📚 Mastering Fractions, Decimals and Percentages for IGCSE Maths | 掌握IGCSE数学的分数、小数与百分数
Fractions, decimals and percentages are three different ways of representing the same idea: a part of a whole. In the IGCSE Mathematics syllabus, you are expected to convert fluently between these forms, compare them, calculate with them, and apply them to real-life problems such as discounts, interest and reverse percentage questions. This article breaks down the essential methods, common pitfalls and exam techniques you need to master.
分数、小数和百分数是表示同一概念（整体的一部分）的三种不同方式。在 IGCSE 数学大纲中，你需要熟练地在这些形式之间进行转换、比较大小、进行计算，并将它们应用于折扣、利息和逆百分数等实际问题。本文将分解你需要掌握的基本方法、常见错误和考试技巧。
1. Why Fractions, Decimals and Percentages Matter | 为什么分数、小数和百分数很重要
In IGCSE Maths, questions involving fractions, decimals and percentages appear in both the Core and Extended papers. They often appear inside larger problems such as probability, ratio, financial mathematics and data handling. Being confident with the relationships between these forms saves time and reduces arithmetic errors.
在 IGCSE 数学中，涉及分数、小数和百分数的题目在核心卷和扩展卷中都会出现。它们常常出现在概率、比率、金融数学和数据处理等更大的问题中。掌握这些形式之间的关系可以节省时间并减少计算错误。
For example, the value 0.2, the fraction 1/5 and the percentage 20% all describe exactly the same quantity. Recognising that they are interchangeable is the key to solving many exam questions quickly.
例如，数值 0.2、分数 1/5 和百分数 20% 都描述完全相同的量。认识到它们可以互相替换是快速解决许多考试题目的关键。
The ability to move between these forms also helps you make sense of data. If a shop offers 1/4 off, you can instantly understand that the discount is 25%, or that you pay 75% of the original price. These conversions are examined regularly.
在这些形式之间转换的能力也帮助你理解数据。如果一家商店提供 1/4 的折扣，你可以立刻明白折扣是 25%，也就是说你支付原价的 75%。这些转换经常出现在考试中。
2. Equivalent Forms: The Core Idea | 等价形式：核心思想
All three representations express the same proportional relationship. A percentage is simply a fraction with denominator 100, and a decimal is another way of writing a fraction whose denominator is a power of 10. The central skill is to convert between them accurately and to simplify fractions where possible.
这三种表示都表达相同的比例关系。百分数就是分母为 100 的分数，小数则是分母为 10 的幂的分数的另一种写法。核心技能是在它们之间准确转换，并尽可能化简分数。
The diagram below shows the same shaded amount expressed in three equivalent ways:
下图展示了同一个阴影量用三种等价方式表示：
1/4 = 0.25 = 25%
This equivalence means that when you divide the top number by the bottom number in a fraction, you produce its decimal form. To change a decimal into a percentage, you multiply by 100. To change a percentage into a fraction, you place it over 100 and simplify.
这种等价关系意味着当你用分数的分子除以分母时，就得到它的小数形式。要把小数转换为百分数，需要乘以 100。要把百分数转换为分数，把它写在 100 的上面然后化简。
Always simplify fractions to their lowest terms unless the question states otherwise. For example, 50/100 should be written as 1/2.
除非题目另有说明，否则始终把分数化简到最简形式。例如，50/100 应写成 1/2。
3. Converting Fractions to Decimals | 将分数转换为小数
To convert a fraction to a decimal, divide the numerator by the denominator. This works for both proper fractions, where the numerator is smaller than the denominator, and improper fractions, where the numerator is larger.
要将分数转换为小数，用分子除以分母。这适用于真分数（分子小于分母）和假分数（分子大于分母）。
either short division or a calculator, but in the non-calculator paper you should be able to divide simple cases by hand.
例如，要将 3/8 转换为小数，计算 3 ÷ 8 = 0.375。你可以使用短除法或计算器，但在非计算器试卷中，你应该能手动计算简单的情况。
= 0.333... The recurring part is often shown with a dot or a bar over the repeating digit.
有些分数产生有限小数，例如 1/4 = 0.25，而另一些分数产生循环小数，例如 1/3 = 0.333... 循环部分通常用重复数字上的点或横线表示。
In the IGCSE exam, recurring decimals may be written using a dot notation, for example 0.3̇ means 0.333... and 0.1̇2̇ means 0.121212... You are expected to understand this notation.
在 IGCSE 考试中，循环小数可能用点记号表示，例如 0.3̇ 表示 0.333...，0.1̇2̇ 表示 0.121212... 你需要理解这种记号。
Knowing common fraction– = 0.2, 1/8 = 0.125 and 1/10 = 0.1. Memorising these will help you check your work quickly.
