P4.1 Fractions | P4.1 分数

📚 P4.1 Fractions | P4.1 分数

Fractions are a core part of the Cambridge KS3 Mathematics curriculum. They appear in everyday situations – sharing a pizza, measuring ingredients, or expressing a part of a whole. In this unit, you will learn how to simplify fractions, find equivalent forms, and perform addition, subtraction, multiplication and division with confidence. A strong grasp of fractions is essential before moving on to decimals, percentages, ratios and algebra.

分数是剑桥 KS3 数学课程的核心部分。它们出现在日常生活的各种情境中——分享披萨、称量食材或是表示整体中的一部分。本单元将带你学会如何约分、寻找等值分数，并自信地进行分数的加减乘除。扎实掌握分数相关知识，是进一步学习小数、百分数、比与比例以及代数的重要基础。

1. What Is a Fraction? | 什么是分数？

A fraction represents a part of a whole. It is written with a numerator (top number) and a denominator (bottom number). The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have. For example, 3/4 means 3 parts out of 4 equal parts.

分数表示整体中的一部分，写作分子（上方的数）和分母（下方的数）。分母表示整体被分成了多少等份，分子表示取了几份。例如 3/4 表示从 4 等份中取 3 份。

Proper fraction: numerator < denominator, e.g. 2/5
Improper fraction: numerator ≥ denominator, e.g. 7/3
Mixed number: whole number and proper fraction, e.g. 1 ½

真分数：分子 < 分母，如 2/5
假分数：分子 ≥ 分母，如 7/3
带分数：整数与真分数的组合，如 1 ½

2. Equivalent Fractions | 等值分数

Equivalent fractions have different numerators and denominators but represent the same value. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non‑zero number. For example, 1/2 = 2/4 = 4/8. Simplifying a fraction means finding the equivalent form with the smallest possible whole numbers.

等值分数虽然分子、分母不同，但表示相同的值。你可以将分子和分母同时乘或除以同一个非零数来构造等值分数。例如 1/2 = 2/4 = 4/8。约分就是找到分子与分母尽可能小的等值分数。

a/b = (a × k)/(b × k), k ≠ 0

To simplify 12/16, divide both by 4 to get 3/4. This process is called cancelling.

要将 12/16 约分，分子分母都除以 4，得到 3/4。这个过程也称作约分。

3. Ordering and Comparing Fractions | 分数的大小比较与排序

To compare fractions like 2/3 and 3/5, rewrite them with a common denominator. The common denominator is a common multiple of 3 and 5, such as 15. Then 2/3 = 10/15 and 3/5 = 9/15, so 2/3 > 3/5. Always convert to the same denominator before comparing numerators.

比较 2/3 和 3/5 这样分数的大小时，需要将它们改写成分母相同的分数。公分母可以是两个分母的公倍数，例如 15。那么 2/3 = 10/15，3/5 = 9/15，所以 2/3 > 3/5。记得始终先通分再比较分子。

Find a common denominator (LCM works best).
Convert each fraction.
Compare numerators directly.

找到公分母（最小公倍数最优）。
将每个分数化为同分母。
直接比较分子大小。

4. Adding and Subtracting Fractions | 分数的加法和减法

When the denominators are the same, simply add or subtract the numerators and keep the denominator. For example, 2/7 + 3/7 = 5/7. When denominators are different, find a common denominator first, then add or subtract the numerators. Always simplify the answer if possible.

当分母相同时，直接将分子相加或相减，分母不变。例如 2/7 + 3/7 = 5/7。如果分母不同，先通分，再对分子进行加减运算。最后注意将结果约分为最简分数。

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

For mixed numbers, convert them to improper fractions before adding or subtracting. Afterwards, you may convert back to a mixed number.

如果遇到带分数，建议先转化为假分数再进行加减，之后可根据需要转回带分数。

5. Multiplying Fractions | 分数的乘法

Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. There is no need for a common denominator. Simplify before multiplying if possible (cross‑cancelling).

分数乘法很简单：分子乘分子，分母乘分母，无需通分。计算前如果可以先约分（交叉约分）的话，会使运算更简便。

a/b × c/d = (a×c)/(b×d)

Example: 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2. Notice that we could cancel the common factor 3 first to get 2/1 × 1/4 = 2/4 = 1/2.

例如：2/3 × 3/4，先做 2×3 得 6，3×4 得 12，结果为 6/12，约分为 1/2。也可以先交叉约去公因数 3，变成 2/1 × 1/4 = 2/4 = 1/2。

6. Dividing Fractions | 分数的除法

To divide by a fraction, multiply by its reciprocal (flip the second fraction). The reciprocal of a/b is b/a. Then proceed as in multiplication. This rule works for both proper and improper fractions.

