📚 IGCSE Maths: Number, Ratio and Percentages (G-1 Teacher's Resource 600) | IGCSE 数学：数、比与百分比（G-1 教师用书 600）
This revision guide covers the essential skills in IGCSE Mathematics for working confidently with numbers, ratios, percentages and proportional reasoning. These topics appear frequently on both Core and Extended papers, often in real-life contexts such as shopping, finance, measurement and travel. Understanding these foundations is vital for success across the entire IGCSE Mathematics syllabus.
本复习指南涵盖 IGCSE 数学中处理数、比、百分比和比例推理的核心技能。这些主题在 Core 和 Extended 试卷中频繁出现，常用于购物、金融、测量和旅行等实际情境。掌握这些基础对于整个 IGCSE 数学课程的成功至关重要。
1. Integers, Negatives and Order of Operations | 整数、负数与运算顺序
In IGCSE Mathematics, the order in which operations are carried out decides whether an answer is right or wrong. BIDMAS (also written BODMAS) is the rule: Brackets first, then Indices (powers and roots), then Division and Multiplication working from left to right, and finally Addition and Subtraction, also from left to right. Division and multiplication share the same priority, and so do addition and subtraction.
在 IGCSE 数学中，运算的先后顺序直接决定答案的对错。BIDMAS（也写作 BODMAS）规则为：先算括号，再算指数（乘方与开方），然后从左到右算除法与乘法，最后同样从左到右算加法与减法。除法与乘法优先级相同，加法与减法优先级也相同。
BIDMAS: Brackets → Indices → Division/Multiplication → Addition/Subtraction | 括号 → 指数 → 除法/乘法 → 加法/减法
he addition to happen first. Worked example 3: 2 + 3 × (8 − 5)² ÷ 3 = 2 + 3 × 9 ÷ 3 = 2 + 9 = 11.
例 1：计算 3 + 4 × 5。先算乘法，4 × 5 = 20，再加 3 得到 23，而不是 35。例 2：(3 + 4) × 5 = 7 × 5 = 35，括号强制先做加法。例 3：2 + 3 × (8 − 5)² ÷ 3 = 2 + 3 × 9 ÷ 3 = 2 + 9 = 11。
applies only to the 3 and the minus sign is applied afterwards. When a calculation mixes division and multiplication, such as 24 ÷ 4 × 3, work strictly from left to right: 6 × 3 = 18.
指数表示重复相乘：2³ = 2 × 2 × 2 = 8。处理负数时要注意：(−3) × (−4) = 12，(−3)² = 9，但 −3² = −9，因为指数只作用于 3，负号在最后才加上。当算式中同时出现除法和乘法时，例如 24 ÷ 4 × 3，必须严格从左到右计算：6 × 3 = 18。
- Brackets are the strongest priority, so always simplify inside them first. | 括号优先级最高，永远先算括号内的内容。
- Fraction bars and root signs act like brackets: in (2 + 4)/(3 + 1), simplify the top and the bottom separately. | 分数线与根号相当于括号：计算 (2 + 4)/(3 + 1) 时，分子分母要分别化简。
- A calculator will follow BIDMAS automatically, but you must still enter brackets exactly as they appear. | 计算器会自动遵循 BIDMAS，但你输入括号时必须与原式完全一致。
2. Fractions, Decimals and Percentages: Conversion Skills | 分数、小数与百分比：互化技能
Fractions, decimals and percentages are three ways of representing the same proportion, so you must be able to convert between them quickly and accurately. To turn a fraction into a decimal, divide the numerator by the denominator. To turn a decimal into a percentage, multiply by 100. To turn a percentage into a fraction, write it over 100 and simplify.
分数、小数和百分比是表示同一比例的三种方式，因此你必须能够快速而准确地在它们之间转换。分数化小数：分子除以分母。小数化百分比：乘以 100。百分比化分数：写成分母为 100 的分数再化简。
| Fraction 分数 | Decimal 小数 | Percentage 百分比 |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 1/3 | 0.333... | 33.3% (3 s.f.) |
| 2/3 | 0.666... | 66.7% (3 s.f.) |
| 7/20 | 0.35 | 35% |
| 1/25 | 0.04 | 4% |
For example, 3/4 = 3 ÷ 4 = 0.75 = 75%, and 0.6 = 6/10 = 3/5 = 60% the final stage of a question, and you should state the degree of accuracy you have used.
例如，3/4 = 3 ÷ 4 = 0.75 = 75%；0.6 = 6/10 = 3/5 = 60%。像 1/3 这样的循环小数只应在题目的最后一步才取近似值，并且要写明你所保留的精确度。
- Compare quantities by converting everything to the same form first, usually decimals. | 比较大小前，先把所有数量化为同一种形式，通常化成小数。
- When a fraction has a denominator of 100, it is already a percentage: 37/100 = 37%. | 当分母为 100 时，分数本身就是百分比：37/100 = 37%。
- To convert a percentage to a decimal, divide by 100, which moves the decimal point two places left. | 百分比化小数即除以 100，也就是小数点向左移两位。
3. Factors, Multiples, Primes, HCF and LCM | 因数、倍数、质数、最大公因数与最小公倍数
Factors of a number divide into it exactly, while multiples are obtained by multiplying the number by 1, 2, 3, ... A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note that 1 is not prime and 2 is the only even prime.
一个数的因数是可以整除它的数，而倍数则是这个数分别乘以 1、2、3……所得的结果。质数恰好有两个因数：1 和它本身，例如 2、3、5、7、11、13、17、19、23、29。注意 1 不是质数，而 2 是唯一的偶质数。
mon factor (HCF) is built from the prime factors the two numbers share, and the lowest common multiple (LCM) is built from all the prime factors, taking the highest power of each.
