📚 IGCSE Maths Topic 1: Fractions, Decimals and Percentages | IGCSE 数学主题1：分数、小数和百分比
In IGCSE Mathematics, fractions, decimals and percentages are fundamental number skills that appear in almost every paper. They are used in arithmetic, estimation, financial problems, data interpretation and everyday calculations. This article explains the key methods, common conversions and exam techniques you need to master.
在 IGCSE 数学中，分数、小数和百分比是几乎每份试卷都会出现的基础数字技能。它们广泛应用于算术、估算、金融问题、数据解释和日常计算中。本文讲解你需要掌握的关键方法、常见转换以及考试技巧。
1. Understanding Fractions, Decimals and Percentages | 认识分数、小数和百分比
A fraction represents a part of a whole and is written as a numerator over a denominator. A decimal uses place value to represent parts of a whole, such as 0.25. A percentage is a number expressed out of 100, so 25% means 25 parts per hundred. These three forms are equivalent ways of describing the same quantity.
分数表示整体的一部分，写为分子在分母之上的形式。小数使用位值来表示整体的一部分，例如 0.25。百分比是以 100 为基准表示的数，因此 25% 表示每一百份中的 25 份。这三种形式是描述同一数量的等价方式。
For example, one quarter can be written as the fraction 1/4, the decimal 0.25 or the percentage 25%. Understanding how to move between these forms is essential for IGCSE number work.
例如，四分之一可以写成分数 1/4、小数 0.25 或百分比 25%。理解如何在这几种形式之间转换对于 IGCSE 数字部分至关重要。
2. Equivalent Fractions and Simplifying | 等价分数与化简
Equivalent fractions are created by multiplying or dividing the numerator and denominator by the same non-zero number. This does not change the value of the fraction. To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF).
等价分数通过将分子和分母同时乘以或除以同一个非零数得到，这不会改变分数的值。化简分数时，需要用分子和分母的最大公因数（HCF）同时除它们。
a/b = (a × k)/(b × k) = (a ÷ m)/(b ÷ m)
For example, 12/18 can be simplified by dividing both 12 and 18 by 6, giving 2/3. Always give your final answer in its simplest form unless the question asks otherwise, as marks may be lost for unsimplified fractions.
例如，12/18 可以通过将 12 和 18 同时除以 6 化简为 2/3。除非题目另有要求，否则最终答案应始终化简为最简形式，因为未化简的分数可能会被扣分。
3. Converting between Fractions, Decimals and Percentages | 分数、小数和百分比的转换
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a fraction, write it over 100 and simplify. To convert a percentage to a decimal, divide by 100.
将分数转换为小数，用分子除以分母。将小数转换为百分比，乘以 100。将百分比转换为分数，先写成以 100 为分母的分数再化简。将百分比转换为小数，除以 100。
| Fraction to decimal | Divide numerator by denominator | 3/8 = 0.375 |
| Decimal to percentage | Multiply by 100 | 0.375 × 100 = 37.5% |
| Percentage to fraction | Write over 100 and simplify | 37.5% = 37.5/100 = 3/8 |
| Percentage to decimal | Divide by 100 | 37.5% = 0.375 |
You must be comfortable with these conversions without a calculator, especially common values such as 1/2 = 0.5 = 50%, 1/3 = 0.333… = 33.3%, 2/3 = 0.666… = 66.7%, 1/4 = 0.25 = 25% and 3/4 = 0.75 = 75%.
你必须能够不依赖计算器熟练完成这些转换，尤其是常见值：1/2 = 0.5 = 50%、1/3 = 0.333… = 33.3%、2/3 = 0.666… = 66.7%、1/4 = 0.25 = 25% 以及 3/4 = 0.75 = 75%。
4. Recurring Decimals as Fractions | 循环小数化为分数
A recurring decimal repeats one or more digits forever, such as 0.444… or 0.181818… . To convert a recurring decimal to a fraction, set the decimal equal to x, multiply by a power of 10 to move the repeating block, then subtract to eliminate the repeating part.
循环小数是一个或多个数字无限重复的小数，例如 0.444… 或 0.181818…。要将循环小数化为分数，设该小数等于 x，乘以 10 的幂以移动重复节，然后相减消去循环部分。
For a single repeated digit, use 10 subtraction. For example, let x = 0.777… . Then 10x = 7.777… . Subtracting gives 9x = 7, so x = 7/9.
对于单个重复数字，使用乘以 10 再相减。例如，设 x = 0.777…。则 10x = 7.777…。两式相减得到 9x = 7，所以 x = 7/9。
x = 0.777… → 10x = 7.777… → 9x = 7 → x = 7/9
If two digits repeat, multiply by 100. If three digits repeat, multiply by 1000. Always check your answer by dividing the numerator by the denominator on your calculator.
如果两个数字重复，则乘以 100；如果三个数字重复，则乘以 1000。最后务必用计算器将分子除以分母来验证答案。
5. Ordering and Comparing Quantities | 数量的大小排序与比较
To compare fractions, decimals and percentages, first convert all quantities into the same form. Decimals are usually the easiest to compare because you can align decimal points and compare digit by digit. Percentages can also be useful when the denominators are different.
