📚 IGCSE Mathematics: Fractions, Decimals and Percentages | IGCSE 数学：分数、小数与百分数
In IGCSE Mathematics, fractions, decimals and percentages are essential number skills that appear in almost every topic, from ratio and proportion to probability and compound interest. This article explains the key methods, common conversions and exam strategies you need to master.
在 IGCSE 数学中，分数、小数和百分数是几乎每个主题都会用到的基本数感技能，从比和比例到概率和复利。本文讲解你需要掌握的关键方法、常见转换和考试策略。
1. Understanding Fractions, Decimals and Percentages | 理解分数、小数与百分数
A fraction is written as a/b, where a is the numerator and b is the non-zero denominator. It represents a equal parts out of b equal parts. A decimal uses the base-10 place value system, so 0.47 means 4 tenths plus 7 hundredths. A percentage is a fraction with denominator 100; for example, 47% means 47/100.
分数写作 a/b，其中 a 是分子，b 是不为零的分母。它表示 b 等份中的 a 份。小数使用以 10 为基数的位值系统，因此 0.47 表示 4 个十分之一加 7 个百分之一。百分数是分母为 100 的分数；例如 47% 表示 47/100。
In exam questions, you often need to interpret these forms in real contexts: a test score of 17/20, a discount of 0.25, or an interest rate of 3% per year. Fluency in converting between them is the foundation for more complex calculations.
在考试题中，你经常需要在真实情境中理解这些形式：测试得分 17/20、折扣 0.25 或年利率 3%。在三者之间熟练转换是进行更复杂计算的基础。
2. Converting Between the Three Forms | 三种形式之间的转换
The central conversion rules are simple but very important:
核心转换规则简单但非常重要：
- Fraction to decimal: divide the numerator by the denominator, so a/b = a ÷ b. / 分数转小数：分子除以分母，因此 a/b = a ÷ b。
- Decimal to percentage: multiply the decimal by 100, so d × 100%. / 小数转百分数：小数乘以 100，即 d × 100%。
- Percentage to decimal: divide the percentage by 100, so p% = p/100. / 百分数转小数：百分数除以 100，即 p% = p/100。
- Percentage to fraction: write the percentage over 100 and simplify, so p% = p/100. / 百分数转分数：将百分数写成分母为 100 的分数并化简，即 p% = p/100。
Fraction → Decimal: a/b = a ÷ b | Decimal → Percent: d × 100% | Percent → Fraction: p/100
The table below shows common conversions that are regularly tested in IGCSE papers:
下表列出了 IGCSE 试卷中经常考查的常见转换：
| 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% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333… | 33⅓% |
| 2/3 | 0.666… | 66⅔% |
3. Ordering and Comparing Quantities | 数量的大小比较与排序
To compare fractions, decimals and percentages, convert all values to the same form. Decimals are usually the fastest choice because they use a single place value scale.
要比较分数、小数和百分数，需要先把所有数值转换成同一种形式。小数通常是最快的方法，因为它使用统一的位值标度。
For example, arrange 2/5, 0.45 and 38% in ascending order. Convert: 2/5 = 0.4, 0.45 = 0.45, 38% = 0.38. The order is 0.38, 0.4, 0.45, so 38%, 2/5, 0.45.
例如，将 2/5、0.45 和 38% 按升序排列。转换：2/5 = 0.4，0.45 = 0.45，38% = 0.38。顺序为 0.38、0.4、0.45，因此是 38%、2/5、0.45。
If a question asks you to compare fractions with different denominators, find a common denominator or use equivalent decimals. Always show your conversion steps for method marks.
如果题目要求比较分母不同的分数，应找出公分母或使用等值小数。始终写出转换步骤以获得方法分。
4. Calculating Fractions of Amounts | 求一个数的几分之几
To find a fraction of an amount, divide the amount by the denominator and multiply the result by the numerator. Equivalently, multiply the amount by the fraction itself.
要找一个数的几分之几，先将该数除以分母，再把结果乘以分子。等价地，也可以用该数直接乘以这个分数。
Fraction of amount = (a/b) × N = N ÷ b × a
For mixed numbers, first convert to an improper fraction. For example, 1 1/4 of 20 means 5/4 × 20 = 25.
