📚 Year 7 WJEC Further Maths: Formula & Theorem Quick Reference Guide | Year 7 WJEC 进阶数学：公式定理速查手册
This quick reference guide covers the essential formulas and theorems for Year 7 WJEC Further Maths. Use it to revise key concepts, memorise equations, and build confidence before tests. Each section provides clear explanations in English followed by matching Chinese translations, ensuring you grasp the theory from both linguistic angles.
本速查手册涵盖了 Year 7 WJEC 进阶数学的核心公式与定理。用它来复习关键概念、记忆方程，并在考试前建立信心。每个部分都先用英文清晰解释，再配上对应的中文翻译，确保从双语角度理解理论。
1. Algebraic Expressions and Simplification | 代数表达式与化简
In algebra, letters (variables) stand for unknown numbers. Simplifying an expression means combining like terms – terms that have exactly the same variable and exponent.
在代数中，字母（变量）代表未知数。化简表达式意味着合并同类项——即变量和指数完全相同的项。
e variable x is identical. 4y - y = 3y, and 2a + 3b cannot be combined into a single term.
例如：3x + 2x = 5x，因为变量 x 相同。4y - y = 3y，而 2a + 3b 不能合并为一项。
kets: a(b + c) = ab + ac. Every term inside the bracket is multiplied by the term outside.
分配律可用来展开括号：a(b + c) = ab + ac。括号内的每一项都要乘以外面的项。
a(b + c) = ab + ac
When multiplying two binomials, use the FOIL method: (a + b)(c + d) = ac + ad + bc + bd.
两个二项式相乘时，使用 FOIL 方法：(a + b)(c + d) = ac + ad + bc + bd。
of expanding. For instance, 2x + 6 = 2(x + 3). Always look for the highest common factor.
因式分解是展开的逆运算。例如，2x + 6 = 2(x + 3)。始终寻找最大公因数。
2. Solving Linear Equations | 解线性方程
A linear equation can be solved by performing the same operation on both sides to keep it balanced. The goal is to isolate the variable, usually x.
线性方程可以通过在等号两边进行相同的运算来求解，保持平衡。目标是分离出变量，通常是 x。
If x + 5 = 12, subtract 5 from both sides: x = 7. Check: 7 + 5 = 12.
若 x + 5 = 12，两边同时减去 5：x = 7。检验：7 + 5 = 12。
For equations like 3x = 18, divide both sides by 3: x = 6.
对于 3x = 18 这样的方程，两边同时除以 3：x = 6。
ltiplication. Solve 2x + 3 = 11 by first subtracting 3, then dividing by 2: 2x = 8, x = 4.
两步方程结合了加法和乘法。解 2x + 3 = 11，先减去 3，再除以 2：2x = 8，x = 4。
ble appears on both sides, collect like terms. Example: 5x - 4 = 2x + 8 → 3x = 12 → x = 4.
当变量出现在等号两边时，先合并同类项。例如：5x - 4 = 2x + 8 → 3x = 12 → x = 4。
Always verify your solution by substituting it back into the original equation.
始终将解代回原方程进行验证。
3. Number Properties and Sequences | 数的性质与数列
Factors are numbers that divide exactly into another number with no remainder. The highest common factor (HCF) is the largest shared factor.
因数是能整除另一个数且没有余数的数。最大公因数（HCF）是几个数共有的最大因数。
Multiples are the results of multiplying a number by integers. The lowest common multiple (LCM) is the smallest shared multiple.
倍数是某个数与整数相乘的结果。最小公倍数（LCM）是几个数共有的最小倍数。
Prime numbers have exactly two distinct factors: 1 and themselves. The first few primes are 2, 3, 5, 7, 11, 13…
质数只有两个不同的因数：1 和它本身。最前面的几个质数是 2, 3, 5, 7, 11, 13……
number by itself: 1² = 1, 2² = 4, 3² = 9, 4² = 16. Cube numbers: 1³ = 1, 2³ = 8, 3³ = 27.
平方数是某个数自乘的结果：1² = 1, 2² = 4, 3² = 9, 4² = 16。立方数：1³ = 1, 2³ = 8, 3³ = 27。
l term: nth term = a + (n - 1)d, where a is the first term and d is the common difference.
等差数列中，相邻两项的差为常数。通项公式：第 n 项 = a + (n - 1)d，其中 a 是首项，d 是公差。
For the sequence 3, 7, 11, 15… the nth term is 4n - 1.
对于数列 3, 7, 11, 15……，第 n 项公式为 4n - 1。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
tion to decimal: divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75.