比较分数、小数和百分比时，首先将所有数量转换为同一种形式。小数通常最容易比较，因为可以对齐小数点后逐位比较。当分母不同时，使用百分比也很方便。
For example, to order 2/3, 0.62 and 63%, convert 2/3 ≈ 0.6667 and 63% = 0.63. The correct ascending order is 0.62, 63%, 2/3. Always remember to write your final answer using the original forms given in the question.
例如，要排序 2/3、0.62 和 63%，先将 2/3 转换为约 0.6667，63% 转换为 0.63。正确的升序是 0.62、63%、2/3。务必记住最终答案要使用题目中给出的原始形式来书写。
6. Percentage Calculations: Increase, Decrease and Reverse | 百分比计算：增加、减少与逆运算
To increase a quantity by r%, multiply by the multiplier (1 + r/100). To decrease a quantity by r%, multiply by (1 – r/100). These multipliers are very efficient in IGCSE exam questions involving repeated percentage changes.
将某个数量增加 r%，乘以乘数 (1 + r/100)。将某个数量减少 r%，乘以 (1 – r/100)。在 IGCSE 考试中涉及重复百分比变化的问题时，使用乘数非常高效。
Increase: new = original × (1 + r/100)
Decrease: new = original × (1 – r/100)
For a reverse percentage problem, you are given the final value after a percentage change and must find the original value. Divide by the same multiplier instead of multiplying. Always check whether the question is asking for the original amount or the final amount.
对于逆向百分比问题，题目会给出百分比变化后的最终值，要求你求出原始值。此时需要除以相同的乘数而不是乘以。务必确认题目要求的是原始量还是最终量。
7. Compound Interest and Depreciation | 复利与折旧
Compound interest is calculated on the original amount plus the interest already earned. The formula for compound interest is A = P(1 + r/100)ⁿ, where A is the final amount, P is the principal, r is the rate per period, and n is the number of periods.
复利是在原始本金加上已经产生的利息的基础上计算的。复利公式为 A = P(1 + r/100)ⁿ，其中 A 是最终金额，P 是本金，r 是每期利率，n 是期数。
A = P(1 + r/100)ⁿ
Depreciation uses the same idea but subtracts the rate: A = P(1 – r/100)ⁿ. Both formulas appear regularly in IGCSE financial maths questions, so learn how to substitute values and use brackets correctly on your calculator.
折旧使用相同的思路，但要减去折旧率：A = P(1 – r/100)ⁿ。这两个公式经常出现在 IGCSE 金融数学题目中，因此要学会正确代入数值并在计算器中正确使用括号。
8. Ratio and Proportion in Real-Life Problems | 现实生活中的比与比例问题
A ratio compares two or more quantities in the same unit. Ratios can be simplified by dividing all parts by their highest common factor. A proportion is an equation that states two ratios are equal, often used to scale quantities up or down.
比用来比较两个或多个同类量。比可以通过将所有部分除以它们的最大公因数来化简。比例是一个等式，表示两个比相等，常用于按比例放大或缩小数量。
The unitary method is a reliable approach: first find the value of one unit, then multiply to find the required amount. For example, if 5 pens cost $3, then one pen costs $3 ÷ 5 = $0.60, so 8 pens cost $0.60 × 8 = $4.80.
单位法是可靠的方法：先求出一个单位的值，再乘以所需的数量。例如，如果 5 支笔花费 $3，则一支笔花费 $3 ÷ 5 = $0.60，因此 8 支笔花费 $0.60 × 8 = $4.80。
9. Estimation, Significant Figures and Bounds | 估算、有效数字与界限
Estimation is a quick way to check whether an answer is reasonable. Round each number to one significant figure before calculating. For example, estimate 48.7 × 3.14 by rounding to 50 × 3 = 150.
估算是快速检查答案是否合理的方法。计算前先将每个数四舍五入到一位有效数字。例如，估算 48.7 × 3.14 时，先四舍五入为 50 × 3 = 150。
Significant figures are counted from the first non-zero digit. For a measurement, the lower and upper bounds are the smallest and largest possible values that would round to that measurement. This concept is important for exactness in IGCSE problems involving rounding and measurement.
有效数字从第一个非零数字开始计数。对于一个测量值，下界和上界是四舍五入到该测量值的最小和最大可能值。这一概念在 IGCSE 涉及四舍五入和测量的问题中对于精确性非常重要。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Many marks are lost in IGCSE number questions because students forget to simplify fractions, use the wrong percentage multiplier, or round too early in multi-step calculations. Always carry full precision through intermediate steps and only round at the final answer.
在 IGCSE 数字题中，许多学生因为忘记化简分数、使用错误的百分比乘数或在多步计算中过早四舍五入而失分。中间步骤应始终保持完整精度，仅在最终答案处四舍五入。
- Simplify fractions fully unless the question asks for a mixed number or a specific form.
- Use multipliers correctly: increase is plus, decrease is minus, reverse is divide.
- Check sign changes in inequalities and reverse percentage problems.
- Write units clearly and use the same units throughout a ratio or rate problem.
- Practise without a calculator for basic conversions, as non-calculator papers test these skills directly.
- 务必充分化简分数，除非题目要求带分数或特定形式。
- 正确使用乘数：增加是加号，减少是减号，逆向是除以乘数。
- 检查符号变化，尤其是逆向百分比问题中的乘除关系。
- 清楚写出单位，并在比或比率问题中
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