KS3 Cambridge Maths: Fractions, Decimals and Percentages (p39.2) | KS3 剑桥数学：分数、小数与百分数（p39.2）

📚 KS3 Cambridge Maths: Fractions, Decimals and Percentages (p39.2) | KS3 剑桥数学：分数、小数与百分数（p39.2）

This guide covers the core skills in KS3 Cambridge mathematics that appear on p39.2, focusing on fractions, decimals and percentages. You will learn how to convert between the three forms, compare values, and apply percentage calculations in real contexts.

本指南涵盖 KS3 剑桥数学 p39.2 中的核心技能，重点讲解分数、小数与百分数。你将学习三种形式之间的互化、数值比较，以及百分数在实际情境中的应用。

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1. Understanding Fractions | 理解分数

A fraction represents a part of a whole. The numerator is the top number and shows how many equal parts are taken; the denominator is the bottom number and shows the total number of equal parts. For example, in ¾, 3 is the numerator and 4 is the denominator.

分数表示整体的一部分。分子是上面的数，表示取了多少等份；分母是下面的数，表示整体被分成多少等份。例如在 ¾ 中，3 是分子，4 是分母。

There are three main types of fraction: a proper fraction has a numerator smaller than the denominator, an improper fraction has a numerator larger than or equal to the denominator, and a mixed number combines a whole number with a proper fraction.

分数主要有三种类型：真分数的分子小于分母，假分数的分子大于或等于分母，带分数由一个整数和一个真分数组成。

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2. Equivalent Fractions and Simplifying | 等值分数与化简

Equivalent fractions are written differently but have the same value. You can multiply or divide both the numerator and the denominator by the same non-zero number without changing the value of the fraction. To simplify a fraction, divide both parts by their highest common factor.

等值分数写法不同但值相同。你可以将分子和分母同时乘或除以同一个非零数，而不会改变分数的值。化简分数时，将分子和分母同时除以它们的最大公因数。

a/b = (a × k) / (b × k), where k ≠ 0

a/b = (a × k) / (b × k)，其中 k ≠ 0

For example, 18/24 simplifies to ¾ because both 18 and 24 can be divided by 6. Always check whether the numerator and denominator have any common factor before deciding that a fraction is in its simplest form.

例如 18/24 可以化简为 ¾，因为 18 和 24 都能被 6 整除。在判断一个分数是否为最简形式之前，一定要检查分子和分母是否有公因数。

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3. Converting Between Fractions and Decimals | 分数与小数的互化

0.375. If the division does not terminate, the decimal will repeat, such as 1/3 = 0.333...

将分数化为小数，用分子除以分母。例如 3/8 = 3 ÷ 8 = 0.375。如果除法无法除尽，小数就会循环，例如 1/3 = 0.333...

mal as a fraction over a power of 10 and then simplify. For example, 0.45 = 45/100 = 9/20.

将有限小数化为分数，先把小数写成以 10 的幂为分母的分数，然后化简。例如 0.45 = 45/100 = 9/20。

Fraction to decimal: a/b = a ÷ b

分数化小数：a/b = a ÷ b

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4. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化

Percentage means 'out of 100'. To convert a fraction to a percentage, first change it to a decimal, then multiply by 100%. To convert a percentage to a fraction, write the percentage over 100 and simplify if possible.

百分数表示“每一百份”。将分数化为百分数，先转为小数，再乘以 100%。将百分数化为分数，把百分数写在 100 上，然后尽可能化简。

Fraction	Decimal	Percentage
½	0.5	50%
¼	0.25	25%
¾	0.75	75%
⅕	0.2	20%
⅓	0.333...	33⅓%

These common conversions should be memorised because they appear frequently in KS3 tests and Checkpoint papers.

这些常见转换应该熟记，因为它们在 KS3 测验和 Checkpoint 试卷中经常出现。

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5. Ordering and Comparing | 排序与比较

When ordering fractions, decimals and percentages, change all values into the same form. Converting to decimals is usually the fastest method. Then compare the digits from left to right, starting with the largest place value.

在对分数、小数和百分数排序时，把所有数值转化为同一种形式。通常转化为小数最快。然后从左到右比较数位，从最大的数位开始。

For example, to order ⅖, 0.45 and 38%, first convert each value: ⅖ = 0.4, 0.45 = 0.45, 38% = 0.38. The order from smallest to largest is 38% < ⅖ < 0.45.

例如，要对 ⅖、0.45 和 38% 进行排序，先转换每个数值：⅖ = 0.4，0.45 = 0.45，38% = 0.38。从小到大排列为 38% < ⅖ < 0.45。

A useful check is to imagine a number line from 0 to 1. Fractions and decimals close to 1 are larger than those closer to 0.

一种有效的检查方法是想象一条从 0 到 1 的数轴。接近 1 的分数和小数比接近 0 的更大。

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6. Adding and Subtracting Fractions | 分数加减

To add or subtract fractions, first find a common denominator. Rewrite each fraction with that denominator, then add or subtract the numerators and keep the denominator the same. For mixed numbers, convert them to improper fractions first.

