📚 Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three different ways of representing parts of a whole. In the Cambridge Key Stage 3 mathematics curriculum, you are expected to move fluently between these forms, compare quantities, and apply them in real‑life contexts such as calculating discounts, comparing test scores or measuring ingredients. Understanding the link between them helps build number sense and lays the groundwork for ratio, proportion and algebra.
分数、小数和百分数是表示整体部分的三种不同方式。在剑桥初中(KS3)数学课程中，你需要能在这些形式之间灵活转换、比较数量，并在实际情境中加以应用，如计算折扣、比较考试分数或衡量配料。理解它们之间的联系有助于培养数感，并为比、比例和代数学习打下基础。
1. What Are Fractions? | 什么是分数？
f a whole or a part of a set. It is written as a numerator over a denominator, for example ⅗ be proper (numerator < denominator), improper (numerator ≥ denominator) or mixed numbers.
分数表示一个整体或一个集合的部分。它写成分子在上、分母在下的形式，例如 ⅗，表示 5 等份中的 3 份。分母告诉我们整体被分成多少等份，分子告诉我们取了多少份。分数可以是真分数(分子 < 分母)、假分数(分子 ≥ 分母)或带分数。
In KS3, you will work with unit fractions (⅛, ½), non‑unit fractions (¾, ⅘) and also fractions greater than 1 such as ⁷⁄₄ or 1 ³⁄₄. Using diagrams like fraction bars or circles helps you visualise these parts clearly.
在 KS3 阶段，你会接触到单位分数(⅛、½)、非单位分数(¾、⅘)以及大于 1 的分数，如 ⁷⁄₄ 或 1 ³⁄₄。借助分数条或圆形图等图示，可以帮助你清楚地看到这些部分。
2. Equivalent Fractions | 等值分数
You can find them by multiplying or dividing the numerator and denominator by the same non‑zero integer. For example, ½ = ²⁄₄ = ⁴⁄₈ = ⁵⁰⁄₁₀₀.
等值分数的分子和分母不同，但表示相同的值。你可以通过将分子和分母同时乘以或除以同一个非零整数来求得等值分数。例如 ½ = ²⁄₄ = ⁴⁄₈ = ⁵⁰⁄₁₀₀。
mbridge Checkpoint, you might be asked to fill in missing numbers in statements like ³⁄₅ = ?/₂₀, where you multiply numerator and denominator by 4 to get ³×⁴⁄₅×⁴ = ¹²⁄₂₀.
理解等值对于约分和比较分数至关重要。在剑桥 Checkpoint 考试中，你可能会被要求填写类似 ³⁄₅ = ?/₂₀ 的空缺，你需要将分子和分母同时乘以 4，得到 ³×⁴⁄₅×⁴ = ¹²⁄₂₀。
3. Simplifying Fractions | 约分
simplify, divide both by their greatest common divisor (GCD). For example, ⁸⁄₁₂ ÷ 4 = ²⁄₃.
约分就是把分数化为最简形式，即分子和分母除了 1 以外没有其他公因数。约分时，用分子和分母的最大公约数(GCD)同时去除。例如 ⁸⁄₁₂ ÷ 4 = ²⁄₃。
3, GCD = 6, so simplified to ³⁄₄. Always check if the fraction can be simplified further.
回想乘法表和质因数会很有帮助。对于 ¹⁸⁄₂₄，质因数分解：18 = 2 × 3²，24 = 2³ × 3，最大公约数为 6，因此约简为 ³⁄₄。要经常检查分数是否还能继续约简。
4. Comparing and Ordering Fractions | 比较和排序分数
multiple (LCM) of 3 and 4 is 12, so ⅔ = ⁸⁄₁₂ and ¾ = ⁹⁄₁₂. Since 8 < 9, we see that ⅔ < ¾.
要比较如 ⅔ 和 ¾ 这样的分数，我们需要将它们改写为公分母。3 和 4 的最小公倍数(LCM)是 12，所以 ⅔ = ⁸⁄₁₂，¾ = ⁹⁄₁₂。由于 8 < 9，可见 ⅔ < ¾。
When ordering a mix of proper fractions, improper fractions and mixed numbers, convert all to improper fractions with a common denominator. For ascending order, list from smallest to largest. Diagrams or number lines can also support reasoning.
当排列真分数、假分数和带分数混合序列时，将它们全部化为具有相同分母的假分数。由小到大的顺序即为升序排列。图表或数轴也有助于推理。
5. Adding and Subtracting Fractions | 分数的加法和减法
denominators differ, find a common denominator first, convert, then perform the operation.
如果分数分母相同，只需将分子相加或相减，分母不变：⅛ + ³⁄₈ = ⁴⁄₈ = ½。如果分母不同，先找出公分母，进行转换，再执行运算。
rs, convert to improper fractions, then add or subtract, and finally convert back to a mixed number if required. Remember to simplify the result.
对于 ½ + ⅓，2 和 3 的最小公倍数是 6：½ = ³⁄₆，⅓ = ²⁄₆，因此和为 ⁵⁄₆。对于带分数，先化成假分数，再加减，必要时最后再化回带分数。记得化简结果。
6. Multiplying and Dividing Fractions | 分数的乘法和除法
g if possible, by cancelling any common factors between any numerator and any denominator.
