📚 Fractions, Decimals and Percentages | 分数、小数和百分数
At Key Stage 3, Cambridge Mathematics expects learners to move between fractions, decimals and percentages with confidence. These three forms describe the same idea - parts of a whole - but each is useful in different situations. This revision guide covers the essential skills, common misconceptions and exam-style applications you need to master.
在剑桥初中数学阶段，学生需要自信地在分数、小数和百分数之间进行转换。这三种形式描述的是同一个概念——整体的部分——但它们在不同场景下各有优势。本复习指南涵盖你需要掌握的核心技能、常见误区以及考试题型应用。
1. Equivalent Fractions and Simplifying | 等值分数与约分
To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). For example, 12/18 simplifies to 2/3 because the HCF of 12 and 18 is 6. Always write the final answer in its simplest form unless the question says otherwise.
化简分数时，用分子和分母的最大公因数（HCF）同时相除。例如，12/18 化简为 2/3，因为 12 和 18 的最大公因数是 6。除非题目另有说明，最终答案都要写成最简形式。
Equivalent fractions are made by multiplying or dividing both parts by the same non-zero number. This is useful for comparing fractions and for adding and subtracting. For example, 1/2, 2/4 and 5/10 are all equivalent fractions.
等值分数是通过将分子和分母同时乘以或除以同一个非零数得到的。这在比较分数以及加减分数时非常有用。例如，1/2、2/4 和 5/10 都是等值分数。
e fractions are equal. This method is very useful when you cannot see the multiplier easily.
检查等值分数的快速方法是交叉相乘。对于 3/4 和 9/12，3 × 12 = 36，4 × 9 = 36，所以这两个分数相等。当你不容易看出倍数时，这个方法非常有用。
Remember that simplifying a fraction does not change its value; it only changes the way the number is written. This is why 18/24 and 3/4 represent exactly the same proportion of a whole.
记住，化简分数不会改变它的值；它只是改变数字的书写方式。这就是为什么 18/24 和 3/4 表示整体的完全相同的比例。
2. Mixed Numbers and Improper Fractions | 带分数与假分数
An improper fraction has a numerator larger than its denominator, such as 11/4. A mixed number combines a whole number with a proper fraction, such as 2 3/4. Both forms are common in real life, so you need to convert between them quickly.
假分数的分子大于分母，例如 11/4。带分数由一个整数和一个真分数组成，例如 2 3/4。这两种形式在生活中都很常见，因此你需要快速地在它们之间进行转换。
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and put this over the original denominator. This works because the whole number is simply a group of the chosen size.
将带分数转换为假分数时，用整数乘以分母，加上分子，再将结果放在原分母上。这是因为整数部分就是一组所选大小的整体。
2 3/4 = (2 × 4 + 3)/4 = 11/4
er and the remainder is the new numerator. For 11/4, 11 ÷ 4 = 2 remainder 3, giving 2 3/4.
转换回带分数时，用分子除以分母；商为整数部分，余数为新的分子。对于 11/4，11 ÷ 4 = 2 余 3，得到 2 3/4。
When adding, subtracting, multiplying or dividing mixed numbers, it is usually easier to change them to improper fractions first. After calculating, you can convert the answer back if the question asks for a mixed number.
对带分数进行加、减、乘、除运算时，通常先把它们转换为假分数更简单。计算完成后，如果题目要求带分数，你可以再把答案转换回去。
3. Adding and Subtracting Fractions | 分数的加法与减法
To add or subtract fractions, they must have the same denominator. Find the lowest common multiple (LCM) of the denominators, convert each fraction, then add or subtract the numerators only. The denominator stays the same.
分数的加减法要求分母相同。先求分母的最小公倍数（LCM），将每个分数转换，然后只对分子进行加减。分母保持不变。
= 2/12. Therefore, 3/12 + 2/12 = 5/12. This method is reliable for any pair of fractions.
示例：1/4 + 1/6。4 和 6 的最小公倍数是 12，所以 1/4 = 3/12，1/6 = 2/12。因此 3/12 + 2/12 = 5/12。这个方法对任意两个分数都可靠。
r, convert it and carry the whole part across. For example, 1 2/3 + 2 2/3 = 3 4/3 = 4 1/3.
对于带分数，先加整数部分，再加分数部分。如果分数和为假分数，要转换并把整数部分进位。例如，1 2/3 + 2 2/3 = 3 4/3 = 4 1/3。
. The correct approach is to convert to a common denominator: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
一个常见错误是将分母直接相加。例如，1/2 + 1/3 不等于 2/5。正确的做法是转换成公分母：1/2 + 1/3 = 3/6 + 2/6 = 5/6。
4. Multiplying Fractions | 分数乘法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. Simplify before or after multiplying to keep numbers small. This skill is used in many problem-solving questions.
分数乘法很直接：分子乘分子，分母乘分母。在乘法前后约分可以保持数值较小。这个技能会用在许多解决问题的题目中。
3/5 × 2/7 = (3 × 2)/(5 × 7) = 6/35
3 = 4/1 × 2/3 = 8/3 = 2 2/3. This reminds you that multiplication and division are linked.
分数乘以整数时，先将整数写成以 1 为分母的分数。例如，4 × 2/3 = 4/1 × 2/3 = 8/3 = 2 2/3。
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