📚 Fractions, Decimals and Percentages | 分数、小数与百分比
In mathematics, fractions, decimals and percentages are three different ways to represent parts of a whole. Understanding how to convert between them and solve problems using all three forms is a core skill in the Cambridge KS3 curriculum. This article will guide you through the key concepts, conversion methods, and real-world applications, with plenty of examples to build your confidence.
在数学中，分数、小数和百分比是表示整体的一部分的三种不同方式。掌握它们之间的转换并解决涉及这三种形式的问题是剑桥 KS3 课程的核心技能。本文将通过大量的示例，带你梳理关键概念、转换方法和实际应用，帮助你树立信心。
1. Understanding Fractions | 理解分数
A fraction represents a part of a whole or a number of equal parts. It is written as a numerator over a denominator, separated by a horizontal bar. For example, in ⅗, 3 is the numerator and 5 is the denominator, meaning 3 out of 5 equal parts.
分数表示整体的一部分或若干等份。它写作分子在上、分母在下的形式，中间用横线隔开。例如，在 ⅗ 中，3 是分子，5 是分母，表示 5 等份中的 3 份。
quivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. Simplifying a fraction means dividing both by their highest common factor (HCF).
等值分数是表示相同值的不同分数，例如 ½ = ²⁄₄ = ³⁄₆。你可以通过将分子和分母同时乘以或除以同一个非零数来找到等值分数。化简分数意味着将分子和分母同时除以它们的最大公因数 (HCF)。
2. Working with Decimals | 处理小数
Decimals are another way to write fractions whose denominators are powers of 10. The decimal point separates the whole number part from the fractional part. For example, 0.7 means 7 tenths, and 0.25 means 25 hundredths.
小数是分母为 10 的幂的分数的另一种写法。小数点将整数部分和小数部分分开。例如，0.7 表示十分之七，0.25 表示一百分之二十五。
. Adding zeros to the right of a decimal does not change its value, so 0.5 = 0.50 = 0.500.
小数可以是有限小数（如 0.125）或循环小数（如 0.333...）。比较小数时，将小数点对齐，从左到右依次比较各位数字。在小数右边添加零不会改变数值，因此 0.5 = 0.50 = 0.500。
3. Introduction to Percentages | 百分比入门
A percentage is a number expressed as a fraction of 100. The symbol % means 'per 100' or 'out of 100'. For instance, 45% means 45 out of 100, which can be written as the fraction ⁴⁵⁄₁₀₀ or the decimal 0.45.
百分比是以 100 为分母的分数。符号 % 表示“百分之…”或“每 100 份中的”。例如，45% 表示 100 份中的 45 份，可以写成分数 ⁴⁵⁄₁₀₀ 或小数 0.45。
Percentages are widely used in everyday life: discounts in shops, test scores, interest rates, and data displays. Being able to find a percentage of an amount without a calculator (e.g. 10% by dividing by 10) is a useful mental maths skill.
百分比在日常生活中广泛使用：商店折扣、考试分数、利率以及数据展示。能够不依赖计算器求出一个数的某个百分比（例如，通过除以 10 求 10%）是一项实用的心算技能。
4. Converting Fractions to Decimals | 分数转换为小数
ou can use short division or recognise common conversions. For example, ⅜ = 3 ÷ 8 = 0.375.
要将分数转换为小数，用分子除以分母。你可以使用短除法，或者熟记常见的转换。例如，⅜ = 3 ÷ 8 = 0.375。
point in the correct place. For instance, ⁷⁄₁₀ = 0.7, ¹³⁄₁₀₀ = 0.13. Mixed numbers like 2 ⅗ rted by keeping the whole number and changing only the fractional part: 2 + (3 ÷ 5) = 2.6.
如果分母是 10、100、1000 这样的 10 的幂，可以直接将分子的小数点放在正确位置。例如 ⁷⁄₁₀ = 0.7，¹³⁄₁₀₀ = 0.13。带分数如 2 ⅗ 应保留整数部分，只将分数部分转换：2 + (3 ÷ 5) = 2.6。
| Fraction | Divide | Decimal |
|---|---|---|
| ½ | 1 ÷ 2 | 0.5 |
| ⅔ | 2 ÷ 3 | 0.666... (0.6̇) |
| ⅘ | 4 ÷ 5 | 0.8 |
5. Converting Decimals to Fractions | 小数转换为分数
Write the decimal as a fraction with denominator 10, 100, 1000 depending on the number of decimal places, then simplify if possible. For a terminating decimal like 0.64, there are two decimal places, so write ⁶⁴⁄₁₀₀ and simplify to ¹⁶⁄₂₅.
