📚 IGCSE WJEC Maths: Calculation Questions Intensive Training | IGCSE WJEC 数学：计算题专项训练
Calculation questions sit at the heart of the IGCSE WJEC Mathematics specification, appearing on both the non-calculator and calculator papers. Building fluency in arithmetic, algebraic manipulation, geometry and statistics is the surest way to raise your grade. This revision article walks you through core calculation techniques, common examiner traps and quick-check methods, all tailored to the WJEC style.
计算题是 IGCSE WJEC 数学考试的核心，无论在非计算器卷还是计算器卷中都会大量出现。熟练掌握算术、代数变形、几何与统计的计算方法是提高分数的最可靠途径。本文带你系统梳理核心计算技巧、常见考官陷阱和快速验证方法，所有内容均针对 WJEC 考试风格设计。
1. Integers and Fractions | 整数与分数
When adding a negative number, visualise moving left on the number line: 8 + (–3) = 5. Subtracting a negative is equivalent to addition: 8 – (–3) = 8 + 3 = 11.
加上一个负数时，可以想象在数轴上向左移动：8 + (–3) = 5。减去一个负数相当于做加法：8 – (–3) = 8 + 3 = 11。
The product or quotient of two negatives is positive: (–9) ÷ (–3) = 3. If signs differ, the answer is negative.
两个负数相乘或相除结果为正：(–9) ÷ (–3) = 3。如果符号不同，结果为负。
For adding and subtracting fractions, find the lowest common denominator first. For example, 2/5 + 3/10 = 4/10 + 3/10 = 7/10.
分数的加减法，先找到最小公分母。例如 2/5 + 3/10 = 4/10 + 3/10 = 7/10。
Multiply fractions by multiplying the numerators and the denominators: 3/8 × 4/9 = 12/72, which simplifies to 1/6. Cancel early to save time.
分数相乘时分子与分母分别相乘：3/8 × 4/9 = 12/72，约分后得 1/6。提前约分可以节省时间。
Always apply BIDMAS: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). 10 – 2 × 3² = 10 – 2 × 9 = 10 – 18 = –8.
任何时候都要遵守 BIDMAS 运算顺序：括号、指数、除法和乘法（从左到右）、加法和减法（从左到右）。10 – 2 × 3² = 10 – 2 × 9 = 10 – 18 = –8。
2. Decimals and Percentages | 小数与百分比
To multiply decimals, first ignore the decimal points and multiply as integers, then count the total decimal places. 0.3 × 0.04 has 1 + 2 = 3 decimal places, giving 0.012.
小数相乘时，先忽略小数点按整数相乘，再数出总的小数位数。0.3 × 0.04 有 1 + 2 = 3 位小数，结果为 0.012。
Converting a percentage to a decimal: divide by 100. 7% = 0.07. To find a percentage of an amount, multiply by the decimal. 15% of 80 = 0.15 × 80 = 12.
百分数化小数：除以 100。7% = 0.07。求某个数的百分之几，用此小数乘以该数。80 的 15% = 0.15 × 80 = 12。
Percentage increase: new value = original × (1 + percentage/100). A 20% increase on £45 gives 45 × 1.2 = £54. For decrease, multiply by (1 – percentage/100).
百分比增加：新值 = 原值 × (1 + 百分数/100)。45 英镑增加 20% 为 45 × 1.2 = 54 英镑。减少则乘以 (1 – 百分数/100)。
Compound interest: A = P(1 + r/100)ⁿ
复利公式：A = P(1 + r/100)ⁿ，其中 P 为本金，r 为每期利率，n 为期数。
3. Ratio and Proportion | 比和比例
Simplify a ratio by dividing all parts by their highest common factor. 24 : 36 : 60 simplifies to 2 : 3 : 5 after dividing by 12.
化简比时，将各部分同除以最大公因数。24 : 36 : 60 除以 12 后简化为 2 : 3 : 5。
To share £200 in the ratio 3:7, first add the parts (3 + 7 = 10). One part is £200 ÷ 10 = £20, so the shares are 3 × £20 = £60 and 7 × £20 = £140.
按 3:7 的比例分配 200 英镑，先求总份数（3+7=10）。一份为 200 ÷ 10 = 20 英镑，所以分摊为 3×20=60 英镑和 7×20=140 英镑。
Direct proportion: if y ∝ x, then y = kx. When x = 4, y = 20 gives k = 5. The equation is y = 5x.
