📚 Mastering Fractions, Decimals and Percentages | 分数、小数与百分比精讲
Fractions, decimals and percentages are three interconnected ways of representing parts of a whole. In Key Stage 3 mathematics, you will learn to convert smoothly between these forms, compare their values, perform arithmetic operations, and apply them to real-life problems such as discounts, proportions and probability. This article provides a thorough revision of the core concepts, worked examples and step-by-step methods, aligned to the Cambridge Lower Secondary curriculum. By the end, you will be able to tackle any fraction, decimal or percentage question with confidence.
分数、小数和百分比是表示整体之部分的三种相互关联的方式。在 KS3 数学中，你将学习如何在这几种形式之间流畅转换、比较其大小、进行运算，并将它们应用于折扣、比例和概率等实际问题。本文将系统复习核心概念，提供范例与分步方法，紧扣剑桥初中课程。读完本文，你将能够自信地解答任何分数、小数与百分比的题目。
1. Understanding the Three Forms | 理解三种形式
A fraction expresses a number as a numerator over a denominator, such as 3/4. A decimal uses the base‑10 place value system, writing parts after a decimal point, for example 0.75. A percentage is a fraction with a denominator of 100, so 75% means 75 out of 100. All three are equivalent; they simply present the same quantity in different notations.
分数用分子与分母表示一个数，如 3/4。小数使用十进制位值系统，小数点后写出部分，例如 0.75。百分比是分母为 100 的分数，因此 75% 表示百分之七十五。这三种形式是等价的，只是同一数量的不同写法。
2. Converting Fractions to Decimals | 分数转小数
If the division does not terminate, you get a recurring decimal. For example, 1/3 = 0.333…, written as 0.3̅. Recognise common equivalents: 1/2 = 0.5, 1/5 = 0.2, 3/4 = 0.75, 1/8 = 0.125.
将分数变为小数的方法是分子除以分母。对于像 1/4 这样的简单分数，1 ÷ 4 = 0.25。如果除法不能终止，会得到循环小数。例如 1/3 = 0.333…，记作 0.3̅。记住常见等价关系：1/2 = 0.5，1/5 = 0.2，3/4 = 0.75，1/8 = 0.125。
3. Converting Decimals to Fractions | 小数转分数
divide numerator and denominator by 5). For recurring decimals, use algebraic methods or recognise patterns.
将小数写成分母为 10 的幂的分数。如果小数点后有一位数字，分母为 10；两位则为 100；三位则为 1000，以此类推。然后约分。例如 0.45 = 45/100 = 9/20（分子分母同时除以 5）。对于循环小数，可使用代数方法或辨认规律。
4. Converting Fractions and Decimals to Percentages | 分数与小数转百分比
To convert a decimal to a percentage, multiply by 100 and add the % sign: 0.63 × 100 = 63% ernatively, write the fraction with a denominator of 100. For instance, 7/20 = 35/100 = 35%. Learn common percentage–fraction links: 1/4 = 25%, 1/3 ≈ 33.3%, 1/2 = 50%, 2/3 ≈ 66.7%, 3/4 = 75%.
小数转百分比，只需乘以 100 并加上百分号：0.63 × 100 = 63%。分数则先通过除法化为小数，再乘以 100；或者写成分母为 100 的分数。例如 7/20 = 35/100 = 35%。记住常见百分比与分数的对应：1/4 = 25%，1/3 ≈ 33.3%，1/2 = 50%，2/3 ≈ 66.7%，3/4 = 75%。
5. Converting Percentages to Fractions and Decimals | 百分比转分数与小数
To change a percentage to a fraction, write it over 100 and simplify: 40% = 40/100 = 2/5. For a decimal, simply divide the percentage by 100: 7% = 0.07, 125% = 1.25. When a percentage includes a fraction, e.g. 12½%, first turn it into an improper fraction (25/2 %) then write with denominator 100: (25/2)/100 = 25/200 = 1/8.
百分比化分数，将其写成分母为 100 的形式并约分：40% = 40/100 = 2/5。化小数则直接将百分数除以 100：7% = 0.07，125% = 1.25。当百分比含有分数时，如 12½%，先将其化为假分数（25/2%），再写分母 100：(25/2)/100 = 25/200 = 1/8。
6. Comparing and Ordering | 比较与排序
When comparing fractions, decimals and percentages, convert all numbers to the same form — ally decimals or percentages with common denominators. For example, to order 3/5, 0.58, 59% from smallest to largest, convert: 3/5 = 0.60 = 60%, 0.58 = 58%, 59% = 0.59. So the order is 0.58 (58%), 59%, 3/5 (60%).