根据小数的位数，将小数写成分母为 10、100、1000 的分数，然后尽可能地化简。对于有限小数如 0.64，它有两位小数，所以写成 ⁶⁴⁄₁₀₀，然后化简为 ¹⁶⁄₂₅。
For recurring decimals like 0.3̇ at KS3 level, common recurring decimals are usually memorised: ⅓ = 0.333..., ⅔ = 0.666..., ⅙ = 0.1666...
对于循环小数如 0.3̇ (0.333...)，需要使用不同的方法，但在 KS3 阶段，常见的循环小数通常需要熟记：⅓ = 0.333...，⅔ = 0.666...，⅙ = 0.1666...
6. Converting Fractions to Percentages | 分数转换为百分比
First convert the fraction to a decimal by dividing, then multiply by 100 and add the % sign. Alternatively, find an equivalent fraction with denominator 100. For example, ⅗ = 3 ÷ 5 = 0.6 → 0.6 × 100% = 60%. Or ⅗ = ⁶⁰⁄₁₀₀ = 60%.
首先通过除法将分数转换为小数，然后乘以 100 并加上百分号。另一种方法是找到分母为 100 的等值分数。例如，⅗ = 3 ÷ 5 = 0.6 → 0.6 × 100% = 60%。或者 ⅗ = ⁶⁰⁄₁₀₀ = 60%。
ominator does not easily multiply to 100, use the decimal method: ⅜ = 3 ÷ 8 = 0.375 → 37.5%. Learn common equivalents like ½ = 50%, ¼ = 25%, ¾ = 75%, ⅕ = 20%.
当分母不容易乘以某个数得到 100 时，使用小数方法：⅜ = 3 ÷ 8 = 0.375 → 37.5%。熟记常见的对应关系，如 ½ = 50%，¼ = 25%，¾ = 75%，⅕ = 20%。
7. Converting Percentages to Fractions | 百分比转换为分数
Write the percentage as a fraction over 100, then simplify. For example, 65% ¹³⁄₂₀ (divide numerator and denominator by 5). For percentages with decimals, such as 12.5% of 100: ¹²·⁵⁄₁₀₀, then multiply top and bottom by 10 to remove the decimal: ¹²⁵⁄₁₀₀₀ = ⅛.
将百分比写成分母为 100 的分数，然后化简。例如，65% = ⁶⁵⁄₁₀₀ = ¹³⁄₂₀（分子分母同时除以 5）。对于带有小数的百分比，如 12.5%，先写成分母为 100 的分数：¹²·⁵⁄₁₀₀，然后将分子分母同时乘以 10 去掉小数：¹²⁵⁄₁₀₀₀ = ⅛。
Remember to simplify fractions fully: find the highest common factor (HCF) of the numerator and denominator and divide both by it. For 60%, ⁶⁰⁄₁₀₀ simplifies by dividing by 20 to get ⅗.
记得将分数化简到最简：找到分子和分母的最大公因数 (HCF)，并除以它。对于 60%，⁶⁰⁄₁₀₀ 可以除以 20 化简为 ⅗。
8. Converting Between Percentages and Decimals | 百分比与小数之间的转换
age to a decimal, divide by 100 (move the decimal point two places left). For instance, 73% = 0.73, 8% = 0.08, and 120% to a percentage, multiply by 100 (move the decimal point two places right). Thus 0.47 = 47%, 0.03 = 3%, and 1.5 = 150%.
要将百分比转换为小数，除以 100（将小数点向左移动两位）。例如，73% = 0.73，8% = 0.08，120% = 1.2。要将小数转换为百分比，乘以 100（将小数点向右移动两位）。因此，0.47 = 47%，0.03 = 3%，1.5 = 150%。
This is the simplest conversion because percentages and decimals both use the base-10 system. Always check that the number of decimal places matches the percentage digits: 0.5 is 50%, not 5%.