正比例：若 y 与 x 成正比，则 y = kx。当 x = 4 时 y = 20，得 k = 5，方程为 y = 5x。
Inverse proportion: if y ∝ 1/x, then y = k/x. Use given values to find k and then answer the question.
反比例：若 y 与 1/x 成正比，则 y = k/x。用已知数值求出 k，再回答问题。
4. Algebraic Substitution | 代数代入
When substituting into an expression, replace the variable with its value exactly, using brackets for negative numbers. If a = –3, then 2a² = 2 × (–3)² = 2 × 9 = 18.
代数代入时，将变量替换为给定值，负数要加括号。若 a = –3，则 2a² = 2 × (–3)² = 2 × 9 = 18。
For a formula like v = u + at, substitute carefully. u = 5, a = –2, t = 3 gives v = 5 + (–2) × 3 = 5 – 6 = –1.
对于公式 v = u + at，代入 u=5, a=–2, t=3，得 v = 5 + (–2)×3 = 5 – 6 = –1。
Substitute into expressions with fractions: if x = ½, then 3x + 4 = 3 × ½ + 4 = 1.5 + 4 = 5.5.
代入含有分数的代数式时同样计算：若 x = ½，则 3x + 4 = 3×½ + 4 = 1.5 + 4 = 5.5。
5. Expanding and Factorising | 展开与因式分解
Expand a single bracket by multiplying each term inside by the term outside. 4(3x – 7) = 12x – 28.
单项式乘以括号：用括号外的项乘括号内的每一项。4(3x – 7) = 12x – 28。
Expand two binomials using FOIL: (x + 5)(x – 3) = x² – 3x + 5x – 15 = x² + 2x – 15.
用 FOIL 法展开两个二项式：(x+5)(x–3) = x² – 3x + 5x – 15 = x² + 2x – 15。
(a + b)² = a² + 2ab + b²
完全平方公式：(a + b)² = a² + 2ab + b²。
Factorising is the reverse: find common factors or use double brackets. x² + 7x + 10 = (x + 2)(x + 5) because 2 × 5 = 10 and 2 + 5 = 7.
因式分解是展开的逆过程：提取公因式或用双括号。x²+7x+10 = (x+2)(x+5)，因为 2×5=10 且 2+5=7。
6. Solving Linear Equations | 解一元一次方程
To solve 5x – 3 = 2x + 9, collect x-terms on one side and numbers on the other. 5x – 2x = 9 + 3 → 3x = 12 → x = 4.
解方程 5x–3=2x+9，把含 x 的项移到一边，常数移到另一边：5x–2x=9+3 → 3x=12 → x=4。
When the equation contains fractions, multiply every term by the common denominator. For x/3 + 1 = 2x/5, multiply by 15: 5x + 15 = 6x → x = 15.
方程中含有分数时，将每一项乘以公分母。对 x/3 + 1 = 2x/5，乘以 15 得 5x+15=6x → x=15。
Always check your solution by substituting back into the original equation.
始终将解代回原方程进行检验。
7. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角函数
a² + b² = c² (where c is the hypotenuse)
勾股定理：a² + b² = c²（c 为斜边）
To find the hypotenuse: c = √(a² + b²). If a = 6 and b = 8, c = √(36 + 64) = √100 = 10.
求斜边：c = √(a² + b²)。若 a=6, b=8，则 c = √(36+64) = √100 = 10。
In right-angled triangles, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use SOH CAH TOA.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。记住 SOH CAH TOA。
When finding an angle, use the inverse trig functions: θ = sin⁻¹(O/H). Always ensure your calculator is in degree mode for WJEC.
求角度时使用反三角函数：θ = sin⁻¹(对边/斜边)。WJEC 考试中务必确保计算器处于角度模式。
8. Perimeter, Area and Volume | 周长、面积与体积
Area of a rectangle = length × width. Perimeter is the sum of all side lengths.
矩形面积 = 长 × 宽。周长为各边长度之和。
Area of triangle = ½ × base × perpendicular height
三角形面积 = ½ × 底 × 垂直高。
Area of a circle = πr², circumference = 2πr. Use the π button on your calculator for accuracy.
圆面积 = πr²，周长 = 2πr。为确保精度，请使用计算器上的 π 键。
Volume of a prism = area of cross-section × length. For a cylinder, V = πr²h.
棱柱体积 = 横截面积 × 长度。圆柱体积 V = πr²h。
Surface area of a sphere: 4πr². Volume of a sphere: 4/3 πr³. These are usually tested on Paper 2.