记住常见的分数与小数转换可以节省时间。例如，1/2 = 0.5，1/4 = 0.25，3/4 = 0.75，1/5 = 0.2，1/8 = 0.125，1/10 = 0.1。记住这些有助于你快速检查答案。
4. Converting Decimals to Fractions | 将小数转换为分数
To convert a terminating decimal to a fraction, first write the decimal as a fraction with denominator 10, 100, 1000 or another power of 10, depending on the number of decimal places. Then simplify the fraction fully.
要将有限小数转换为分数，首先根据小数的位数，把小数写成分母为 10、100、1000 或其他 10 的幂的分数。然后完全化简这个分数。
ifies to 9/20 because both 45 and 100 can be divided by 5. Similarly, 0.08 = 8/100 = 2/25.
例如，0.45 有两位小数，因此写为 45/100。化简为 9/20，因为 45 和 100 都能被 5 整除。类似地，0.08 = 8/100 = 2/25。
recurring decimal, the method is more involved. To convert a recurring decimal such as 0.3̇ minate the recurring part: 10x − x = 3.333... − 0.333..., giving 9x = 3, so x = 3/9 = 1/3.
对于循环小数，方法更复杂一些。要把诸如 0.3̇ 的循环小数转换为分数，设 x = 0.333... 然后将等式两边乘以 10：10x = 3.333... 从这个式子减去原式以消去循环部分：10x − x = 3.333... − 0.333...，得到 9x = 3，所以 x = 3/9 = 1/3。
For a recurring decimal with two repeating digits, such as 0.4̇5̇ 100x: 99x = 45, so x = 45/99 = 5/11. This method is commonly tested in the Extended paper.
对于有两个循环数字的循环小数，例如 0.4̇5̇，设 x = 0.454545... 并乘以 100：100x = 45.454545... 然后用 100x 减去 x：99x = 45，所以 x = 45/99 = 5/11。这种方法在扩展卷中经常考查。
5. Converting Percentages to Fractions and Decimals | 将百分数转换为分数和小数
divide by 100. This is equivalent to moving the decimal point two places to the left. For example, 35% ecimal to a percentage, multiply by 100 or move the decimal point two places to the right.
要将百分数转换为小数，除以 100。这相当于把小数点向左移动两位。例如，35% = 35 ÷ 100 = 0.35。要将小数转换为百分数，乘以 100 或把小数点向右移动两位。
o a fraction, write the percentage as the numerator over 100 and simplify. For example, 65% = 65/100 = 13/20. If the percentage contains a decimal, such as 12.5% multiply the numerator and denominator by 10 to obtain 125/1000, which simplifies to 1/8.
要将百分数转换为分数，把百分数写在 100 的上面作为分子，然后化简。例如，65% = 65/100 = 13/20。如果百分数中含有小数，例如 12.5%，先写成 12.5/100，然后将分子和分母同时乘以 10 得到 125/1000，化简为 1/8。
Common percentage conversions that are useful to memorise include 50% = 1/2, 25% = 1/4, 75% = 3/4, 20% = 1/5, 40% = 2/5, 60% = 3/5, 80% = 4/5, 10% = 1/10, 5% = 1/20 and 1% = 1/100.
值得记住的常见百分数转换包括：50% = 1/2，25% = 1/4，75% = 3/4，20% = 1/5，40% = 2/5，60% = 3/5，80% = 4/5，10% = 1/10，5% = 1/20，1% = 1/100。
l by dividing, then multiply by 100. For example, 3/8 = 0.375, so 3/8 = 0.375 × 100 = 37.5%. This is often the quickest route in the exam.
要将分数转换为百分数，先将分数通过除法转换为小数，再乘以 100。例如，3/8 = 0.375，所以 3/8 = 0.375 × 100 = 37.5%。这通常是考试中最快的路径。
6. Ordering and Comparing | 排序与比较
Exam questions frequently ask you to arrange a mixed list of fractions, decimals and percentages in ascending or descending order. The safest method is to convert all the values to the same form, usually decimals or percentages, and then compare them.
考试题目经常要求你将分数、小数和百分数的混合列表按升序或降序排列。最安全的方法是把所有数值转换成同一种形式，通常是小数或百分数，然后进行比较。
For example, to order 0.3, 2/5 and 35%, convert 2/5 to 0.4 and 35% to 0.35. The values are 0.3, 0.4 and 0.35, so the ascending order is 0.3, 35%, 2/5.
