📚 Year 10 WJEC Maths: Formula & Theorem Quick Reference Handbook | Year 10 WJEC 数学：公式定理速查手册
Master the essential formulas and theorems you will encounter in Year 10 WJEC Mathematics. This quick reference guide covers number operations, algebra, geometry, statistics, and probability, with clear bilingual explanations to help you revise efficiently and build confidence for your assessments.
掌握 Year 10 WJEC 数学中你将遇到的核心公式与定理。本速查手册涵盖数与运算、代数、几何、统计和概率，并配有清晰的双语解释，助你高效复习，为考试树立信心。
1. Number: Fractions, Decimals & Percentages | 数与运算：分数、小数和百分比
Understanding the links between fractions, decimals, and percentages is fundamental. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. The key percentage formula is: Percentage = (Part / Whole) × 100%.
理解分数、小数和百分比之间的联系是基础。将分数转换为小数，用分子除以分母。将小数转换为百分比，乘以 100。核心百分比公式为：百分比 = (部分 / 总量) × 100%。
For percentage increase or decrease, use: New Value = Original × (1 ± Percentage Change as a decimal). When working with ratio, remember to divide a quantity in the ratio a:b by calculating the total parts a+b, then each share is (a/(a+b)) × Quantity and (b/(a+b)) × Quantity.
对于百分比增减，使用：新值 = 原值 × (1 ± 百分比变化化为小数)。在比例问题中，记住按 a:b 分配数量时，先计算总份数 a+b，然后每一份为 (a/(a+b)) × 总量 和 (b/(a+b)) × 总量。
- Fraction to decimal: a/b = a ÷ b
- Decimal to percentage: d × 100%
- Percentage of amount: (p/100) × Amount
- Simple interest: I = P × r × t (r as decimal)
- 分数转小数：a/b = a ÷ b
- 小数转百分比：d × 100%
- 求一个数的百分比：(p/100) × 总量
- 单利：I = P × r × t（r 为小数）
2. Algebraic Expressions & Equations | 代数表达式与方程
Algebra uses letters to represent unknown numbers. You must be confident expanding brackets and factorising expressions. The distributive law, a(b + c) = ab + ac, is essential. To factorise, find the highest common factor (HCF). For example, 6x + 9 = 3(2x + 3).
代数用字母表示未知数。你需要熟练展开括号和因式分解表达式。分配律 a(b + c) = ab + ac 是核心。进行因式分解时，找出最大公因数 (HCF)。例如 6x + 9 = 3(2x + 3)。
Solving linear equations means isolating the variable. Perform inverse operations on both sides of the equation. For equations with brackets, expand first. For equations with fractions, multiply every term by the lowest common denominator (LCD).
解线性方程意味着分离变量。对等式两边执行逆运算。带括号的方程，先展开。带分数的方程，各项乘以最小公分母 (LCD)。
When multiplying two binomials, use FOIL: (x + a)(x + b) = x² + (a+b)x + ab. To factorise a quadratic x² + bx + c, find two numbers that multiply to c and add to b.
将两个二项式相乘时，使用首外内尾法则：(x + a)(x + b) = x² + (a+b)x + ab。因式分解二次式 x² + bx + c 时，找到两个数，相乘得 c，相加得 b。
3. Linear Equations & Inequalities | 线性方程与不等式
A linear equation can be written in the form y = mx + c, where m is the gradient and c is the y-intercept. To find the equation of a line given two points, first compute the gradient m = (y₂ − y₁) / (x₂ − x₁), then use y − y₁ = m(x − x₁).
线性方程可写成 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。已知两点求直线方程，先计算斜率 m = (y₂ − y₁) / (x₂ − x₁)，然后使用 y − y₁ = m(x − x₁)。
Inequalities are solved like equations, but remember: when multiplying or dividing by a negative number, reverse the inequality sign. Represent solutions on a number line using open circles for < or > and closed circles for ≤ or ≥.
解不等式与解方程类似，但要记住：乘以或除以负数时，不等号方向改变。在数轴上表示解集时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
Simultaneous linear equations can be solved by elimination or substitution. Elimination involves adding or subtracting equations to remove one variable. Substitution involves rearranging one equation for a variable and substituting into the other.
二元一次方程组可通过消元法或代入法求解。消元法通过方程相加或相减消去一个变量。代入法先改写一个方程，将一个变量用另一个表示，再代入另一个方程。
4. Indices & Surds | 指数与根式
The laws of indices simplify expressions with powers. Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ. Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Power of a power: (aᵐ)ⁿ = aᵐⁿ. Negative index: a⁻ⁿ = 1/aⁿ. Fractional index: a¹/ⁿ = ⁿ√a and aᵐ/ⁿ = (ⁿ√a)ᵐ.