球表面积 = 4πr²；球体积 = 4/3 πr³。这些通常出现在计算器卷（Paper 2）中。
9. Averages and Range | 平均数与极差
Mean = (sum of values) ÷ (number of values). Median is the middle value when data are ordered; for an even count, average the two middle numbers.
平均数 = 数值总和 ÷ 数据个数。中位数是将数据排序后处于中间位置的数；若个数为偶数，中位数为中间两数的平均值。
Mode is the most frequent value. Range = highest value – lowest value.
众数是最常出现的值。极差 = 最大值 – 最小值。
From a frequency table, mean = Σ(f×x) ÷ Σf. The modal class is the group with the highest frequency.
对于频数表，平均数 = Σ(频数×组中值) ÷ 总频数。众数所在组是频数最高的组。
When an extra value is added, the new mean can be found by recalculating the total sum and dividing by the new count.
当增加一个数据时，重新计算总和并除以新的总个数即可得到新的平均数。
10. Probability Basics | 概率基础
Probability of an event = (number of favourable outcomes) ÷ (total number of outcomes). Values lie between 0 and 1.
事件概率 = 有利结果数 ÷ 所有可能结果总数。概率值在 0 到 1 之间。
The sum of probabilities of all mutually exclusive outcomes is 1. P(not A) = 1 – P(A).
所有互斥结果的概率之和为 1。事件 A 不发生的概率 P(非 A) = 1 – P(A)。
For combined events, use tree diagrams to list all outcomes. Multiply along the branches for ‘and’, add for ‘or’ when appropriate.
对于复合事件，使用树状图列出所有结果。沿着分支相乘代表“且”，根据情况相加代表“或”。
Expected frequency = probability × number of trials. If the probability of rain is 0.3, in 50 days expect 0.3 × 50 = 15 rainy days.
期望频数 = 概率 × 试验次数。若下雨概率为 0.3，则 50 天内预计下雨天数为 0.3×50=15 天。
11. Standard Form and Bounds | 标准形式与界
Standard form writes a number as a × 10ⁿ where 1 ≤ a < 10. 4700 = 4.7 × 10³, 0.032 = 3.2 × 10⁻².
标准形式将数字表示为 a × 10ⁿ，其中 1 ≤ a < 10。4700 = 4.7 × 10³，0.032 = 3.2 × 10⁻²。
To multiply in standard form, multiply the a-values and add the exponents: (2 × 10⁵) × (3 × 10⁷) = 6 × 10¹².
标准形式乘法：a 值相乘，10 的指数相加。(2×10⁵)×(3×10⁷)=6×10¹²。
Upper and lower bounds: a measurement given to the nearest whole number, say 15 cm, lies between 14.5 cm and 15.5 cm. The same principle applies to all rounded values.
上下界：一个量测到整数 15 cm，实际值在 14.5 cm 到 15.5 cm 之间。此原则适用于所有已舍入的数值。
When calculating with bounds, use the smallest possible numerator divided by largest denominator to find the lower bound of a quotient, and vice versa.
利用界计算时，商的下界用最小可能被除数除以最大可能除数，商的上界则相反。
12. Using a Calculator Effectively | 有效使用计算器
On Paper 2, know how to use the fraction key (a b/c or d/c) for exact answers. Enter mixed numbers as 1 ⌟ 2 ⌟ 3 to represent 1 2/3.
在计算器卷中，学会使用分数键（a b/c 或 d/c）得到精确答案。输入带分数时如 1⌟2⌟3 代表 1 2/3。
Use the square and cube buttons, the square root √ and the ANS key for multi-step calculations to avoid rounding errors. Store intermediate results in memory when needed.
使用平方键、立方键、平方根 √ 和 ANS 键进行多步计算，避免四舍五入误差。必要时将中间结果存入存储器。
For trigonometric calculations, always check that the calculator is in degree mode (D or DEG on display). A common mistake is leaving it in radian mode.
进行三角计算时，始终检查计算器是否处于角度模式（显示屏上出现 D 或 DEG）。常见错误是误留在弧度模式。
When evaluating expressions like (2.3 + √5.76) ÷ 1.2³, use brackets carefully: (2.3 + √(5.76)) ÷ (1.2^3).
计算类似 (2.3 + √5.76) ÷ 1.2³ 的表达式时，正确使用括号：(2.3 + √(5.76)) ÷ (1.2^3)。
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