📚 IGCSE Maths G-1: Fractions, Decimals and Percentages | IGCSE 数学 G-1：分数、小数与百分数
This guide covers the essential number skills required for IGCSE Mathematics Topic G-1. You will learn how to move confidently between fractions, decimals and percentages, perform calculations in each form, and apply these skills to real-world problems such as discounts, interest and comparison of quantities. The content is designed for both classroom teaching and independent revision, with worked examples, key equivalences and exam-style tips.
本指南涵盖 IGCSE 数学主题 G-1 所必需的核心数字运算技能。你将学会在分数、小数和百分数之间熟练转换，用三种形式进行计算，并将这些技能应用到折扣、利息、数量比较等实际问题中。内容既适合课堂教学，也适合自主复习，包含例题、必备等价关系与考试技巧。
1. Understanding Fractions, Decimals and Percentages | 理解分数、小数与百分数
A fraction represents a part of a whole and is written as a/b, where a is the numerator and b is the denominator. The denominator shows how many equal parts the whole is divided into, and the numerator shows how many of those parts are taken. For example, 3/4 means 3 parts out of 4 equal parts.
分数表示整体的一部分，写作 a/b，其中 a 是分子，b 是分母。分母表示整体被分成多少等份，分子表示取了多少份。例如，3/4 表示 4 等份中的 3 份。
you the power of ten in the denominator: 0.7 = 7/10, 0.43 = 43/100, and 0.125 = 125/1000.
小数是分母为 10、100、1000 等的分数的另一种表达方式。小数点后的位数告诉你分母是 10 的多少次方：0.7 = 7/10，0.43 = 43/100，0.125 = 125/1000。
A percentage means "per hundred" and is represented by the symbol %. So 47% simply means 47 out of every 100, which is the fraction 47/100 and the decimal 0.47.
百分数表示 "每一百份"，用符号 % 表示。因此 47% 就是每一百份中的 47 份，即分数 47/100，也就是小数 0.47。
The same quantity can be written in all three forms, and none of them is "better" than the others. Choosing the right form usually makes a calculation much easier: percentages are convenient for increases and decreases, fractions are convenient for exact sharing, and decimals are convenient for the calculator.
同一个数量可以用三种形式表示，它们之间没有优劣之分。选对形式通常能让计算容易很多：百分数适合处理增减，分数适合精确分配，小数适合使用计算器。
1/2 = 0.5 = 50%
2. Converting Between Forms | 三种形式之间的换算
75. Use a short division or a calculator, and take care to keep the decimal point aligned.
将分数转换为小数，用分子除以分母。例如，3/8 = 3 ÷ 8 = 0.375。可以用短除法或计算器，注意保持小数点位置对齐。
To convert a decimal to a percentage, multiply by 100 and add the % sign, so 0.375 × 100 = 37.5%. Multiplying by 100 shifts every digit two places to the left.
将小数转换为百分数，乘以 100 并加上百分号，因此 0.375 × 100 = 37.5%。乘以 100 相当于每一位数字向左移两位。
age to a fraction, write the percentage over 100 and simplify if possible. For example, 65% decimal part, first multiply the numerator and denominator by 10 or 100 to clear it: 12.5% = 12.5/100 = 125/1000 = 1/8.
将百分数转换为分数，把百分数写在 100 之上并化简。例如，65% = 65/100 = 13/20。如果百分数带有小数部分，先把分子和分母同乘 10 或 100 消去小数：12.5% = 12.5/100 = 125/1000 = 1/8。
To convert a percentage to a decimal, divide by 100, so 65% e value of the last digit: 0.375 = 375/1000, then divide top and bottom by 125 to get 3/8.
将百分数转换为小数，除以 100，因此 65% = 0.65。将小数转换为分数，用最后一位数字的数位：0.375 = 375/1000，上下同除 125 得 3/8。
A useful summary of the six possible conversions is shown below. Notice that every pair of forms can be linked directly, so you never have to take a long route through a third form.
