📚 Mastering Fractions, Decimals and Percentages for IGCSE Mathematics | IGCSE 数学：分数、小数与百分数综合突破
Fractions, decimals and percentages are three different ways of expressing the same idea: a part of a whole. In IGCSE Mathematics, you must be able to convert confidently between these forms, compare them, and apply them to real problems such as discounts, interest, and data interpretation. This revision guide covers the essential methods, common errors, and exam-style strategies needed for both Core and Extended papers.
分数、小数和百分数是表达同一个概念的三种不同形式：整体的一部分。在 IGCSE 数学中，你必须能够在这些形式之间熟练转换、比较大小，并把它们应用到折扣、利息和数据分析等实际问题中。本复习指南涵盖核心与扩展试卷所需的基本方法、常见错误以及考试型解题策略。
1. The Three Equivalent Forms | 三种等价形式
A fraction shows a part out of a total number of equal parts, such as 3/4 meaning 3 parts out of 4. A decimal uses place value after the decimal point, so 3/4 can be written as 0.75. A percentage means ‘per hundred’, so 3/4 is the same as 75%.
分数表示整体中若干等份的一部分，例如 3/4 表示 4 份中的 3 份。小数利用小数点后的位值，因此 3/4 可以写成 0.75。百分数表示“每一百份中的多少”，所以 3/4 等同于 75%。
The key idea is that they are interchangeable: 1/2 = 0.5 = 50%. When a question mixes all three forms, your first step should usually be to convert everything to the same form, because comparing using mixed notation is a common source of error.
关键思想是它们可以互相转换：1/2 = 0.5 = 50%。当题目混合出现这三种形式时，第一步通常是把所有量都转换成同一种形式，因为混合使用不同记法进行比较是常见的失分原因。
2. Converting Between Fractions and Decimals | 分数与小数互化
To change a fraction to a decimal, divide the numerator by the denominator. For example, 7/8 means 7 ÷ 8 = 0.875. You can use short division, long division, or a calculator, but you should recognise common decimal equivalents such as 1/4 = 0.25 and 3/8 = 0.375.
要把分数化成小数，用分子除以分母。例如，7/8 表示 7 ÷ 8 = 0.875。你可以使用短除法、长除法或计算器，但也应熟记常见的小数等值，如 1/4 = 0.25 和 3/8 = 0.375。
Some fractions produce terminating decimals because the denominator has only prime factors 2 and 5 after simplification. For example, 3/20 = 0.15 terminates. Other fractions such as 1/3 = 0.333… and 5/11 = 0.454545… give recurring decimals, which are shown with a dot or bar over the repeating digit or block.
有些分数会化成有限小数，因为化简后的分母只含有质因数 2 和 5。例如 3/20 = 0.15 是有限小数。另一些分数如 1/3 = 0.333… 和 5/11 = 0.454545… 会得到循环小数，通常在循环数字或循环块上加点或横线表示。
7/8 = 0.875, 1/3 = 0.333…
3. Converting Between Decimals and Percentages | 小数与百分数互化
Because percent means ‘out of 100’, the conversion is simple: multiply a decimal by 100 to get a percentage, and divide a percentage by 100 to get a decimal. For example, 0.62 × 100 = 62%, and 8% ÷ 100 = 0.08.
由于百分数表示“每一百份”，因此转换非常简单：小数乘以 100 得到百分数，百分数除以 100 得到小数。例如 0.62 × 100 = 62%，而 8% ÷ 100 = 0.08。
Remember that 1.0 corresponds to 100%, so decimals greater than 1 give percentages greater than 100%. This is useful for increases: an increase of 15% can be written as a multiplier of 1.15, because the original 100% plus 15% equals 115%, or 1.15 as a decimal.
请记住 1.0 对应 100%，因此大于 1 的小数会得到大于 100% 的百分数。这在处理增长时很有用：增长 15% 可以写成乘数 1.15，因为原来的 100% 加上 15% 等于 115%，即小数 1.15。
0.05 = 5%, 1.2 = 120%, 0.375 = 37.5%
4. Converting Between Fractions and Percentages | 分数与百分数互化
To change a fraction to a percentage, first convert the fraction to a decimal by dividing, then multiply by 100. For example, 3/5 = 3 ÷ 5 = 0.6, so 3/5 = 0.6 × 100 = 60%.
要把分数化成百分数，先用除法把分数化成小数，再乘以 100。例如 3/5 = 3 ÷ 5 = 0.6，因此 3/5 = 0.6 × 100 = 60%。
An alternative method is to find an equivalent fraction with denominator 100, if possible. For instance, 7/20 = 35/100 = 35%. This only works when the denominator divides 100 exactly, but it is a quick mental method for common fractions.