比较分数、小数和百分比时，先将所有数转化为同一种形式——通常是小数或公分母的百分比。例如，要将 3/5、0.58、59% 从小到大排序，先转化：3/5 = 0.60 = 60%，0.58 = 58%，59% = 0.59。因此顺序为 0.58 (58%)、59%、3/5 (60%)。
7. Adding and Subtracting Fractions | 分数的加减
umbers, convert to improper fractions first or add whole numbers and fractions separately.
分数加减时，要保证它们有相同的分母。计算 2/3 + 1/4，求 3 和 4 的最小公倍数 12。改写成：2/3 = 8/12，1/4 = 3/12。相加得 8/12 + 3/12 = 11/12。对于带分数，可先化为假分数，或将整数部分与分数部分分别相加。
8. Multiplying and Dividing Fractions | 分数的乘除
y its reciprocal: 2/5 ÷ 3/4 = 2/5 × 4/3 = 8/15. Always simplify the answer where possible.
分数乘法，分子相乘作分子，分母相乘作分母：2/5 × 3/4 = (2×3)/(5×4) = 6/20，约分后为 3/10。分数除法，乘以除数的倒数：2/5 ÷ 3/4 = 2/5 × 4/3 = 8/15。结果尽可能化简。
9. Operations with Decimals | 小数的运算
t a whole number and do the same shift in the dividend. For example, 0.6 × 0.2: 6 × 2 = 12, two decimal places in total gives 0.12.
小数加减要对齐小数点。乘法可先忽略小数点，按整数相乘，再根据因数中小数位的总数在积中点上小数点。除法可把除数的小数点右移为整数，被除数也做相同移动。例如 0.6 × 0.2：6 × 2 = 12，总共有两位小数，得 0.12。
10. Finding a Percentage of a Quantity | 求一个数量的百分之几
of an amount, write the percentage as a fraction or decimal and multiply. For example, 30% of £240 = 30/100 × 240 = 0.3 × 240 = £72. For mental work, break it down: 10% is 24, so 30% is 3 × 24 = 72. This method is useful for discounts, tax and interest problems.
求一个数量的百分比，将百分比写成分数或小数再相乘。例如，£240 的 30% = 30/100 × 240 = 0.3 × 240 = £72。心算时可拆分：10% 是 24，那么 30% 就是 3 × 24 = 72。此方法适用于折扣、税款和利息问题。
11. Increasing and Decreasing by a Percentage | 百分比的增减
To increase an amount by 15%, multiply by (100% + 15%) = 115% = 1.15. To decrease by 15%, multiply by (100% − 15%) = 85% = 0.85. For example, a coat costing £80 with a 20% discount: New price = 0.80 × £80 = £64. Conversely, a price after a 25% mark‑up: New price = 1.25 × original.
增加 15%，乘以 (100% + 15%) = 115% = 1.15；减少 15%，乘以 (100% − 15%) = 85% = 0.85。例如，一件外套原价 £80，打 20% 折扣：新价格 = 0.80 × £80 = £64。反之，加价 25% 后，新价格 = 1.25 × 原价。
12. Problem Solving with Fractions, Decimals and Percentages | 分数、小数与百分比的综合应用
ne these representations. A typical problem: In a class, 3/5 of the students are girls; 40% of the girls wear glasses. What fraction of the class are girls who wear glasses? 3/5 of class are girls → 60% are girls. 40% of these wear glasses: 0.40 × 60% = 24% of the class. As a fraction, 24% rcentage or fraction, and convert everything to a single form to simplify the calculation.
许多考题会综合这些表示方式。典型问题：一个班级中 3/5 是女生，女生中有 40% 戴眼镜。戴眼镜的女生占全班的几分之几？3/5 是女生 → 60% 是女生。其中 40% 戴眼镜：0.40 × 60% = 24%，即全班的 24%。化为分数，24% = 24/100 = 6/25。解题时务必认清每个百分比或分数的基准量，并统一为一种形式以简化计算。
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