除以一个分数，等于乘以它的倒数（将第二个分数的分子分母颠倒）。a/b 的倒数就是 b/a。之后按照乘法规则计算即可。这一法则对真假分数都适用。

a/b ÷ c/d = a/b × d/c

Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.

例如：3/4 ÷ 2/5，变成 3/4 × 5/2 = 15/8，化成带分数为 1 7/8。

7. Fractions and Mixed Numbers | 分数与带分数

A mixed number combines a whole part and a fractional part. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the sum over the original denominator. To convert an improper fraction to a mixed number, divide the numerator by the denominator: the quotient is the whole number, the remainder is the numerator, and the denominator stays the same.

带分数包含整数部分和真分数部分。将它化为假分数时，用整数乘分母，加上原分子，结果作为新分子，分母不变。假分数转带分数时，分子除以分母，商作为整数部分，余数作为分子，分母不变。

a b/c = (a×c + b)/c

Mastering these conversions is vital for addition, subtraction, multiplication and division involving mixed numbers.

掌握这些互化技巧，对含有带分数的加减乘除运算至关重要。

8. Fractions of Quantities | 求一个量的几分之几

To find a fraction of an amount, divide by the denominator and multiply by the numerator. For example, to calculate 3/4 of 60 kg, first divide 60 by 4 to get 15, then multiply by 3 to get 45 kg. This method works because the fraction tells you how many parts of the whole you need.

求一个数量的几分之几，先除以分母，再乘以分子。例如计算 60 kg 的 3/4，先将 60 除以 4 得 15，再乘以 3 得 45 kg。这种方法之所以有效，是因为分数本身就指明了需要从整体中取多少份。

Divide amount by denominator
Multiply the result by numerator
Keep the original unit

数量 ÷ 分母
结果 × 分子
保留原单位

9. Fractions and Decimals | 分数与小数

Every fraction can be expressed as a decimal by dividing the numerator by the denominator. Some fractions produce terminating decimals (e.g. 1/4 = 0.25), while others produce recurring decimals (e.g. 1/3 = 0.333…). Knowing common fraction‑decimal equivalents is useful: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/10 = 0.1, etc.

每一个分数都可以通过分子除以分母化为小数。有的分数得到有限小数（如 1/4 = 0.25），有的则得到循环小数（如 1/3 = 0.333…）。熟记常见的分数小数转换很有帮助：1/2 = 0.5，1/4 = 0.25，3/4 = 0.75，1/10 = 0.1 等。

10. Fraction Problems and Real‑Life Contexts | 分数应用题与实际情境

Fraction skills are applied in sharing problems, recipe adjustments, time calculations and measurements. For example: ‘A cake is cut into 12 equal slices. Lucy eats 2 slices and Tom eats 3 slices. What fraction of the cake remains?’ Total eaten = 5/12, so remaining = 7/12. Always read the problem carefully to decide which operation is needed.

分数技能可用于分配问题、食谱调整、时间计算和测量。例如：“一块蛋糕被切成 12 等份，露西吃了 2 块，汤姆吃了 3 块，蛋糕还剩下几分之几？”总共吃掉 5/12，所以剩余 7/12。解题时务必仔细读题，判断需要使用哪种运算。

Key words	Operation	示例
of, product	Multiply	2/3 of 90
sum, total	Add	1/4 + 2/3
difference, how many more	Subtract	3/5 – 1/2
share equally, split	Divide	3/4 ÷ 5

11. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Forgetting to find a common denominator before adding or subtracting – always check denominators first.
Forgetting to simplify final answers – make it a habit to look for common factors.
Mishandling mixed numbers – convert to improper fractions for calculations, then convert back if needed.
Inverting the wrong fraction in division – only flip the second fraction, not both.

加减前忘记通分——务必先检查分母是否相同。
忘记约分最终结果——养成寻找公因数的习惯。
带分数处理不当——计算时先化为假分数，必要时再转回。
除法时颠倒了错误的分数——只翻转第二个分数，而不是两个都翻。

12. Summary and Revision Tips | 总结与复习建议

Fractions are built around a few fundamental ideas: equivalence, common denominators, and the link between multiplication and division. Practise regularly with visual models (fraction bars, number lines) before moving to abstract calculations. Test yourself on mixed operations and word problems to build confidence. Remember: every fraction problem can be solved by applying the correct operation in the right order.

分数的核心在于几个基本概念：等值、通分，以及乘除法之间的联系。建议先借助视觉模型（分数条、数轴）进行练习，再过渡到抽象计算。通过混合运算和文字题自测，能够有效提升信心。记住：只要按照正确顺序使用正确的运算，所有分数问题都能迎刃而解。

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P4.1 Fractions | P4.1 分数