📚 Cambridge KS3 Mathematics: Fractions, Decimals and Percentages | 剑桥 KS3 数学：分数、小数与百分比
Welcome to this Cambridge KS3 revision note. We will build confidence in switching between fractions, decimals and percentages, and in using these forms to solve real exam-style problems. This is one of the most heavily assessed number topics at KS3, so careful practice of the methods below will raise your accuracy and speed.
欢迎阅读本份剑桥 KS3 数学复习笔记。我们将建立分数、小数和百分比之间熟练转换的能力，并学会运用这些形式解决真实的考试型问题。这是 KS3 阶段考查最频繁的“数”主题之一，因此认真练习以下方法将提高你的准确度和速度。
1. Key Equivalences Between FDP | 分数、小数与百分比的常用等值
itten in three ways: as a fraction, a decimal or a percentage. For example, 1/2 = 0.5 = 50%. Knowing the common equivalences saves time in ordering and comparing questions.
同一个数量可以用三种方式表示：分数、小数或百分数。例如 1/2 = 0.5 = 50%。记住常用等值能帮你在排序和比较题中节省时间。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333... | 33⅓% |
| 2/3 | 0.666... | 66⅔% |
| 1/8 | 0.125 | 12.5% |
You should be able to recall these quickly without a calculator. Rapid recall of these pairs makes later work on percentage increase, probability and pie charts much easier.
你应该能够不借助计算器快速回忆这些等值。快速记忆这些对应关系，会让后面有关百分比增减、概率和饼图的内容简单得多。
2. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator by the denominator. For a mixed number, keep the whole number part and convert the fractional part only.
要将分数转换为小数，用分子除以分母。对于带分数，保留整数部分，只转换分数部分。
fraction → decimal: numerator ÷ denominator
hould set out the division carefully if the decimal is recurring, such as 1/3 = 0.333... .
例如，7/8 = 7 ÷ 8 = 0.875。如果小数循环，如 1/3 = 0.333...，你应该认真写出除法过程。
3/20 = 15/100 = 0.15. This is especially useful with fractions such as 1/4, 3/5 and 7/25.
另一种方法是把分数改写成分母为 10、100 或 1000 的等值分数。例如 3/20 = 15/100 = 0.15。这种方法对 1/4、3/5 和 7/25 这样的分数尤其有用。
- Divide the numerator by the denominator when exact division is easy. 当可以整除时，用分子除以分母。
- Use equivalent fractions over 10, 100 or 1000 for simple conversions. 对简单转换，使用分母为 10、100 或 1000 的等值分数。
3. Converting Decimals to Percentages | 小数转百分数
Multiply the decimal by 100, or move the decimal point two places to the right, and add the percent symbol. This method works for all decimals, including values greater than 1.
将小数乘以 100，或将小数点向右移动两位，再加上百分号。这个方法适用于所有小数，包括大于 1 的数。
decimal × 100 = percentage
For example, 0.47 = 47%, 0.03 = 3% and 1.25 = 125% two-place move clearly, you can first write 0.03 as 03, then insert the percent sign as 3%.
例如 0.47 = 47%，0.03 = 3%，1.25 = 125%。为了看清小数点的两位移动，你可以先把 0.03 写成 03，再加百分号得到 3%。
Be careful with decimals such as 0.7. Since 0.7 is the same as 0.70, it becomes 70%, not 7%. Writing a zero in the hundredths place can help avoid errors.
要小心像 0.7 这样的小数。因为 0.7 等于 0.70，所以它应转换为 70%，而不是 7%。在百分位补写一个零可以避免错误。
4. Converting Percentages to Fractions | 百分数转分数
Write the percentage as a fraction over 100, then simplify fully. If the percentage contains a decimal part, multiply the numerator and denominator by 10 or 100 to clear the decimal before simplifying.
将百分数写成分母为 100 的分数，然后约分到最简。如果百分数含有小数部分，先给分子和分母同乘 10 或 100 去掉小数点，再约分。
percentage → fraction: percentage ÷ 100
For example, 45% = 45/100 = 9/20. For 37.5% ite 37.5/100, then multiply top and bottom by 10 to get 375/1000, which simplifies to 3/8.