下面给出六种换算方向的总结。注意任意两种形式都可以直接互化，因此你不需要"绕道"第三种形式。
| From | To | Method | Example |
| Fraction | Decimal | Numerator ÷ denominator | 3/8 = 0.375 |
| Decimal | Fraction | Use place value, then simplify | 0.375 = 3/8 |
| Decimal | Percentage | × 100 | 0.375 = 37.5% |
| Percentage | Decimal | ÷ 100 | 65% = 0.65 |
| Percentage | Fraction | Write over 100, simplify | 65% = 13/20 |
| Fraction | Percentage | × 100 after dividing | 3/8 = 37.5% |
3. Key Equivalences to Memorise | 必须记住的常见等价关系
Examiners expect you to recognise common equivalences instantly, without a calculator. Learning the table below saves time and reduces errors in longer questions.
考官期望你能不借助计算器、一眼认出常见等价关系。记熟下表可以节省时间，并减少长题中的错误。
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333... | 33.3% (1 d.p.) |
iples of 0.125. Once you know 1/5 = 0.2 you can immediately write 2/5 = 0.4 and 4/5 = 0.8.
如果忘记了表格，有两条规律可以帮你快速重建。五分之一系列是 0.2 的倍数，八分之一系列是 0.125 的倍数。只要记住 1/5 = 0.2，就能立刻写出 2/5 = 0.4 和 4/5 = 0.8。
4. Terminating and Recurring Decimals | 有限小数与循环小数
A terminating decimal stops after a finite number of digits, such as 0.375 or 0.04. A decimal terminates only when the fraction in simplest form has a denominator built from the prime factors 2 and 5. The denominators 4, 8, 20, 25 and 50 all work; 3, 6, 7, 9 and 11 do not.
有限小数在有限位后结束，例如 0.375 或 0.04。只有当最简分数的分母只由质因数 2 和 5 组成时，小数才是有限的。分母 4、8、20、25、50 都可以；3、6、7、9、11 不行。
u may also write it out with dots following: 1/3 = 0.333..., 1/6 = 0.1666..., 1/7 = 0.142857142857... and 5/11 = 0.454545...
循环小数会无限重复一串数字。标准写法是在循环节的首尾数字上方各加一个点，也可以直接写省略号表示：1/3 = 0.333...，1/6 = 0.1666...，1/7 = 0.142857142857...，5/11 = 0.454545...
git over 9 gives the fraction: 0.777... = 7/9, 0.444... = 4/9, 0.121212... = 12/99 = 4/33.
将简单的循环小数化为分数，可以用 9 和 99 作分母。任何一个循环数字写在 9 上就是对应分数：0.777... = 7/9，0.444... = 4/9，0.121212... = 12/99 = 4/33。
0.777... = 7/9 0.121212... = 12/99 = 4/33
Exam questions usually say "give your answer as a fraction" or "correct to 3 significant figures", so always read the rounding instruction before you start. Rounding 0.1666... to 2 decimal places gives 0.17, but the exact value is 1/6.
考题通常会说 "答案写成分数" 或 "保留 3 位有效数字"，所以动笔前一定要看清取整要求。0.1666... 保留两位小数是 0.17，但精确值是 1/6。
5. Comparing and Ordering Quantities | 比较与排序
To compare fractions, decimals and percentages, convert everything to the same form. Decimals are usually the quickest choice because you can line up the place values.
比较分数、小数和百分数时，先全部化成同一种形式。一般小数最快，因为可以直接对齐数位。
Example: arrange 3/5, 62%, 0.59 and 7/12 in ascending order. As decimals: 3/5 = 0.6, 62% = 0.62, 0.59 = 0.59 and 7/12 = 0.5833... (4 d.p.). So the order is 7/12, 0.59, 3/5, 62%.
例：把 3/5、62%、0.59 和 7/12 按从小到大排列。化为小数：3/5 = 0.6，62% = 0.62，0.59 = 0.59，7/12 = 0.5833...（保留 4 位小数）。因此顺序为 7/12、0.59、3/5、62%。
When two values look almost equal, keep more decimal places before deciding. Showing your converted values in the working earns method marks even if the final order is wrong.
当两个值看起来几乎相等时，多保留几位小数再判断。把换算结果写在过程里，即使最终顺序写错也有过程分。
A number line is a useful check. Mark 0, 0.5 and 1 first, then place each value; any value you place on the wrong side of 0.5 is a signal to recalculate.