对于带分数，应首先转换为假分数。例如，20 的 1 1/4 等于 5/4 × 20 = 25。
Worked example: Find 3/8 of 64. Divide 64 by 8 to get 8, then multiply by 3 to get 24. So 3/8 of 64 is 24.
例题：求 64 的 3/8。先将 64 除以 8 得到 8，再乘以 3 得到 24。所以 64 的 3/8 是 24。
5. Percentage Increase and Decrease | 百分数增加与减少
A percentage increase or decrease is calculated by changing the original amount by a given percentage. Instead of finding the change separately, you can use a multiplier.
百分数增加或减少通过按给定百分数改变原量来计算。除了先单独求出变化量，你也可以使用乘数。
New value = Original × (1 + r/100) for increase | New value = Original × (1 − r/100) for decrease
For example, increase 80 by 15%: multiplier is 1 + 15/100 = 1.15, so new value = 80 × 1.15 = 92.
例如，将 80 增加 15%：乘数为 1 + 15/100 = 1.15，因此新值 = 80 × 1.15 = 92。
Decrease 200 by 30%: multiplier is 1 − 30/100 = 0.7, so new value = 200 × 0.7 = 140.
将 200 减少 30%：乘数为 1 − 30/100 = 0.7，因此新值 = 200 × 0.7 = 140。
Always check whether the question says ‘percentage increase’ or ‘percentage of’ because they require different steps. ‘15% of 80’ is 80 × 0.15 = 12, whereas a 15% increase of 80 is 92.
始终检查题目说的是“增加百分之几”还是“百分之几是多少”，因为它们需要不同的步骤。“80 的 15%”是 80 × 0.15 = 12，而 80 增加 15% 是 92。
6. Reverse Percentages | 反向百分数
Reverse percentage problems give you the final amount after a percentage change and ask you to find the original amount. Do not simply subtract or add the percentage to the final value.
反向百分数问题给出百分数变化后的最终量，要求找出原量。不要简单地从最终值中减去或加上百分数。
Original = Final ÷ multiplier, where multiplier = 1 + r/100 or 1 − r/100
For example, a coat is reduced by 20% and now costs £48. The multiplier for a decrease is 0.8, so original price = 48 ÷ 0.8 = £60.
例如，一件外套降价 20% 后售价为 48 英镑。减少的乘数是 0.8，所以原价 = 48 ÷ 0.8 = 60 英镑。
Similarly, if a salary of £23,000 is after a 15% increase, the original salary = 23000 ÷ 1.15 = £20,000. Showing the multiplier in your working makes the method clear to the examiner.
类似地，如果 23000 英镑是加薪 15% 后的工资，原工资 = 23000 ÷ 1.15 = 20000 英镑。在解题过程中写出乘数能让阅卷人清楚地看到你的方法。
7. Repeated Percentage Change and Compound Interest | 重复百分数变化与复利
When a percentage change is applied several times, multiply by the same multiplier repeatedly. This is equivalent to using a power or exponent.
当百分数变化多次发生时，需要反复乘以同一个乘数。这相当于使用幂或指数。
A = P(1 + r/100)ⁿ for compound interest | A = P(1 − r/100)ⁿ for repeated decrease
In this formula, P is the principal or starting amount, r is the rate per period, and n is the number of periods.
在这个公式中，P 是本金或起始金额，r 是每期利率，n 是期数。
Worked example: £500 is invested at 4% per year for 3 years. The amount is A = 500(1 + 4/100)³ = 500(1.04)³ = 500 × 1.124864 = £562.43.
例题：500 英镑以年利率 4% 投资 3 年。到期金额为 A = 500(1 + 4/100)³ = 500(1.04)³ = 500 × 1.124864 = 562.43 英镑。
For depreciation, such as a car losing 15% of value each year, use A = P(1 − 15/100)ⁿ = P(0.85)ⁿ.
对于折旧，例如汽车每年贬值 15%，使用 A = P(1 − 15/100)ⁿ = P(0.85)ⁿ。
8. Converting Recurring Decimals to Fractions | 循环小数化分数
Some decimals repeat forever, such as 0.333… or 0.272727…. These recurring decimals can be written as fractions using a simple algebraic method.
有些小数无限循环，例如 0.333… 或 0.272727…。这些循环小数可以用简单的代数方法写成分数。
Worked example: Write 0.272727… as a fraction. Let x = 0.272727…. Multiply by
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