📚 IGCSE Maths: Fractions, Decimals and Percentages | IGCSE 数学：分数、小数与百分数
In IGCSE Mathematics, fractions, decimals and percentages form the core of numerical fluency. You need to move confidently between these three forms, apply percentage change, and solve problems involving proportional reasoning.
在 IGCSE 数学中，分数、小数和百分数是数字运算能力的核心。你需要在三种形式之间灵活转换，应用百分数变化，并解决涉及比例推理的问题。
1. Understanding Fractions | 理解分数
A fraction shows a part of a whole. The top number is the numerator, and the bottom number is the denominator. In IGCSE questions, you will meet proper fractions such as 3/5, improper fractions such as 9/4, and mixed numbers such as 2 1/4. You must be able to change between improper fractions and mixed numbers.
分数表示一个整体的一部分。上面的数是分子，下面的数是分母。在 IGCSE 题目中，你会遇到真分数如 3/5、假分数如 9/4，以及带分数如 2 1/4。你必须能够在假分数和带分数之间互相转换。
hole number by the denominator and add the numerator: 2 × 4 + 1 = 9, so the result is 9/4.
例如，带分数 2 1/4 表示两个整体加上一个四分之一。要把它写成假分数，用整数乘以分母再加上分子：2 × 4 + 1 = 9，因此结果是 9/4。
2 1/4 = (2 × 4 + 1) / 4 = 9/4
2. Equivalent Fractions and Simplifying | 等值分数与化简
To find an equivalent fraction, multiply or divide both the numerator and the denominator by the same non-zero number. For example, 18/24 simplifies to 3/4 because both 18 and 24 divide by 6. Always give your answer in the lowest terms unless the question says otherwise.
要找出等值分数，把分子和分母同时乘以或除以同一个非零数。例如，18/24 可以化简为 3/4，因为 18 和 24 都可以除以 6。除非题目另有说明，答案总是要写成最简形式。
st rewrite them with a common denominator. For example, 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6.
当分数相加或相减时，先把它们改写为同分母。例如，1/2 + 1/3 变为 3/6 + 2/6 = 5/6。
18/24 = 3/4, 1/2 + 1/3 = 3/6 + 2/6 = 5/6
3. Converting Between Fractions and Decimals | 分数与小数互化
A fraction can be converted to a decimal by division. For example, 3/8 means 3 ÷ 8, giving 0.375. A terminating decimal ends after a finite number of digits. If the denominator has only prime factors 2 and 5 after simplifying, the decimal will terminate. Otherwise, the decimal recurs, such as 0.333... for 1/3.
分数可以通过除法转换为小数。例如，3/8 表示 3 ÷ 8，得到 0.375。有限小数在小数点后有限位数处结束。如果化简后分母只含有质因数 2 和 5，那么小数就会终止。否则，小数会循环，例如 1/3 是 0.333...。
3/8 = 3 ÷ 8 = 0.375
write the decimal as the numerator over a power of 10. For example, 0.65 = 65/100 = 13/20.
要把有限小数转换回分数，把小数作为分子，分母为 10 的幂。例如，0.65 = 65/100 = 13/20。
0.65 = 65/100 = 13/20
4. Recurring Decimals to Fractions | 循环小数化分数
27.272727... . Subtract the original equation from this, which removes the repeating part.
循环小数需要一种简短的代数方法。要把 0.272727... 转换为分数，设 x = 0.272727...。因为循环节是两位数，所以乘以 100：100x = 27.272727...。用这个等式减去原等式，循环部分就会消去。
x = 0.272727..., 100x = 27.272727..., 99x = 27, x = 27/99 = 3/11
Use the same method for any repeating block. If one digit repeats, multiply by 10; if three digits repeat, multiply by 1000.
对任何循环节都可以使用同样的方法。如果循环节是一位数，乘以 10；如果循环节是三位数，乘以 1000。
5. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数互化
Percent means "per hundred" by first dividing to get a decimal, then multiplying by 100. For example, 7/20 = 0.35 = 35%.
百分数表示每一百份中的多少。把小数转换为百分数要乘以 100，把百分数转换为小数要除以 100。把分数转换为百分数，先用分子除以分母得到小数，再乘以 100。例如，7/20 = 0.35 = 35%。
7/20 = 0.35 = 35%
These common conversions are worth memorising because they appear frequently in IGCSE papers.
以下常见转换值得记住，因为它们在 IGCSE 试卷中经常出现。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/3 | 0.333... | 33.3% approx. |
6. Percentage Increase and Decrease | 百分数增减
To increase an amount by p%, multiply by (1 + p/100). To decrease an amount by p%, multiply by (1 - p/100). For example, increasing 80 by 15% gives 80 × 1.15 = 92, while decreasing 80 by 15% gives 80 × 0.85 = 68.
把一个数增加 p%，乘以 (1 + p/100)。把一个数减少 p%，乘以 (1 - p/100)。例如，80 增加 15% 得到 80 × 1.15 = 92，而 80 减少 15% 得到 80 × 0.85 = 68。
Increase: 80 × 1.15 = 92, Decrease: 80 × 0.85 = 68
Do not simply add or subtract the percentage as a fixed amount unless the original value is known. A percentage multiplier keeps the change proportional to the starting value.
不要简单地把百分数当作固定数量直接加减，除非已知原来的值。百分数乘数可以让变化与原值保持比例关系。
7. Reverse Percentages | 逆百分数
ages find the original value before a percentage change. If an amount of 92 represents 115% equation with the final amount equal to the original multiplied by the percentage multiplier.
逆百分数用于求百分数变化前的原始值。如果数值 92 代表原始值的 115%，那么除以 1.15 就可以得到原始值：92 ÷ 1.15 = 80。把最终量写成原始值乘以百分数乘数的等式即可。
Original = 92 ÷ 1.15 = 80
A common error is to subtract 15% from 92 by calculating 92 × 0.85. That gives 78.2, which is wrong because 15% is taken from 92 rather than from the unknown original value.
一个常见错误是从 92 中减去 15%，即计算 92 × 0.85。这样得到 78.2，这是错误的，因为 15% 是从 92 中扣除的，而不是从未知的原始值中扣除的。
8. Ratio, Proportion and Percentage Change | 比率、比例与百分比变化
arts, which is 5. One part is 120 ÷ 5 = 24, so the shares are 3 × 24 = 72 and 2 × 24 = 48.
百分数与比率和比例密切相关。要把 120 按 3:2 分配，首先求总份数 5。一份是 120 ÷ 5 = 24，因此两部分分别是 3 × 24 = 72 和 2 × 24 = 48。
120 ÷ 5 = 24, 3 × 24 =
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