Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分比

📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分比

Numbers appear in many forms, and understanding the links between fractions, decimals and percentages is a cornerstone of KS3 mathematics. This guide breaks down each concept step by step, showing you how to convert fluently between them and perform calculations with confidence. By the end, you will be able to tackle problems ranging from sharing pizzas to calculating discounts in a sale.

数字以多种形式出现，理解分数、小数和百分比之间的联系是 KS3 数学的基石。本指南逐步分解每个概念，教你如何熟练地在它们之间进行转换，并自信地进行计算。学完之后，你将能够解决从分享披萨到计算折扣等各种问题。

1. Understanding Fractions | 理解分数

A fraction represents a part of a whole. The top number (numerator) shows how many parts we have, while the bottom number (denominator) shows how many equal parts the whole is divided into. For example, in ¾, the whole is divided into 4 equal parts, and we have 3 of those parts.

分数表示整体的一部分。分子（上面的数字）表示我们拥有多少份，分母（下面的数字）表示整体被分成了多少等份。例如，在 ¾ 中，整体被分成 4 等份，我们拥有其中的 3 份。

Proper fractions: numerator < denominator, e.g. ⅔.
Improper fractions: numerator ≥ denominator, e.g. ⁵⁄₃.
Mixed numbers: a whole number and a fraction, e.g. 1½.

真分数：分子小于分母，例如 ⅔。
假分数：分子大于或等于分母，例如 ⁵⁄₃。
带分数：一个整数和一个分数，例如 1½。

2. Equivalent Fractions | 等值分数

Equivalent fractions have the same value even though they look different. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number. For example, ½ = ²⁄₄ = ⁴⁄₈.

等值分数虽然看起来不同，但数值相同。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得到等值分数。例如，½ = ²⁄₄ = ⁴⁄₈。

This idea is essential when simplifying fractions or finding common denominators for addition and subtraction.

在简化分数或为加减法寻找公分母时，这一概念至关重要。

3. Simplifying Fractions | 简化分数

To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). When the HCF is 1, the fraction is in its simplest form. For example, to simplify ¹²⁄₁₆, the HCF of 12 and 16 is 4, so ¹²⁄₁₆ = ³⁄₄.

要简化分数，用分子和分母的最大公因数（HCF）去除。当 HCF 为 1 时，分数就是最简形式。例如，简化 ¹²⁄₁₆，12 和 16 的 HCF 是 4，所以 ¹²⁄₁₆ = ³⁄₄。

Step 1: List factors of both numbers.
Step 2: Identify the highest factor they share.
Step 3: Divide both by that number.

步骤 1：列出两个数的所有因数。
步骤 2：找出它们共有的最大因数。
步骤 3：用该数除分子和分母。

4. Converting Fractions to Decimals | 分数化小数

To convert a fraction to a decimal, divide the numerator by the denominator. You can do this using short division or a calculator. For example, ¾ as a decimal is 3 ÷ 4 = 0.75.

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计算器。例如，¾ 化为小数是 3 ÷ 4 = 0.75。

Some fractions produce recurring decimals, such as ⅓ = 0.333… We indicate this with a dot above the repeating digit, e.g. 0.3̇.

有些分数会得到循环小数，例如 ⅓ = 0.333…，我们在重复数字上方加点表示，如 0.3̇。

Fraction	Decimal
½	0.5
¼	0.25
⅕	0.2
⅛	0.125

5. Converting Decimals to Percentages | 小数化百分比

Percent means ‘out of 100’, so to convert a decimal to a percentage, multiply by 100. For instance, 0.45 × 100 = 45%. The shortcut is to move the decimal point two places to the right.

百分比表示“每一百”，因此要将小数转换为百分比，只要乘以 100。例如，0.45 × 100 = 45%。简便方法是把小数点向右移两位。

Conversely, to convert a percentage to a decimal, divide by 100 (move the point two places left). For example, 72% = 0.72.

反过来，将百分比转换为小数，除以 100（小数点向左移两位）。例如，72% = 0.72。

0.07 = 7%
1.2 = 120%
0.3% = 0.003

6. Converting Fractions to Percentages | 分数化百分比

There are two main approaches. Method A: convert the fraction to a decimal first, then multiply by 100. Method B: find an equivalent fraction with denominator 100. For example, ⅗ as a decimal is 0.6, then × 100 = 60%. Alternatively, ⅗ = ⁶⁰⁄₁₀₀ = 60%.

有两种主要方法。方法 A：先将分数化为小数，再乘以 100。方法 B：找一个分母为 100 的等值分数。例如，⅗ 化为小数是 0.6，再 × 100 = 60%。或者，⅗ = ⁶⁰⁄₁₀₀ = 60%。

For fractions like ⅔, method A is easier: 2 ÷ 3 ≈ 0.6667, so 66.67% or 66⅔%.

对于像 ⅔ 这样的分数，方法 A 更简单：2 ÷ 3 ≈ 0.6667，得到 66.67% 或 66⅔%。

7. Adding and Subtracting Fractions | 分数的加减

When denominators are the same, simply add or subtract the numerators. For example, ⅜ + ²⁄₈ = ⅝. When denominators are different, first find a common denominator using the lowest common multiple (LCM). For ½ + ⅓, the LCM of 2 and 3 is 6, so ½ = ³⁄₆ and ⅓ = ²⁄₆, giving ⁵⁄₆.