例如，要排列 0.3、2/5 和 35%，将 2/5 转换为 0.4，将 35% 转换为 0.35。这些值是 0.3、0.4 和 0.35，因此升序排列为 0.3、35%、2/5。
e 3/4 and 5/6, use denominator 12: 3/4 = 9/12 and 5/6 = 10/12, so 5/6 is greater than 3/4.
在没有计算器的情况下比较分数，你也可以用公分母重写它们。例如，比较 3/4 和 5/6，使用分母 12：3/4 = 9/12，5/6 = 10/12，所以 5/6 大于 3/4。
Always read the question carefully: 'ascending' means smallest to largest, while 'descending' means largest to smallest. Losing marks because you ordered in the wrong direction is easily avoidable.
务必仔细读题：“升序”表示从小到大，“降序”表示从大到小。因为排列方向错误而丢分是很容易避免的。
7. Finding a Fraction or Percentage of an Amount | 求一个量的几分之几或百分之几
For example, to find 3/5 of 80, first calculate 80 ÷ 5 = 16, then multiply by 3 to get 48.
要找到一个量的一部分，先用分母除以这个量，再乘以分子。例如，求 80 的 3/5，先计算 80 ÷ 5 = 16，再乘以 3 得到 48。
ods are valid, but the divide-then-multiply approach is often easier without a calculator.
或者，你可以直接将这个量乘以分数：80 × 3/5 = 240/5 = 48。两种方法都有效，但在没有计算器的情况下，先除后乘的方法通常更容易。
st convert the percentage to a decimal or fraction, then multiply. For example, to find 15% of 240, calculate 0.15 × 240 = 36, or find 10% = 24 and 5% = 12, then add to get 36.
要找到一个量的百分之几，先把百分数转换为小数或分数，然后相乘。例如，求 240 的 15%，计算 0.15 × 240 = 36，或者先求 10% = 24 和 5% = 12，然后相加得到 36。
For common percentages, using the fraction form can be quicker. For example, 25% of a value is the same as dividing by 4, and 10% is the same as dividing by 10. This is especially useful when answering non-calculator questions.
对于常见的百分数，使用分数形式可以更快。例如，一个值的 25% 等于除以 4，10% 等于除以 10。这在回答非计算器问题时特别有用。
These skills are the building blocks for more complex problems involving profit, loss and discounts. Always show your working so that method marks can be awarded even if the final arithmetic is incorrect.
这些技能是涉及利润、亏损和折扣等更复杂问题的基础。始终展示你的解题过程，这样即使最终算术有误，也能获得方法分。
8. Percentage Increase and Decrease | 百分数增加与减少
e percentage of the amount, then add it to the original. For example, to increase 80 by 15%, calculate 15% of 80 = 12, then add: 80 + 12 = 92.
要将一个量增加某个百分数，先求出这个百分数对应的量，然后加到原量上。例如，将 80 增加 15%，计算 80 的 15% = 12，然后相加：80 + 12 = 92。
nt by a percentage, find the percentage and subtract it. For example, to decrease 80 by 15%, calculate 15% of 80 = 12, then subtract: 80 − 12 = 68.
要将一个量减少某个百分数，求出百分数对应的量然后减去。例如，将 80 减少 15%，计算 80 的 15% = 12，然后相减：80 − 12 = 68。
A more efficient method is to use a single multiplier. For an increase of 15%, the original amount becomes 100% + 15% = 115%, so multiply by 1.15. For a decrease of 15%, multiply by 100% − 15% = 85%, so multiply by 0.85.
更高效的方法是使用单一乘数。对于增加 15%，原量变为 100% + 15% = 115%，所以乘以 1.15。对于减少 15%，乘以 100% − 15% = 85%，即乘以 0.85。
For example, increasing 200 by 12% can be done as 200 × 1.12 = 224. Decreasing 200 by 12% can be done as 200 × 0.88 = 176. This method is essential for repeated percentage changes.
例如，将 200 增加 12% 可以计算为 200 × 1.12 = 224。将 200 减少 12% 可以计算为 200 × 0.88 = 176。这种方法对于重复百分数变化至关重要。
Always identify whether the question asks for the final amount or just the change. If a shirt costing $40 is reduced by 25%, the discount is $10, but the new price is $30. Many candidates stop after finding the discount and forget to subtract it.
始终要判断题目要求的是最终量还是仅仅是变化量。如果一件衬衫原价 40 美元，降价 25%，折扣是 10 美元，但新价格是 30 美元。许多考生在求出折扣后就停下来，忘了减去它。
9. Reverse Percentages | 逆百分数
Reverse percentage questions give you the value after a percentage change and ask you to find the original value. Instead of multiplying by the multiplier, you divide by it. The key step is to identify what percentage the given value represents.