例如 45% = 45/100 = 9/20。对于 37.5%，先写成 37.5/100，然后分子分母同乘 10，得到 375/1000，再约分为 3/8。
You should always give fractions in their simplest form unless a question says otherwise. Simplifying fully is part of the final answer in Cambridge KS3 marking.
除非题目另有说明，你总是要把分数化简到最简形式。完整约分是剑桥 KS3 评卷中最终答案的一部分。
5. Ordering and Comparing FDP | 分数、小数和百分比的排序与比较
Convert all quantities to the same form, usually decimals, then compare them. After ordering as decimals, rewrite the values in their original forms as required by the question.
将所有数量转换为同一种形式，通常是小数，再进行比较。按小数排好序后，再根据题目要求用原来的形式写出。
For example, order 2/5, 0.45 and 38% from smallest to largest. Convert to decimals: 2/5 = 0.4, 0.45 = 0.45 and 38% = 0.38. The correct order is 0.38, 0.4, 0.45, so the answer is 38%, 2/5, 0.45.
例如，将 2/5、0.45 和 38% 从小到大排列。转换为小数：2/5 = 0.4，0.45 = 0.45，38% = 0.38。正确的顺序是 0.38、0.4、0.45，因此答案是 38%、2/5、0.45。
Do not compare fractions by looking only at numerators or denominators, because fractions with different denominators cannot be compared directly. Always convert first.
不要只通过分子或分母来比较分数，因为分母不同的分数不能直接比较。一定要先转换。
6. Finding a Fraction of an Amount | 求一个数量的几分之几
To find a fraction of an amount, divide the amount by the denominator, then multiply by the numerator. This works for whole numbers, money, measures and other real-world quantities.
求一个数量的几分之几时，先用分母除这个数量，再乘以分子。这个方法适用于整数、货币、计量单位以及其他现实生活中的量。
a/b of N = N ÷ b × a
3 = 144. For mixed numbers, convert to an improper fraction first, then use the same rule.
例如，240 的 3/5 = 240 ÷ 5 × 3 = 48 × 3 = 144。对于带分数，要先化为假分数，再使用同样的规则。
- Divide by the denominator first. 先除以分母。
- Multiply by the numerator second. 再乘以分子。
- Write the correct units in your answer, such as kg, m or pounds. 答案中要写正确的单位，例如 kg、m 或英镑。
7. Percentage of an Amount and Percentage Change | 一个数量的百分数与百分比变化
To find a percentage of an amount, convert the percentage to a decimal or fraction, then multiply. For percentage increase or decrease, calculate the change and then add it to or subtract it from the original amount.
求一个数量的百分数时，先将百分数转换为小数或分数，再相乘。求百分比增加或减少时，先算出变化量，再把它加到或从原量中减去。
increase: new = original × (1 + p/100)
decrease: new = original × (1 − p/100)
For example, a 15% decrease on £80 is calculated as £80 × 0.85 = £68. The multiplier 0.85 comes from 1 − 0.15.
例如，£80 减少 15% 的计算方法是 £80 × 0.85 = £68。乘数 0.85 来自 1 − 0.15。
Using multipliers is faster than finding the change first, but both methods are acceptable. Always show your multiplier or change clearly in your working.
使用乘数比先求变化量更快捷，但两种方法都可以接受。解题过程中要清楚地写出乘数或变化量。
8. Reverse Percentages | 逆推百分数
If a number is given after a percentage change, find the original value by dividing by the multiplier. Do not simply add or subtract the same percentage, because the percentage applies to the original amount, not the final amount.
如果已知百分比变化后的数，要求原数，应除以对应的乘数。不要直接加上或减去同样的百分数，因为百分数是针对原数，而不是最终数。
original = final amount ÷ multiplier
For example, a price after a 20% increase is £96. The multiplier for a 20% increase is 1.2, so the original price = 96 ÷ 1.2 = £80.
例如，某商品价格增加 20% 后为 £96。增加 20% 的乘数是 1.2，因此原价 = 96 ÷ 1.2 = £80。
Check your answer by applying the percentage change forward: £80 × 1.2 = £96, which confirms the reverse calculation.
通过正向百分比变化来检查答案：£80 × 1.2 = £96，这可以验证逆推计算是否正确。
9. Fractions, Decimals and Percentages in Context | 分数、小数和百分数的实际应用
Many exam questions mix FDP with money, measures
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