另一种方法是尽可能找到分母为 100 的等值分数。例如 7/20 = 35/100 = 35%。只有当分母能整除 100 时这种方法才适用，但对于常见分数而言它是一种快速的巧算方法。
To change a percentage to a fraction, write the percentage over 100 and simplify. For example, 45% = 45/100 = 9/20. Always check whether the fraction can be reduced using common factors.
要把百分数化成分数，将百分数写在 100 上方并化简。例如 45% = 45/100 = 9/20。一定要检查分数是否能用公因数约分。
5. Ordering Mixed Quantities | 混合数值的大小比较
IGCSE questions often ask you to arrange a list such as 3/8, 0.4, 35%, and 2/5 in ascending order. The safest method is to convert all values to decimals to two or three decimal places, then compare.
IGCSE 考试中经常要求你将一组数如 3/8、0.4、35% 和 2/5 按升序排列。最稳妥的方法是把所有数值都化成保留两位或三位的小数，然后进行比较。
For this example, 3/8 = 0.375, 0.4 = 0.400, 35% = 0.350, and 2/5 = 0.400. Writing all decimals to three decimal places makes the order clear: 0.350, 0.375, 0.400, 0.400. Therefore the ascending order is 35%, 3/8, 0.4 and 2/5, with the last two equal.
以这个例子来说，3/8 = 0.375，0.4 = 0.400，35% = 0.350，2/5 = 0.400。把所有小数都写成三位小数后，大小顺序就很清楚：0.350、0.375、0.400、0.400。因此升序排列为 35%、3/8、0.4 和 2/5，其中后两个相等。
Do not compare by looking only at numerators or denominators, because 1/3 is smaller than 1/2 even though 3 is larger than 2. Convert first, then decide.
不要只凭分子或分母的大小进行比较，因为 1/3 小于 1/2，尽管 3 比 2 大。要先转换，再做判断。
6. Recurring Decimals as Fractions | 循环小数转分数
For Extended IGCSE Mathematics, you need to convert a recurring decimal such as 0.272727… into a fraction. The algebraic method works by multiplying by a power of 10 so that the repeating block lines up for subtraction.
在 IGCSE 数学扩展试卷中，你需要把循环小数如 0.272727… 化为分数。代数方法的原理是将小数乘以 10 的幂，使循环块对齐后相减。
Let x = 0.272727… . Because two digits repeat, multiply by 100: 100x = 27.272727… . Now subtract x from 100x: 100x – x = 27.272727… – 0.272727… = 27. So 99x = 27, which gives x = 27/99 = 3/11.
设 x = 0.272727…。因为循环节有两位，所以乘以 100：100x = 27.272727…。现在用 100x 减去 x：100x – x = 27.272727… – 0.272727… = 27。因此 99x = 27，得到 x = 27/99 = 3/11。
0.272727… = 3/11
If only one digit repeats, multiply by 10 instead. If three digits repeat, multiply by 1000. Always simplify the resulting fraction to its lowest terms.
如果只有一位数字循环，则乘以 10。如果有三位数字循环，则乘以 1000。最后一定要把得到的分数化成最简形式。
7. Fractions of Quantities | 求一个数的几分之几
To find a fraction of an amount, divide by the denominator and multiply by the numerator. For example, to find 3/5 of 120, first calculate 120 ÷ 5 = 24, then multiply by 3 to get 72.
要求一个数的几分之几，先除以分母，再乘以分子。例如，要求 120 的 3/5，先计算 120 ÷ 5 = 24，再乘以 3 得到 72。
This process is especially useful in word problems. If a class of 30 students has 2/3 wearing glasses, then 30 ÷ 3 = 10 and 10 × 2 = 20 students wear glasses. You can also think of it as multiplying the fraction by the amount: 2/3 × 30 = 60/3 = 20.
这个过程在文字题中特别有用。如果一个班级有 30 名学生，其中 2/3 戴眼镜，那么 30 ÷ 3 = 10，10 × 2 = 20 名学生戴眼镜。你也可以把它理解为分数乘以数量：2/3 × 30 = 60/3 = 20。
When the numerator is larger than the denominator, the answer is more than the original amount. For instance, 5/4 of 80 is 80 ÷ 4 = 20, then 20 × 5 = 100.
当分子大于分母时，结果会大于原数量。例如，80 的 5/4 是 80 ÷ 4 = 20，再 20 × 5 = 100。
8. Percentage Increase, Decrease and Reverse Percentages | 增减百分数与逆向百分数
To increase an amount by a percentage, multiply by 1 plus the percentage as a decimal. To decrease an amount, multiply by 1 minus the percentage as a decimal. For example, increasing 200 by 12% uses the multiplier 1.12: 200 × 1.12 = 224.
要把一个数增加某个百分数，用 1 加上百分数对应的小数作为乘数。要把一个数减少某个百分数，用 1 减去百分数对应的小数作为乘数。例如，将 200 增加 12%，使用乘数 1.12：200 × 1.12 = 224。
A decrease of 8% uses 0.92 as the multiplier, because 100% – 8% = 92%. So decreasing 150 by 8% gives 150 × 0.92 = 138. The multiplier method is faster than finding the change and adding or subtracting, and it is essential for compound problems.