指数法则可以简化幂的表达式。乘法：aᵐ × aⁿ = aᵐ⁺ⁿ。除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。幂的乘方：(aᵐ)ⁿ = aᵐⁿ。负指数：a⁻ⁿ = 1/aⁿ。分数指数：a¹/ⁿ = ⁿ√a，aᵐ/ⁿ = (ⁿ√a)ᵐ。
Surds are irrational roots. Simplify surds by extracting square factors: √(a²b) = a√b. Rationalise the denominator of 1/√a by multiplying numerator and denominator by √a to get √a/a.
根式是无理的根数。化简根式时提取平方因子：√(a²b) = a√b。分母有理化，如 1/√a，需分子分母同乘 √a，得到 √a/a。
Standard form expresses numbers as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. For multiplication, multiply the A values and add the powers. For division, divide the A values and subtract the powers.
标准形式将数字表示为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。乘法运算时，A 系数相乘，10 的指数相加。除法运算时，A 系数相除，10 的指数相减。
5. Sequences | 数列
A sequence is an ordered list of numbers. Arithmetic sequences have a constant difference, d, between terms. The nth term of an arithmetic sequence is given by aₙ = a₁ + (n − 1)d, where a₁ is the first term. For example, the sequence 5, 8, 11, 14… has nth term 3n + 2.
数列是有序的数字列表。等差数列的相邻两项之差为常数 d。第 n 项公式为 aₙ = a₁ + (n − 1)d，其中 a₁ 是首项。例如数列 5, 8, 11, 14… 的第 n 项为 3n + 2。
Quadratic sequences have a second difference that is constant. The nth term is of the form an² + bn + c. To find a, halve the second difference. Then use the first few terms to solve for b and c.
二次数列的二级差为常数。第 n 项公式为 an² + bn + c。求 a 时，将二级差除以 2。然后利用前几项解出 b 和 c。
Recognising special sequences such as square numbers (n²), cube numbers (n³), triangular numbers (n(n+1)/2), and Fibonacci-like sequences is also important.
识别特殊数列也很重要，如平方数 (n²)、立方数 (n³)、三角形数 (n(n+1)/2) 以及类斐波那契数列。
6. Ratio, Proportion & Rates of Change | 比、比例与变化率
Ratio compares parts of a whole. Simplify ratios by dividing all parts by their HCF. Direct proportion means y ∝ x or y = kx, where k is the constant of proportionality. The graph is a straight line through the origin.
比用于比较整体中的各个部分。化简比时将所有项除以它们的最大公因数。正比例意味着 y ∝ x 或 y = kx，其中 k 为比例常数，图像为一条过原点的直线。
Inverse proportion means y ∝ 1/x or y = k/x. Its graph is a rectangular hyperbola. Speed, distance, and time are linked by: Speed = Distance / Time. Density = Mass / Volume. Pressure = Force / Area.
反比例意味着 y ∝ 1/x 或 y = k/x，图像是一条矩形双曲线。速度、距离和时间的关系为：速度 = 距离 / 时间。密度 = 质量 / 体积。压强 = 力 / 面积。
Compound measures like density and speed require consistent units. Convert all units before substituting into formulas. For best value problems, find the unit price (cost per gram, per litre, etc.).
密度、速度等复合度量要求单位统一。代入公式前需转换所有单位。求解最优价值问题时，应计算单位价格（每克、每升的成本等）。
7. Geometry: Angles & Polygons | 几何：角与多边形
Angle facts: angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal. In a triangle, the sum of interior angles is 180°. An exterior angle equals the sum of the two opposite interior angles.
角的基本定理：平角为 180°，周角为 360°，对顶角相等。三角形内角和为 180°。一个外角等于与它不相邻的两个内角之和。
For polygons, the sum of interior angles = (n − 2) × 180° where n is the number of sides. The sum of exterior angles of any convex polygon is always 360°. Each interior angle of a regular polygon = (n−2)×180° / n.
对于多边形，内角和 = (n − 2) × 180°，其中 n 为边数。任何凸多边形的外角和恒为 360°。正多边形的每个内角为 (n−2)×180° / n。
Angles in parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°. Use these to find unknown angles in diagrams.
平行线中的角：同位角相等，内错角相等，同旁内角相加为 180°。利用这些性质求解图形中的未知角。
8. Perimeter, Area & Volume | 周长、面积与体积
Learn these area formulas: Rectangle = length × width; Triangle = ½ × base × height; Parallelogram = base × height; Trapezium = ½(a + b)h, where a and b are parallel sides; Circle area = πr², circumference = 2πr or πd.
