📚 Fractions, Decimals and Percentages: Conversions and Calculations | 分数、小数与百分数：转换与计算
Fractions, decimals and percentages are three ways of representing the same amount. In the Cambridge KS3 Mathematics curriculum, you need to convert between these forms fluently and apply them to real-life problems such as discounts, interest, test scores and probability. This article explains the key methods and common mistakes to avoid.
分数、小数和百分数是表示同一个数量的三种方式。在剑桥 KS3 数学课程中，你需要熟练地在这三种形式之间进行转换，并把它们应用到折扣、利息、考试成绩和概率等实际问题中。本文讲解关键方法和需要避免的常见错误。
1. The Three Representations | 三种表示方式
A fraction shows a part out of a whole using a numerator and a denominator. For example, 3/4 means 3 parts out of 4 equal parts. A decimal uses place value based on tenths, hundredths and thousandths. For example, 0.75 means 75 hundredths. A percentage is a number out of 100, so 75% means 75 per 100.
分数用分子和分母表示整体中的一部分。例如，3/4 表示 4 等份中的 3 份。小数使用十分位、百分位和千分位的位值。例如，0.75 表示 75 个百分之一。百分数是以 100 为基准的数，所以 75% 表示每 100 份中有 75 份。
2. Equivalent Fractions and Simplification | 等值分数与化简
ded by 4. To find an equivalent fraction, multiply or divide both parts by the same number. Always check whether the fraction can be reduced to its lowest terms.
化简分数时，将分子和分母同时除以相同的公因数。例如，12/16 = 3/4，因为 12 和 16 都能被 4 整除。要找到等值分数，可以将分子和分母同时乘或除以同一个数。始终检查分数是否能化为最简形式。
12/16 = (12 ÷ 4)/(16 ÷ 4) = 3/4
3. Converting Fractions to Decimals | 分数转小数
alue, such as 7/100 = 0.07. For fractions like 1/3, the decimal repeats, so 1/3 = 0.333...
将分数转换为小数，用分子除以分母。例如，3/8 表示 3 ÷ 8，结果为 0.375。如果分母是 10、100 或 1000，可以直接按小数位值写出分子，例如 7/100 = 0.07。对于像 1/3 这样的分数，小数会循环，所以 1/3 = 0.333...
3/8 = 3 ÷ 8 = 0.375
7/100 = 0.07
4. Converting Decimals to Percentages and Back | 小数与百分数互化
To convert a decimal to a percentage, multiply by 100. For example, 0.6 × 100 = 60%. To convert a percentage to a decimal, divide by 100. For example, 8% mal point two places to the right for percentages and two places to the left for decimals.
将小数转换为百分数，乘以 100。例如，0.6 × 100 = 60%。将百分数转换为小数，除以 100。例如，8% = 8 ÷ 100 = 0.08。一个快速方法是：转百分数时小数点向右移动两位，转小数时小数点向左移动两位。
0.6 × 100 = 60%
8% = 8 ÷ 100 = 0.08
5. Converting Fractions to Percentages and Back | 分数与百分数互化
ecimal or find an equivalent fraction with denominator 100. For example, 3/5 = 60/100 = 60% age to a fraction, write the percentage over 100 and simplify if possible. For example, 40% = 40/100 = 2/5.
将分数转换为百分数，可以先转为小数，或者找到分母为 100 的等值分数。例如，3/5 = 60/100 = 60%。将百分数转换为分数时，先写成百分数除以 100，然后化简。例如，40% = 40/100 = 2/5。
3/5 = 60/100 = 60%
40% = 40/100 = 2/5
| Fraction | Decimal | Percentage | 中文说明 |
|---|---|---|---|
| 1/10 | 0.1 | 10% | 十分之一 |
| 1/5 | 0.2 | 20% | 五分之一 |
| 1/4 | 0.25 | 25% | 四分之一 |
| 3/4 | 0.75 | 75% | 四分之三 |
6. Finding a Percentage of an Amount | 求一个数的百分之几
f an amount, write the percentage as a fraction or decimal, then multiply. For example, 15% of 80 = 0.15 × 80 = 12. For common percentages, use mental strategies: 10% is one tenth, 5% is half of 10%, and 1% is one hundredth.
