为小数，再乘以该数量。例如，计算 £80 的 15%：15% = 0.15，于是 0.15 × 80 = 12。因此 £80 的 15% 是 £12。另一种方法使用分数：15% = 15/100 = 3/20，然后 80 × 3/20 = 12。

For more complex percentages, such as 17.5% of 240, convert 17.5% to 0.175 and multiply: 0.175 × 240 = 42. You can also break it down: 10% of 240 is 24, 5% is 12, 2.5% is 6, so 17.5% = 24 + 12 + 6 = 42. This mental approach is very useful in non-calculator assessments.

对于更复杂的百分比，例如求 240 的 17.5%，可将 17.5% 转换为 0.175 再相乘：0.175 × 240 = 42。也可以拆分计算：240 的 10% 是 24，5% 是 12，2.5% 是 6，因此 17.5% = 24 + 12 + 6 = 42。这种心算方法在无计算器测试中非常实用。

6. Percentage Increase and Decrease | 百分比增长与减少

Percentage change compares the difference between a new and old value to the original value. The formula is:

百分比变化将新值与旧值之差与原始值进行比较。公式如下：

Percentage change = (change ÷ original value) × 100%

If the price of a game rises from £40 to £50, the change is £10. Percentage increase = (10 ÷ 40) × 100% = 25%. If a class size drops from 30 to 27, the decrease is 3; percentage decrease = (3 ÷ 30) × 100% = 10%. Always use the original amount as the denominator.

如果一个游戏的价格从 £40 涨到 £50，变化金额为 £10。百分比增长 = (10 ÷ 40) × 100% = 25%。如果一个班级人数从 30 人减少到 27 人，减少量为 3；百分比减少 = (3 ÷ 30) × 100% = 10%。请始终使用原始值作为分母。

A common pitfall is to use the new value as the denominator by mistake. To avoid this, underline the ‘original’ number in the question before calculating.

一个常见错误是误将新值作为分母。为避免这一点，在计算前先把问题中的”原始”数值划出来。

7. Using Multipliers for Percentage Change | 使用乘数计算百分比变化

For efficient calculation, percentage changes can be applied using decimal multipliers. To increase an amount by 5%, multiply by 1.05. The multiplier is 1 + (percentage ÷ 100). To decrease by 15%, multiply by 0.85 (since 1 – 0.15 = 0.85). This method is essential when applying successive changes.

为了高效计算，可以使用小数乘数来处理百分比变化。将某个数量增加 5%，乘以 1.05 即可。乘数为 1 + (百分比 ÷ 100)。减少 15%，则乘以 0.85（因为 1 – 0.15 = 0.85）。在进行连续变化时，这一方法至关重要。

Increase by 12%: multiplier = 1.12
Decrease by 8%: multiplier = 0.92
Increase by 100% (double): multiplier = 2.00
Decrease by 40%: multiplier = 0.60

增加 12%：乘数 = 1.12
减少 8%：乘数 = 0.92
增加 100%（翻倍）：乘数 = 2.00
减少 40%：乘数 = 0.60

When a value undergoes more than one percentage change, simply multiply the original by the succession of multipliers. If a £200 investment grows by 10% in the first year and then by 5% in the second, the final amount is 200 × 1.10 × 1.05 = £231. The order of multiplication does not matter.

当某个数值经历不止一次百分比变化时，只需将原始值依次乘以各个乘数。如果一项 £200 的投资在第一年增长 10%，第二年再增长 5%，最终金额为 200 × 1.10 × 1.05 = £231。乘法顺序不影响结果。

8. Reverse Percentages | 逆向百分比

Reverse percentage problems ask you to find the original amount before a percentage increase or decrease. For example, a sale price of £72 includes a 10% discount. What was the original price? Since the sale price represents 90% of the original (100% – 10%), £72 = 0.90 × original. Divide by 0.90: original = 72 ÷ 0.90 = £80.

逆向百分比问题要求找出经过百分比增长或减少之前的原始量。例如，一件商品的折扣价为 £72，包含了 10% 的折扣。原价是多少？由于折扣价代表原价的 90%（100% – 10%），£72 = 0.90 × 原价。两边除以 0.90：原价 = 72 ÷ 0.90 = £80。

The key is to identify the percentage that the given amount corresponds to. If a population increased by 15% to 23,000, then 23,000 represents 115% of the original. The original is 23,000 ÷ 1.15 = 20,000. Always set up the equation: given value = percentage multiplier × original, and rearrange.

关键在于确定给定数量所对应的百分比。如果某地人口增长了 15% 后达到 23,000，那么 23,000 代表原人口的 115%。原人口为 23,000 ÷ 1.15 = 20,000。务必列出方程：已知值 = 百分比乘数 × 原值，然后进行移项求解。

9. Comparing Quantities Using Percentages | 使用百分比比较数量

Percentages allow us to compare proportions from different-sized groups. For instance, in school A, 45 out of 60 pupils passed a test (75%), while in school B, 88 out of 110 pupils passed (80%). School B had a higher pass rate, even though the raw numbers might be misleading.