分数转小数：分子除以分母。例如，3/4 = 3 ÷ 4 = 0.75。
Decimal to percentage: multiply by 100 and add the % sign. 0.6 × 100 = 60%.
小数转百分数：乘以 100 并添加 % 符号。0.6 × 100 = 60%。
Percentage of a quantity: 20% of 80 = (20/100) × 80 = 0.2 × 80 = 16.
求一个数量的百分数：80 的 20% = (20/100) × 80 = 0.2 × 80 = 16。
Increase or decrease by a percentage: increasing 50 by 10% gives 50 × 1.10 = 55.
按百分比增加或减少：将 50 增加 10% 得到 50 × 1.10 = 55。
ding and subtracting fractions requires a common denominator. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
分数加减法需先通分。1/2 + 1/3 = 3/6 + 2/6 = 5/6。
Multiplying fractions: multiply numerators, multiply denominators. Simplify if possible. Dividing fractions: multiply by the reciprocal.
分数乘法：分子乘分子，分母乘分母，尽量约分。分数除法：乘以其倒数。
5. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. It can be written with a colon, as in 3:4, or as a fraction 3/4. Simplify ratios by dividing by common factors.
比用来比较两个或多个数量。可用冒号书写，如 3:4，或写成分数 3/4。通过除以公因数化简比。
d the parts (2+3=5), then one part is 60 ÷ 5 = 12. The shares are 2×12 = 24 and 3×12 = 36.
按 2:3 分配 60 颗弹珠，总份数 2+3=5，每份为 60 ÷ 5 = 12。分配结果为 2×12=24 和 3×12=36。
Proportion describes an equality of two ratios. Direct proportion: as one quantity increases, the other increases at the same rate.
比例描述两个比相等的关系。正比例：一个量增加，另一个量以相同比率增加。
Unitary method: find the value of one unit first. If 5 pens cost £3, one pen costs 3/5 = £0.60, then 8 pens cost 8 × 0.60 = £4.80.
单位法：先求出一个单位的量。若 5 支笔 3 英镑，一支笔 3/5 = 0.60 英镑，则 8 支笔 8 × 0.60 = 4.80 英镑。
Scale drawings and maps use ratios like 1:100, meaning 1 cm on the plan represents 100 cm in reality.
比例图和地图中使用例如 1:100 的比，表示图上 1 厘米代表实际 100 厘米。
6. Perimeter, Area and Volume | 周长、面积与体积
total distance around a shape. For a rectangle: P = 2(l + w), where l = length, w = width.
周长是围绕图形一周的长度。矩形的周长：P = 2(l + w)，其中 l 为长，w 为宽。
tangle: A = l × w. Area of a triangle: A = ½ bh, where b = base, h = perpendicular height.
A = lw | A = ½ bh
矩形面积：A = l × w。三角形面积：A = ½ bh，其中 b 为底，h 为垂直高。
logram: A = bh. Area of a trapezium: A = ½ (a + b)h, where a and b are the parallel sides.
平行四边形面积：A = bh。梯形面积：A = ½ (a + b)h，其中 a 和 b 是两条平行边。
ference of a circle: C = πd or C = 2πr, where π ≈ 3.14 or 22/7. Area of a circle: A = πr².
圆的周长：C = πd 或 C = 2πr，其中 π 约为 3.14 或 22/7。圆的面积：A = πr²。
Volume of a cuboid: V = lwh. For prisms, volume = area of cross-section × length.
长方体体积：V = lwh。棱柱体积 = 横截面积 × 长度。
Volume of a cylinder: V = πr²h. Surface area of a cylinder: 2πrh + 2πr².
圆柱体积：V = πr²h。圆柱表面积：2πrh + 2πr²。
7. Pythagoras' Theorem | 勾股定理
Pythagoras' theorem applies to right-angled triangles only. It states that the square of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the other two sides.
勾股定理只适用于直角三角形。它指出斜边（直角对边中最长的一边）的平方等于另外两条边的平方之和。
a² + b² = c²
Here, c represents the hypotenuse, while a and b represent the two shorter legs.
这里，c 代表斜边，a 和 b 代表两条较短的直角边。
angle with legs 3 cm and 4 cm, the hypotenuse c satisfies 3² + 4² = c² → 9 + 16 = 25 → c = √25 = 5 cm.
例子：一个直角三角形的两条直角边分别为 3 cm 和 4 cm，斜边 c 满足 3² + 4² = c² → 9 + 16 = 25 → c = √25 = 5 cm。
You can also rearrange the theorem to find a shorter side: a
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