记住以下面积公式：矩形 = 长 × 宽；三角形 = ½ × 底 × 高；平行四边形 = 底 × 高；梯形 = ½(a + b)h，其中 a 和 b 为平行边；圆的面积 = πr²，周长 = 2πr 或 πd。
For composite shapes, split them into simpler shapes, calculate individual areas, and then add or subtract. For arcs and sectors, arc length = (θ/360) × 2πr, sector area = (θ/360) × πr², where θ is the angle in degrees.
对于组合图形，将其拆分为简单图形，分别计算面积后再加减。对于圆弧和扇形，弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²，其中 θ 为圆心角度数。
Volume formulas: Cuboid = length × width × height; Prism = area of cross-section × length; Cylinder = πr²h; Pyramid = ⅓ × base area × height; Cone = ⅓πr²h; Sphere = ⁴⁄₃πr³. Surface area of a cylinder = 2πrh + 2πr².
体积公式：长方体 = 长 × 宽 × 高；棱柱 = 横截面积 × 长；圆柱 = πr²h；棱锥 = ⅓ × 底面积 × 高；圆锥 = ⅓πr²h；球体 = ⁴⁄₃πr³。圆柱的表面积 = 2πrh + 2πr²。
9. Pythagoras’ Theorem & Trigonometry | 勾股定理与三角学
Pythagoras’ theorem applies to right-angled triangles: a² + b² = c², where c is the hypotenuse (longest side). To find the hypotenuse, add the squares of the other two sides and square root. To find a shorter side, subtract the square of the known shorter side from c² and square root.
勾股定理适用于直角三角形：a² + b² = c²，其中 c 为斜边（最长边）。求斜边时，将另外两边的平方相加再开平方。求短边时，用斜边的平方减去已知短边的平方再开平方。
The trigonometric ratios for a right-angled triangle are: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Use SOH CAH TOA to remember them. These ratios allow you to find missing sides or angles.
直角三角形的三角比定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。使用 SOH CAH TOA 记忆。这些比值可用来求缺失的边长或角度。
When finding an angle, use the inverse functions: θ = sin⁻¹(O/H), θ = cos⁻¹(A/H), θ = tan⁻¹(O/A). In 3D Pythagoras problems, the length of the space diagonal in a cuboid is √(l² + w² + h²).
求角度时，使用反函数：θ = sin⁻¹(O/H)，θ = cos⁻¹(A/H)，θ = tan⁻¹(O/A)。在三维勾股定理问题中，长方体的空间对角线长度为 √(l² + w² + h²)。
10. Statistics: Averages & Charts | 统计：平均数与图表
The three main averages are mean, median, and mode. Mean = (sum of all values) ÷ (number of values). Median is the middle value when ordered. Mode is the most frequent value. Range = maximum − minimum, a measure of spread.
三种主要的平均数为平均数、中位数和众数。平均数 = (所有值的总和) ÷ (值的个数)。中位数为排序后中间的值。众数为出现次数最多的值。极差 = 最大值 − 最小值，用于衡量离散程度。
For grouped frequency tables, estimate the mean using midpoints: Mean ≈ Σ(f × midpoint) / Σf. The modal class is the class with the highest frequency. The median interval can be found using cumulative frequency.
对于分组频率表，使用组中值来估算平均数：平均数 ≈ Σ(f × 组中值) / Σf。众数组为频率最高的组。中位数区间可通过累积频率来确定。
Charts: bar charts for categorical data, histograms for continuous data with equal class widths (frequency = frequency density × class width), pie charts where angle = (frequency/total) × 360°, and scatter graphs for correlation.
图表：条形图用于分类数据，直方图用于等组距的连续数据（频数 = 频率密度 × 组距），饼图中角度 = (频数/总数) × 360°，散点图用于展示相关关系。
11. Probability | 概率
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). Theoretical probability = (number of favourable outcomes) / (total number of possible outcomes), assuming all outcomes are equally likely.
概率衡量事件发生的可能性，范围从 0（不可能）到 1（必然）。理论概率 = (有利结果的数量) / (所有可能结果的总数)，前提是所有结果等可能。
For combined events, use sample space diagrams, two-way tables, or tree diagrams. The sum of probabilities of all outcomes is 1. The probability of an event not happening = 1 − P(event).
对于组合事件，使用样本空间图、双表或树状图。所有结果的概率之和为 1。事件不发生的概率 = 1 − P(事件)。
For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B). Tree diagrams branch with probabilities, and you multiply along branches and add between them.
对于独立事件，P(A 和 B) = P(A) × P(B)。对于互斥事件，P(A 或 B) = P(A) + P(B)。树状图按概率分支，沿分支相乘，不同分支相加。
Relative frequency is an estimate of probability from an experiment: Relative frequency = (number of successful trials) / (total number of trials). The more trials, the closer it tends to the theoretical probability.
相对频率是基于实验的概率估计：相对频率 = (成功试验次数) / (试验总次数)。试验次数越多，它越接近理论概率。
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