逆百分数题目给出百分数变化后的值，要求你找出原来的值。不是乘以乘数，而是除以乘数。关键步骤是判断给定值代表原值的百分之几。
For example, if a jacket is reduced by 20% and now costs $64, the $64 represents 80% ce. To find the original price, divide 64 by 0.8: 64 ÷ 0.8 = 80. So the original price was $80.
例如，如果一件夹克降价 20% 后现在售价 64 美元，那么 64 美元代表原价的 80%。要找出原价，用 64 除以 0.8：64 ÷ 0.8 = 80。因此原价是 80 美元。
Similarly, if a salary is increased by 5% and the new salary is $2,100, then $2,100 represents 105% alary. Divide by 1.05 to find the original: 2100 ÷ 1.05 = 2000, so the original salary was $2,000.
类似地，如果薪水增加了 5%，新的薪水是 2,100 美元，那么 2,100 美元代表原薪水的 105%。除以 1.05 得到原薪水：2100 ÷ 1.05 = 2000，因此原薪水是 2,000 美元。
A common mistake is to find the percentage of the new amount and subtract it, which is incorrect because the percentage change was applied to the original amount, not the new one. Always use division by the multiplier.
一个常见错误是求出新量的百分数然后减去，这是不正确的，因为百分数变化是针对原量而不是新量应用的。始终要用乘数进行除法。
Table 1 summarises the correct multiplier for different percentage changes:
表 1 总结了不同百分数变化对应的正确乘数：
| Change | 变化 | Multiplier | 乘数 | Reverse operation | 逆运算 |
|---|---|---|
| Increase 10% | 增加 10% | × 1.10 | ÷ 1.10 |
| Decrease 10% | 减少 10% | × 0.90 | ÷ 0.90 |
| Increase 25% | 增加 25% | × 1.25 | ÷ 1.25 |
| Decrease 25% | 减少 25% | × 0.75 | ÷ 0.75 |
10. Compound Interest and Repeated Percentage Change | 复利与重复百分数变化
When an amount is increased or decreased by the same percentage repeatedly, you can multiply by the same decimal multiplier several times. For compound interest, the formula is based on repeated multiplication, not simple addition.
当一个量重复地以同样的百分数增加或减少时，你可以多次乘以相同的小数乘数。对于复利，公式基于重复相乘，而不是简单的加法。
The compound interest formula is:
复利公式为：
FV = PV × (1 + r/100)ⁿ
For depreciation, the formula uses subtraction inside the bracket: FV = PV × (1 − r/100)ⁿ.
其中 FV 是最终值，PV 是现值或本金，r 是每个周期的百分数利率，n 是周期数。对于折旧，公式在括号内使用减法：FV = PV × (1 − r/100)ⁿ。
For example, if $500 is invested at 4% compound interest for 3 years, the final value is 500 × (1.04)³ = 500 × 1.124864 = $562.43. This is different from simple interest, where you would only add 4% of the original amount each year.
例如，如果 500 美元以 4% 的复利投资 3 年，最终值为 500 × (1.04)³ = 500 × 1.124864 = 562.43 美元。这与单利不同，单利每年只增加原金额的 4%。
percentage decreases, such as depreciation, the multiplier is less than 1. If a car worth $12,000 depreciates by 15% each year, after 2 years its value is 12,000 × (0.85)² = 12,000 × 0.7225 = $8,670.
对于重复的百分数减少，例如折旧，乘数小于 1。如果一辆价值 12,000 美元的汽车每年折旧 15%，两年后它的价值为 12,000 × (0.85)² = 12,000 × 0.7225 = 8,670 美元。
In IGCSE questions, you may be asked to find the original value given the final value after repeated changes. In that case, divide by the multiplier raised to the power n. For example, if an investment grew to $800 after 3 years at 5% compound interest, the original investment was 800 ÷ (1.05)³.
在 IGCSE 题目中，你可能会被要求根据重复变化后的最终值求出原值。在这种情况下，除以乘数的 n 次方。例如，如果一笔投资以 5% 的复利增长 3 年后达到 800 美元，那么原投资额为 800 ÷ (1.05)³。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is adding or subtracting percentages as if they were ordinary numbers. For example, a 10% increase followed by a 10% decrease does not return the amount to its original value. If you start with 100, a 10% increase gives 110, and a 10% decrease gives 99, not 100.
一个常见错误是把百分数当作普通数字直接相加或相减。例如，先增加 10% 再减少 10% 并不会使数值回到原始值。如果从 100 开始，增加 10% 得到 110，再减少 10% 得到 99，而不是 100。
converting to percentages. To convert 0.7 to a percentage, multiply by 100: 0.7 × 100 = 70%, not 7%. Likewise,
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