每个大于 1 的整数都可以写成质数之积，这叫质因数分解。例如 360 = 2³ × 3² × 5。最大公因数（HCF）由两个数共有的质因数组成，最小公倍数（LCM）则由全部质因数组成，每个质数取最高次幂。
3 = 12 and the LCM = 2³ × 3² = 72. A useful check is HCF × LCM = 24 × 36 = 864 = 12 × 72.
例：求 24 与 36 的 HCF 和 LCM。因为 24 = 2³ × 3，36 = 2² × 3²，所以 HCF = 2² × 3 = 12，LCM = 2³ × 3² = 72。一个有用的检验是 HCF × LCM = 24 × 36 = 864 = 12 × 72。
- Divisible by 3 if the digit sum is a multiple of 3 (e.g. 4 + 5 + 6 = 15, so 456 is). | 各位数字之和是 3 的倍数就能被 3 整除（如 4 + 5 + 6 = 15，故 456 可被 3 整除）。
- Divisible by 4 if the last two digits form a multiple of 4. | 末两位数字组成的数是 4 的倍数就能被 4 整除。
- Divisible by 9 if the digit sum is a multiple of 9. | 各位数字之和是 9 的倍数就能被 9 整除。
- LCM questions often appear as 'events repeating together', such as two buses that leave every 12 and 18 minutes. | LCM 常出现在"事件同时重复"的题中，例如每 12 分钟和每 18 分钟发车一次的公交车。
4. Calculating with Fractions | 分数的四则运算
12. To multiply, multiply numerators together and denominators together, cancelling first where possible.
分数相加减时，先通分，再合并分子。例如 2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12。分数相乘时，分子乘分子、分母乘分母，能约分的要先约分。
ctions before multiplying or dividing, so 2 1/2 × 1 1/3 = 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3.
除以一个分数等于乘以它的倒数：2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6。带分数在乘除之前必须先化成假分数，所以 2 1/2 × 1 1/3 = 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3。
- Always give fractions in their simplest form, and convert improper fractions to mixed numbers if the question asks for it. | 分数一定要化成最简形式；若题目要求，还要把假分数化成带分数。
- In word problems, the word 'of' usually means multiply: 3/5 of 250 = 250 ÷ 5 × 3 = 150. | 应用题中"of"通常表示相乘：250 的 3/5 = 250 ÷ 5 × 3 = 150。
- Division by zero is undefined, so a denominator must never be zero. | 除以零无意义，因此分母绝不能为零。
5. Ratio: Simplifying, Sharing and Comparing | 比：化简、按比分配与比较
he ratio: 24 : 36 = 2 : 3. Units must match before a ratio can be formed, so 40 minutes to 2 hours becomes 40 : 120 = 1 : 3.
比用来比较同类量，必须写成最简整数比。两部分同时除以它们的 HCF 即可化简：24 : 36 = 2 : 3。写比之前单位必须统一，所以 40 分钟与 2 小时之比应写成 40 : 120 = 1 : 3。
the amount by that total to find the value of one part, then multiply. For example, share $420 in the ratio 3 : 4: there are 7 parts, so one part is 420 ÷ 7 = $60, giving $180 and $240.
按给定比例分配数量时，先把各部分相加得到总份数，用总量除以总份数得到一份的值，再分别相乘。例如把 420 美元按 3 : 4 分配：共 7 份，一份为 420 ÷ 7 = 60 美元，因此两份分别是 180 美元和 240 美元。
tion gives the difference between two shares, subtract the ratio parts: if the ratio is 5 : 3 and the difference is $12, then 2 parts = $12, so 1 part = $6 and the amounts are $30 and $18.
比也可以有三个或更多部分。把 500 克按 2 : 3 : 5 分配，共 10 份，一份为 50 克，于是三份分别是 100 克、150 克和 250 克。若题目给出两份之差，就先做减法：若比为 5 : 3，差为 12 美元，则 2 份 = 12 美元，1 份 = 6 美元，两个量分别是 30 美元和 18 美元。
- A ratio such as 3 : 5 can also be written as the fraction 3/5, which is useful in probability and proportion work. | 比 3 : 5 也可以写成分数 3/5，这在概率和比例问题中很有用。
- Always check that the shares add back to the original total. | 分配完后一定要检验各部分之和是否等于原总量。
- nd model scales are ratios: a scale of 1 : 50 000 means 1 cm represents 50 000 cm = 500 m. | 地图比例尺与模型比例都是比：1 : 50 000 表示图上 1 cm 代表实际 50 000 cm = 500 m。
6. Direct and Inverse Proportion | 正比例与反比例
tities are in inverse proportion when one increases as the other decreases, so their product stays constant: y = k/x, and x × y is fixed.
当两个量按相同速率同时增大时，它们成正比例，此时比值保持不变：y = kx，y/x 为常数。当一个量增大而另一个量减小时，它们成反比例，此时乘积保持不变：y = k/x，x × y 为常数。
Direct proportion example: 5 kg of rice costs $12.50, so 8 kg costs 12.50 ÷ 5 × 8 = $ the value of one unit first and then scales up, which is the safest approach in the exam.
正比例例题：5 千克大米售价 12.50 美元，则 8 千克售价为 12.50 ÷ 5 × 8 = 20 美元。这种"归一法"先求一个单位的值再放大，是考试中最稳妥的方法。
6 × 12 ÷ 9 = 8 days. The total work is 72 worker-days, and this quantity does not change.
反比例例题：6 名工人 12 天砌好一堵墙，则 9 名工人需要 6 × 12 ÷ 9 = 8 天。总工作量为 72 个工日，这个量不变。
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