分母相同时，直接对分子进行加减。例如，⅜ + ²⁄₈ = ⅝。分母不同时，先找到分母的最小公倍数 (LCM)。例如，½ + ⅓，2 和 3 的 LCM 是 6，所以 ½ = ³⁄₆，⅓ = ²⁄₆，相加得 ⁵⁄₆。

Step 1: Find the LCM of the denominators.
Step 2: Convert each fraction to an equivalent fraction with the common denominator.
Step 3: Add or subtract the numerators; keep the denominator the same.
Step 4: Simplify if possible.

步骤 1：找出分母的最小公倍数。
步骤 2：将每个分数转换为以公分母为分母的等值分数。
步骤 3：对分子做加减；分母保持不变。
步骤 4：若可以，化简。

8. Multiplying and Dividing Fractions | 分数的乘除

Multiplication: Multiply the numerators together and multiply the denominators together. For example, ⅔ × ¾ = (2 × 3) / (3 × 4) = ⁶⁄₁₂ = ½. Simplify the result if needed.

乘法：分子相乘，分母相乘。例如，⅔ × ¾ = (2 × 3) / (3 × 4) = ⁶⁄₁₂ = ½。必要时化简。

Division: To divide by a fraction, multiply by its reciprocal (flip the second fraction). For example, ¾ ÷ ⅔ = ¾ × ³⁄₂ = ⁹⁄₈ = 1⅛.

除法：除以一个分数，相当于乘以其倒数（将第二个分数的分子分母互换）。例如，¾ ÷ ⅔ = ¾ × ³⁄₂ = ⁹⁄₈ = 1⅛。

With mixed numbers, convert to improper fractions first.

对于带分数，要先化为假分数。

General rule: a/b ÷ c/d = a/b × d/c

一般法则：a/b ÷ c/d = a/b × d/c

9. Percentage Calculations | 百分比计算

To find a percentage of an amount, convert the percentage to a decimal (or fraction) and multiply. For example, 30% of £240 is 0.3 × 240 = £72. Alternatively, 10% is £24, so 30% is 3 × £24 = £72.

要找出一个数的百分之几，先把百分比转化为小数（或分数）再相乘。例如，£240 的 30% 是 0.3 × 240 = £72。或者，10% 是 £24，所以 30% 就是 3 × £24 = £72。

Increase by a percentage: multiply by (1 + percentage as decimal). Decrease by a percentage: multiply by (1 – percentage as decimal).
增加一个百分比：乘以 (1 + 百分比的小数)。减少一个百分比：乘以 (1 – 百分比的小数)。

Example: A £50 shirt with a 20% discount costs £50 × 0.8 = £40.

例子：一件 £50 的衬衫打八折后价格为 £50 × 0.8 = £40。

Reverse percentage problems: If a price after a 15% increase is £92, the original is £92 ÷ 1.15 = £80.

反向百分比问题：如果价格上涨 15% 后为 £92，原价为 £92 ÷ 1.15 = £80。

10. Ordering Fractions, Decimals and Percentages | 比较分数、小数和百分比的大小

To compare values given in different forms, convert them all to the same form (usually decimals or percentages). For example, rank ⅔, 0.6 and 55% from smallest to largest: ⅔ ≈ 0.667, 0.6 = 0.6, 55% = 0.55, so order is 55%, 0.6, ⅔.

要比较以不同形式给出的数，先把它们都转换成同一种形式（通常是小数或百分比）。例如，按从小到大排列 ⅔、0.6 和 55%：⅔ ≈ 0.667，0.6 = 0.6，55% = 0.55，所以顺序是 55%、0.6、⅔。

Always use the same denominator or decimal places to avoid mistakes.

始终使用相同的分母或小数位数以避免错误。

11. Rounding and Estimation with Percentages and Fractions | 百分比和分数的四舍五入与估算

When dealing with fractions that do not convert neatly to a finite decimal (e.g., ⅔ = 0.6666…), it is common to round to two or three decimal places. For money problems, round to two decimal places (pence). For example, 40.6667% can be rounded to 40.7% depending on the context.

当处理不能转换为有限小数的分数时（例如 ⅔ = 0.6666…），通常会四舍五入到两位或三位小数。涉及金额的题目，应精确到两位小数（便士）。例如，40.6667% 根据上下文可近似为 40.7%。

Estimation helps check answers: ⅞ of 501 ≈ 1 × 500 = 500, while the exact value is 438.375.

估算有助于检查答案：⅞ × 501 ≈ 1 × 500 = 500，而精确值是 438.375。

12. Real-Life Applications | 实际应用

Fractions, decimals and percentages are everywhere: recipe adjustments (¾ cup of flour), discounts (40% off sale), test scores (18 out of 25, or 72%), and probability (½ chance of rain). Understanding these connections allows you to interpret data, budget money, and make informed decisions.

分数、小数和百分比无处不在：食谱调整（¾ 杯面粉）、折扣（打四折）、考试成绩（25题中答对18题，即 72%）以及概率（降雨概率 ½）。理解这些联系使你能够解读数据、管理资金并做出明智的决策。

Practise converting between forms until it feels automatic — this fluency is a key skill assessed throughout KS3 and beyond.

反复练习不同形式之间的转换，直到熟练自如——这一流利度是整个 KS3 及以后评估的关键技能。

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Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分比