这是最简单的转换，因为百分比和小数都使用十进制系统。始终检查小数位数是否与百分比的数字匹配：0.5 是 50%，而不是 5%。
9. Comparing and Ordering | 比较与排序
em all into the same form, usually decimals or percentages. For example, which is largest: ⅖, 0.44, or 41%? Convert: ⅖ = 0.4 = 40%, 0.44 = 44%, 41% = 0.41. So the order is 0.44 > 41% > ⅖.
要比较分数、小数和百分比，最好将它们都转换成同一种形式，通常转化为小数或百分比。例如，哪个最大：⅖、0.44 还是 41%？转换：⅖ = 0.4 = 40%，0.44 = 44%，41% = 0.41。因此顺序为 0.44 > 41% > ⅖。
When ordering on a number line, mark the positions as decimals for clarity. Always double-check your conversions to avoid errors.
在数轴上排序时，为了清晰，可将它们标记为小数。始终反复检查转换，以避免错误。
10. Finding a Percentage of a Quantity | 求一个数量的百分比
To find p% of a quantity, multiply the quantity by p/100. For example, 20% f 250 = (20/100) × 250 = 0.2 × 250 = 50. You can also break it into simpler parts: find 10% first by dividing by 10, then scale up.
要求一个数量的 p%，用这个数量乘以 p/100。例如，250 的 20% = (20/100) × 250 = 0.2 × 250 = 50。你也可以把它拆分成更简单的部分：先除以 10 求 10%，然后再进行缩放。
Common mental methods:
- 50% = half, divide by 2
- 25% = quarter, divide by 4
- 10% = one tenth, divide by 10
- 1% = one hundredth, divide by 100
Combining these allows you to find any percentage: 35% = 25% + 10%.
常见的心算方法：
- 50% = 一半，除以 2
- 25% = 四分之一，除以 4
- 10% = 十分之一，除以 10
- 1% = 百分之一，除以 100
将这些方法组合起来可以求出任意百分比：35% = 25% + 10%。
11. Percentage Increase and Decrease | 百分比增减
nt by a percentage, find the percentage of the amount and add it on. For example, increase £80 by 15%: 15% of £80 = (15/100) × 80 = £12, so new amount = £80 + £12 = £92. Alternatively, use a multiplier: 100% + 15% = 115% = 1.15, so £80 × 1.15 = £92.
要将一个数量增加一个百分比，先求出该数量的百分比，然后加上去。例如，将 £80 增加 15%：£80 的 15% = (15/100) × 80 = £12，因此新金额 = £80 + £12 = £92。或者使用乘数：100% + 15% = 115% = 1.15，所以 £80 × 1.15 = £92。
For a decrease, subtract the percentage: a 20% decrease on £50, find 20% of £50 = £10, so new amount = £50 - £10 = £40. The multiplier method: 100% - 20% = 80% = 0.8, so £50 × 0.8 = £40.
对于减少，减去相应的百分比：将 £50 减少 20%，£50 的 20% = £10，因此新金额 = £50 - £10 = £40。乘数法：100% - 20% = 80% = 0.8，所以 £50 × 0.8 = £40。
12. Real-life Applications and Problem Solving | 实际应用与问题解决
Fractions, decimals and percentages appear in many real-life contexts: discounts and sales (e.g. 30% off), recipes (e.g. ¾ cup of sugar), data representation (e.g. 45% of students like maths), and interest rates. Being able to move flexibly between forms helps you choose the most efficient way to solve problems.
分数、小数和百分比出现在许多实际生活场景中：折扣和促销（例如 30% 折扣）、食谱（例如 ¾ 杯糖）、数据表示（例如 45% 的学生喜欢数学）以及利率。能够在不同形式之间灵活转换，有助于你选择最有效的方法来解决问题。
Key problem-solving strategies include identifying which form the given information is in, deciding what requires conversion, and checking if the answer is reasonable. Always show your steps and, if asked, round decimals to a given precision.
关键的解决问题的策略包括：识别已知信息是哪种形式、判断哪些需要转换、并检查答案是否合理。始终展示你的步骤，如果题目有要求，将小数四舍五入到指定的精度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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