求一个数的百分之几时，先把百分数写成分数或小数，再进行乘法。例如，80 的 15% = 0.15 × 80 = 12。对于常见百分数，可以使用心算策略：10% 是十分之一，5% 是 10% 的一半，1% 是百分之一。
15% of 80 = 0.15 × 80 = 12
10% of 80 = 8, so 5% = 4
7. Percentage Increase and Decrease | 百分比增加与减少
ntage, add the percentage of the amount to the original. For example, increasing 200 by 12% 0.12 × 200 = 224. To decrease, subtract the percentage of the amount. Decreasing 200 by 12% gives 200 - 0.12 × 200 = 176. A multiplier can be used: 1.12 for a 12% increase and 0.88 for a 12% decrease.
将一个数增加某个百分比，先求这个百分比的量，再加上原数。例如，200 增加 12% 得到 200 + 0.12 × 200 = 224。减少时则减去这个百分比的量。200 减少 12% 得到 200 - 0.12 × 200 = 176。也可以使用乘数：增加 12% 用 1.12，减少 12% 用 0.88。
200 × 1.12 = 224
200 × 0.88 = 176
8. Fractions of Quantities | 求一个量的几分之几
identify the total and the fraction, then carry out the calculation in the correct order.
求一个量的几分之几，先除以分母再乘以分子。例如，35 的 2/5 = (35 ÷ 5) × 2 = 14。在应用题中，先识别总量和分数，然后按正确顺序进行计算。
2/5 of 35 = (35 ÷ 5) × 2 = 14
9. Comparing and Ordering Mixed Forms | 比较和排序混合形式
re often easiest because they are all out of 100. For example, to compare 0.45, 2/5 and 38%, write 2/5 = 0.40 = 40%, 38% = 0.38, so 38% < 2/5 < 0.45.
比较分数、小数和百分数时，把所有数值转化为同一种形式。百分数通常最容易，因为都以 100 为基准。例如，比较 0.45、2/5 和 38%，将 2/5 = 0.40 = 40%，38% = 0.38，因此 38% < 2/5 < 0.45。
38% < 2/5 < 0.45
10. Word Problems and Real-Life Contexts | 应用题与生活情境
KS3 exam questions often ask about sales discounts, bank interest, test marks and surveys. Read the problem carefully and identify the original amount, the fraction or percentage, and whether you need an increase or decrease. Write down your working step by step to avoid missing units.
KS3 考试题常涉及折扣、银行利息、考试分数和调查数据。仔细读题，确定原始量、分数或百分数，以及需要增加还是减少。逐步写出计算过程，避免遗漏单位。
11. Common Misconceptions | 常见误区
common error is adding or subtracting percentages directly, for example thinking that a 20% increase followed by a 20% al amount. In fact, 100 × 1.20 × 0.80 = 96, not 100. Another error is confusing 0.5 with 5% or writing 1/2 as 1.2.
常见错误是直接对百分数进行加减，例如认为先增加 20% 再减少 20% 会回到原数。实际上，100 × 1.20 × 0.80 = 96，而不是 100。另一个错误是把 0.5 与 5% 混淆，或把 1/2 写成 1.2。
100 × 1.20 × 0.80 = 96
12. Practice Questions and Key Takeaways | 练习题与关键要点
Test yourself: convert 7/20 to a percentage, find 35% of 240, and work out the final price of a £60 item after a 15% discount. Check your answers: 7/20 = 35%, 35% of 240 = 84, and £60 × 0.85 = £51. Remember to always show your working.
自测：把 7/20 转化为百分数，求 240 的 35%，并计算 £60 的商品打 15% 折后的价格。参考答案：7/20 = 35%，240 的 35% = 84，£60 × 0.85 = £51。记住始终写出计算过程。