百分比使我们能够比较来自不同规模群体的比例。例如，在 A 校，60 名学生中有 45 人通过测试（75%）；而在 B 校，110 名学生中有 88 人通过（80%）。尽管绝对数字可能造成误导，但 B 校的通过率更高。

When comparing fractions or ratios, converting to percentages puts them on a common scale of 100, making decisions straightforward. Always write percentages with one decimal place when precise comparison matters, e.g. 78.3% vs 79.6%.

在比较分数或比率时，将它们转换为百分比就可以置于统一的 100 分制尺度上，从而让判断一目了然。当需要精确比较时，百分比应保留一位小数，例如 78.3% 与 79.6% 的对比。

10. Percentage Word Problems | 百分比应用题

Word problems often combine multiple concepts. Read the question carefully and identify what you know (the whole, the part, the percentage, or the change). Then decide whether you need to find a percentage of an amount, a percentage change, or an original value. Underline key figures and words like ‘of’, ‘extra’, ‘discount’, ‘including VAT’.

应用题常会综合多个概念。仔细读题，弄清已知信息（整体、部分、百分比或变化量）。然后判断你需要计算的是一个数量的百分之几、百分比变化，还是原始值。画出关键数字，以及如”的”、”额外”、”折扣”、”含增值税”等关键词语。

Example: A bike costs £180 plus 20% VAT. What is the total cost? The 20% is based on the pre-tax price. VAT amount = 0.20 × 180 = £36. Total = 180 + 36 = £216. Using a multiplier: 180 × 1.20 = £216. Example: A sofa is reduced by 30% in a sale and now costs £420. The original price was 420 ÷ 0.70 = £600.

示例：一辆自行车售价 £180，另加 20% 增值税。总价是多少？20% 以税前价格为基础。增值税金额 = 0.20 × 180 = £36。总价 = 180 + 36 = £216。使用乘数：180 × 1.20 = £216。示例：一张沙发在促销中降价 30%，现价 £420。原价为 420 ÷ 0.70 = £600。

11. Common Mistakes and Tips | 常见错误与提示

Many learners mix up percentage points and percentages. An increase from 20% to 30% is a rise of 10 percentage points, but it represents a 50% increase relative to the original 20%. The context determines which is meaningful.

许多学生容易混淆百分点和百分比。从 20% 上升到 30% 是提升了 10 个百分点，但相对于原来的 20%，它代表了 50% 的增长。具体意义要视上下文而定。

Another frequent error is applying percentage increase and decrease symmetrically. If a price rises by 20% and then falls by 20%, it does not return to the original value. Starting at £100: after a 20% rise it is £120; a 20% decrease on £120 is £24 off, leaving £96. Always re-calculate based on the new amount.

另一个常见错误是认为百分比增减可以对称抵消。如果价格上涨 20% 后再下跌 20%，并不能回到原值。以 £100 为例：上涨 20% 后为 £120；然后下跌 £120 的 20%，即减少 £24，剩下 £96。务必基于新值重新计算。

Tips: (1) Convert every percentage to a decimal or fraction before multiplying. (2) In word problems, write ‘original’ and ‘new’ labels. (3) Use estimation to check the plausibility of your answer – e.g., 19% of 200 should be a little less than 40.

提示：（1）在进行乘法前，将每个百分比转换为小数或分数。（2）解应用题时，标注”原值”和”新值”。（3）用估算来检查答案的合理性——例如，200 的 19% 应该略小于 40。

12. Summary and Checklist | 总结与检查清单

Mastering percentages at KS3 means you can confidently:

在 KS3 阶段掌握百分比意味着你能自信地完成以下任务：

Define a percentage as ‘out of 100’ and relate it to fractions and decimals.
Convert fluently between fractions, decimals and percentages, including 1/3 and other recurring decimals.
Find a percentage of a quantity using both decimal multiplication and mental fraction strategies.
Calculate percentage increase and decrease using the formula and decimal multipliers.
Solve reverse percentage problems by identifying the correct multiplier and dividing.
Compare proportions from different data sets using percentages.
Interpret and solve multi-step word problems involving VAT, discounts and profit.

将百分比定义为”每一百”，并将其与分数和小数联系起来。
流畅地在分数、小数和百分比间转换，包括 1/3 及其他循环小数。
分别使用小数乘法和心算分数策略来计算一个数量的百分之几。
运用公式和小数乘数计算百分比增长与减少。
通过确定正确的乘数并作除法，解决逆向百分比问题。
使用百分比比较不同数据集的比例。
解读并解决涉及增值税、折扣和利润的多步骤应用题。

Practice these skills with past Checkpoint questions, and soon percentages will be one of your strongest topics!

通过以往的 Checkpoint 试题练习这些技能，百分比很快就会成为你最拿手的主题之一！

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Percentages: Conversions and Calculations | 百分比：转换与计算