减少 8% 则使用乘数 0.92，因为 100% – 8% = 92%。因此将 150 减少 8% 得到 150 × 0.92 = 138。乘数法比先求变化量再加或减更快，而且在复合问题中不可或缺。
Reverse percentage questions give the final amount after a percentage change and ask you to find the original. To undo an increase of 20%, divide by 1.20, not by 0.80. For example, if a price after a 15% increase is $46, then the original price is 46 ÷ 1.15 = $40.
逆向百分数问题给出百分数变化后的最终量，要求你求原来的量。要还原一次 20% 的增长，应除以 1.20，而不是除以 0.80。例如，如果一次 15% 涨价后的价格是 46 美元，那么原价是 46 ÷ 1.15 = 40 美元。
Original price = 46 ÷ 1.15 = 40
9. Compound Interest and Depreciation | 复利与折旧
Compound interest means that interest is added to the account each period, and the next interest is calculated on the new total. The amount after n periods is given by multiplying the principal by the growth factor n times.
复利意味着每个周期都把利息加入账户，下一次利息按新的总额计算。经过 n 个周期后的金额等于本金乘以增长因子的 n 次方。
A = P × (1 + r/100)n
For example, if $500 is invested at 4% compound interest for 3 years, then A = 500 × 1.043 = 500 × 1.124864 = $562.43, correct to two decimal places.
例如，如果 500 美元以 4% 的年复利投资 3 年，那么 A = 500 × 1.043 = 500 × 1.124864 = 562.43 美元，精确到两位小数。
Depreciation works the same way but with a decay factor. If a car loses 15% of its value each year, the multiplier is 0.85. After n years, the value is original value × 0.85n. Do not use simple subtraction repeatedly unless the question says simple interest or flat rate.
折旧的计算方式相同，但使用的是衰减因子。如果一辆汽车每年贬值 15%，则乘数为 0.85。经过 n 年后，价值为原价值 × 0.85n。除非题目说明是单利或固定利率，否则不要反复使用简单减法。
10. Exam Strategy and Common Pitfalls | 考试策略与常见失分点
A common mistake is to confuse ‘increase by 10%’ with ‘increase to 10%’. An increase by 10% means adding 10% of the original, while an increase to 10% means the final value is 10% of the original.
一个常见错误是把“增加 10%”与“增加到 10%”混淆。增加 10% 表示加上原值的 10%，而增加到 10% 表示最终值是原值的 10%。
Another frequent error occurs when using reverse percentages. If a value has decreased by 25% to 75, the original is 75 ÷ 0.75 = 100, not 75 × 1.25 = 93.75. Always identify whether you are moving forward or backward before choosing the multiplier.
另一个常见错误出现在逆向百分数中。如果一个数减少 25% 后为 75，那么原数是 75 ÷ 0.75 = 100，而不是 75 × 1.25 = 93.75。在选择乘数之前，一定要先判断自己是在正向变化还是反向还原。
Also remember that percentage changes applied in sequence are multiplicative, not additive. A 10% increase followed by a 10% decrease does not return the original value, because the second change is taken from a different base.
还要记住，依次发生的百分数变化是乘法关系，而不是加法关系。先增加 10% 再减少 10% 不会回到原值，因为第二次变化是以不同的基数为基准计算的。
11. Worked Exam-Style Question | 考试型例题精讲
A shop increases the price of a jacket by 15%, then offers a discount of 20% in a sale. The original price was $80. Find the final sale price and the overall percentage change.
一家商店将一件夹克的价格提高 15%，然后在促销活动中提供 20% 的折扣。原价为 80 美元。求最终售价和总体百分比变化。
First apply the 15% increase: 80 × 1.15 = 92. Then apply the 20% discount: 92 × 0.80 = 73.60. The final price is $73.60.
首先应用 15% 的涨价：80 × 1.15 = 92。然后应用 20% 的折扣：92 × 0.80 = 73.60。最终价格为 73.60 美元。
The overall change is 73.60 – 80 = -6.40. As a percentage of the original, this is -6.40 ÷ 80 × 100 = -8%. So the overall effect is an 8% decrease, not a 5% decrease as some students expect from 15% – 20%.
总体变化为 73.60 – 80 = -6.40。按原价计算百分比，变化为 -6.40 ÷ 80 × 100 = -8%。因此总体效果是减少 8%，而不是一些学生根据 15% – 20% 所预期的减少 5%。
Overall change = 1.15 × 0.80 = 0.92, which is a decrease of 8%
This demonstrates that you should multiply the multipliers together to find the single equivalent multiplier, then convert back to a percentage if required.
这个例子说明，应把各个乘数相乘得到单一的等效乘数，然后在需要时再转换回百分数。
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