分数加减时，先找到公分母。将每个分数改写为该分母，然后加减分子，分母保持不变。对于带分数，先转化为假分数。

a/c + b/c = (a + b) / c

a/c + b/c = (a + b) / c

1/4 + 2/5. The common denominator is 20, so 1/4 = 5/20 and 2/5 = 8/20. Adding gives 13/20.

示例：1/4 + 2/5。公分母为 20，因此 1/4 = 5/20，2/5 = 8/20。相加得到 13/20。

Do not add denominators together. This is one of the most common mistakes in KS3 fraction work.

不要将分母相加。这是 KS3 分数学习中最常见的错误之一。

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7. Multiplying and Dividing Fractions | 分数乘除

To multiply fractions, multiply the numerators together and the denominators together. Simplify before multiplying if possible to keep numbers small. To divide by a fraction, multiply by its reciprocal.

分数相乘时，分子乘分子，分母乘分母。如果可能，在相乘前先约分，以减小数字。除以一个分数等于乘以它的倒数。

(a/b) × (c/d) = ac / bd

(a/b) × (c/d) = ac / bd

(a/b) ÷ (c/d) = (a/b) × (d/c) = ad / bc

(a/b) ÷ (c/d) = (a/b) × (d/c) = ad / bc

e: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Always give the final answer in its simplest form.

示例：2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6。最后结果一定要化为最简形式。

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8. Finding a Percentage of an Amount | 求一个数的百分之几

To find a percentage of an amount, write the percentage as a fraction over 100 and multiply by the amount. This skill is often tested with money, distance, time and other real-life quantities.

求一个数的百分之几，把百分数写成分母为 100 的分数，再乘以这个数。这一技能常涉及金钱、距离、时间和其他实际量。

p% of N = N × p/100

p% of N = N × p/100

For example, 15% of 60 = 60 × 15/100 = 9. You can also use a mental method: 10% of 60 is 6, 5% is 3, so 15% is 6 + 3 = 9.

例如，15% of 60 = 60 × 15/100 = 9。你也可以使用心算方法：60 的 10% 是 6，5% 是 3，因此 15% 是 6 + 3 = 9。

When working with percentages greater than 100%, the result will be larger than the original amount. For example, 120% of 80 = 80 × 120/100 = 96.

当百分数大于 100% 时，结果会大于原数。例如 120% of 80 = 80 × 120/100 = 96。

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9. Percentage Increase and Decrease | 百分数增减

To increase an amount by a percentage, multiply by (100 + p)/100. To decrease an amount by a percentage, multiply by (100 - p)/100. The multiplier must match the direction of the change.

将一个数增加百分之 p，乘以 (100 + p)/100；减少百分之 p，乘以 (100 - p)/100。乘数必须与增减方向一致。

Increase: New value = N × (100 + p) / 100

增加：新值 = N × (100 + p) / 100

Decrease: New value = N × (100 - p) / 100

减少：新值 = N × (100 - p) / 100

Example: a shirt costs 40 pounds. In a sale the price is reduced by 15%. The new price is 40 × 85/100 = 34 pounds.

示例：一件衬衫售价 40 英镑。促销时价格降低 15%。新价格为 40 × 85/100 = 34 英镑。

Do not confuse percentage increase with percentage of an amount. If a value increases by 20%, the multiplier is 120/100, not 20/100.

不要混淆百分数增加与求一个数的百分之几。如果数值增加 20%，乘数是 120/100，而不是 20/100。

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10. Common Misconceptions and Exam Tips | 常见误区与考试技巧

Common mistakes include adding denominators when adding fractions, forgetting to simplify, and confusing 0.05 with 5%. Another frequent error is using 0.5 to represent 5%, when 5% is actually 0.05.

常见错误包括分数相加时把分母也相加、忘记化简，以及混淆 0.05 与 5%。另一个常见错误是用 0.5 表示 5%，而实际上 5% 是 0.05。

In exams, show clear working at each step. Write the common denominator when adding or subtracting fractions, and use estimation to check that percentage answers are reasonable. For example, 40% of 90 should be less than 90 but more than 30.

考试中要展示每一步的清晰过程。进行分数加减时写出公分母，并用估算检查百分数答案是否合理。例如 90 的 40% 应小于 90 但大于 30。

When a question gives a mixture of fractions, decimals and percentages, convert everything to decimals first. This reduces errors and makes comparison straightforward.

当题目给出分数、小数和百分数的混合形式时，先将所有数值转换为小数。这样可以减少错误，并使比较更加直接。

Finally, always re-read the question to check whether it asks for an increase, a decrease, or a simple fraction-of-an-amount calculation. One word can change the multiplier entirely.

最后，一定要重新读题，确认题目要求的是增加、减少还是简单求一个数的几分之几。一个词就可能完全改变乘

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