数轴是很好的检验工具。先标出 0、0.5 和 1，再逐个放置数值；如果你把某个值放在了 0.5 的错误一侧，就说明需要重新计算。
6. Calculating with Fractions | 分数的四则运算
merators. For example, 1/3 + 1/4 = 4/12 + 3/12 = 7/12, and 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
分数相加减时，先通分，再合并分子。例如，1/3 + 1/4 = 4/12 + 3/12 = 7/12；3/4 − 1/6 = 9/12 − 2/12 = 7/12。
× 3/5 = 6/15 = 2/5. Cancelling common factors before multiplying keeps the numbers small.
分数相乘时，分子乘分子、分母乘分母，然后化简。例如，2/3 × 3/5 = 6/15 = 2/5。先约分再相乘可以让数字保持较小。
ixed number must be changed into an improper fraction before any calculation: 2 1/2 = 5/2.
除以一个分数，等于乘以它的倒数（上下颠倒）。例如，3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。带分数在计算前必须先化成假分数：2 1/2 = 5/2。
a/b ÷ c/d = a/b × d/c
Always give the final answer in its simplest form, and convert an improper fraction back to a mixed number if the question asks for one. Leaving 15/8 as the answer when the question wants 1 7/8 loses the final mark.
最终答案一定要写成最简形式；如果题目要求带分数，就把假分数换回带分数。题目要 1 7/8 而你留 15/8，会丢掉最后一分。
7. Finding a Fraction or Percentage of a Quantity | 求一个数量的分数或百分数
nator and multiply by the numerator. For example, 3/4 of 240 = 240 ÷ 4 × 3 = 60 × 3 = 180.
求一个数量的几分之几，先除以分母，再乘以分子。例如，3/4 of 240 = 240 ÷ 4 × 3 = 60 × 3 = 180。
percentage of a quantity, change the percentage to a decimal and multiply. For example, 15% of 80 = 0.15 × 80 = 12, and 12.5% of 96 = 0.125 × 96 = 12.
求一个数量的百分之几，先把百分数化为小数再相乘。例如，15% of 80 = 0.15 × 80 = 12；12.5% of 96 = 0.125 × 96 = 12。
Mental methods are often faster: 10% is found by dividing by 10, 5% by halving the 10% value, and 1% by dividing by 100. So 35% of 60 = 3 × 6 + 0.5 × 6 = 18 + 3 = 21.
口算方法往往更快：10% 是除以 10，5% 是取 10% 的一半，1% 是除以 100。因此 35% of 60 = 3 × 6 + 0.5 × 6 = 18 + 3 = 21。
Questions often link this skill to money, measures or ratio. Read carefully to see whether the quantity given is the whole or only a part of the whole.
这类题目常与钱数、度量或比例结合。要仔细判断题目给出的数量是整体，还是只是整体的一部分。
8. Percentage Increase and Decrease | 百分数的增加与减少
The multiplier method is the most reliable technique. To increase a quantity by r%, multiply by 1 + r/100. To decrease a quantity by r%, multiply by 1 − r/100.
乘数法是最可靠的方法。把数量增加 r%，就乘以 1 + r/100；把数量减少 r%，就乘以 1 − r/100。
New value = Original value × (1 ± r/100)
For example, increasing £450 by 12% gives 450 × 1.12 = £504. Decreasing 80 kg by 15% gives 80 × 0.85 = 68 kg.
例如，把 £450 增加 12% 得 450 × 1.12 = £504。把 80 kg 减少 15% 得 80 × 0.85 = 68 kg。
For repeated changes, multiply the multipliers together. A price that rises by 10% and then falls by 10% does not return to its starting value: 1.10 × 0.90 = 0.99, so it ends 1% lower.
若连续变化，把乘数连乘。先涨 10% 再跌 10% 并不会回到原价：1.10 × 0.90 = 0.99，最终比原来低 1%。
To find a percentage change, use the formula below and remember the denominator is always the original value, never the new one.
求百分数变化时使用下面的公式，注意分母永远是原来的值，而不是新值。
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