乘法时，分子相乘，分母相乘：⅔ × ¾ = (2×3)/(3×4) = ⁶⁄₁₂ = ½。如果可能，可以在乘之前约去任意分子与分母之间的公因数，使计算简化。
: ¾ ÷ ²⁄₅ = ¾ × ⁵⁄₂ = ¹⁵⁄₈ = 1 ⁷⁄₈. For mixed numbers, change to improper fractions first.
除以一个分数，要乘以它的倒数。对于 a/b ÷ c/d，将第二个分数上下颠倒变为 d/c，然后相乘：¾ ÷ ²⁄₅ = ¾ × ⁵⁄₂ = ¹⁵⁄₈ = 1 ⁷⁄₈。带分数要先化为假分数。
7. Converting Fractions to Decimals | 分数转小数
enominator is a power of 10, the conversion is straightforward: ⁷⁄₁₀ = 0.7, ²³⁄₁₀₀ = 0.23.
分数可以通过分子除以分母转换为小数。对于 ⅝，用 5 除以 8 得到 0.625。如果分母是 10 的幂，转换更为直接：⁷⁄₁₀ = 0.7，²³⁄₁₀₀ = 0.23。
Some fractions produce recurring decimals, like ⅓ = 0.333... which is written as 0.3̅ mal equivalents helps you move quickly between the two forms: ½ = 0.5, ¼ = 0.25, ¾ = 0.75, ⅕ = 0.2, ⅛ = 0.125, and so on.
有些分数会得到循环小数，例如 ⅓ = 0.333...，记作 0.3̅。熟记常见分数及其对应小数有助于在两种形式之间快速转换：½ = 0.5，¼ = 0.25，¾ = 0.75，⅕ = 0.2，⅛ = 0.125 等。
8. Converting Decimals to Percentages and Vice Versa | 小数与百分数的互换
Percent means 'per hundred'. To convert a decimal to a percentage, multiply by 100 and add the % symbol. For example, 0.47 × 100 = 47%, and 0.025 × 100 = 2.5%. To convert a percentage to a decimal, divide by 100: 63% = 0.63, and 7% = 0.07.
百分数表示“每一百”。将小数转换为百分数时，乘以 100 并加上百分号。例如，0.47 × 100 = 47%，0.025 × 100 = 2.5%。将百分数转换为小数时，除以 100：63% = 0.63，7% = 0.07。
d directly into percentages: first convert the fraction to a decimal, then multiply by 100%. For ⅗, 3 ÷ 5 = 0.6 → 60%. Alternatively, make an equivalent fraction with denominator 100: ⅗ = ⁶⁰⁄₁₀₀ = 60%.
分数也可以直接转换为百分数：先将分数化为小数，再乘以 100%。对于 ⅗，3 ÷ 5 = 0.6 → 60%。或者，找一个分母为 100 的等值分数：⅗ = ⁶⁰⁄₁₀₀ = 60%。
9. Percentages of Amounts | 百分数的计算
ity, you can multiply the quantity by the percentage as a decimal. For example, to find 15% of £240, work out 0.15 × 240 = £36. Non‑calculator methods often use key percentages: 10% is found by dividing by 10, 5% is half of 10%, and 1% by dividing by 100, then combine.
要计算一个数量的百分之几，可以用该数量乘以对应的小数。例如，计算 £240 的 15%，即 0.15 × 240 = £36。非计算器的方法常利用关键百分数：10% 除以 10 求得，5% 是 10% 的一半，1% 是除以 100，然后组合起来。
For example, 17.5% of 300: 10% = 30, 5% = 15, 2.5% nd for solving problems involving percentage increase and decrease, discounts, and profit.
例如，求 300 的 17.5%：10% = 30，5% = 15，2.5% = 7.5，相加得 52.5。这种方法对心算以及解决涉及百分比增减、折扣和利润的问题极其有用。
10. Linking Fractions, Decimals and Percentages as Proportions | 分数、小数和百分数作为比例的联系
nd word problems. For instance, a test score of 18 out of 25 is ¹⁸⁄₂₅ = 72/100 = 0.72 = 72%.
分数、小数和百分数都是描述比例的方式。能够在它们之间切换有助于你解读表格、图表和文字题。比如，一次考试得分 18 分，总分 25 分，即 ¹⁸⁄₂₅ = ⁷²⁄₁₀₀ = 0.72 = 72%。
In real life, a 'half‑price' sale means 50% off, which is equivalent to paying 0.5 of the original price or multiplying by ½. A recipe may ask for 0.75 litres of milk, which is ¾ of a litre. The more fluent you are, the easier it is to solve KS3 problems on ratio and proportion later.
现实生活中的“半价”促销意味着优惠 50%，相当于支付原价的 0.5 倍或乘以 ½。食谱可能要求 0.75 升牛奶，也就是 ¾ 升。你越是熟练，将来解决 KS3 的比和比例问题就会越容易。
Always check whether an answer is reasonable: 20% of a quantity should be less than half. Practise converting mentally so that ⅖ = 0.4 = 40% and 0.125 = ⅛ = 12.5% become automatic.
一定要检查答案是否合理：一个数的 20% 应该小于它的一半。多加练习心算转换，使 ⅖ = 0.4 = 40% 和 0.125 = ⅛ = 12.5% 这样的转换成为条件反射。
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