📚 Year 8 CAIE Further Maths: Formula & Theorem Quick Reference | Year 8 CAIE 进阶数学：公式定理速查手册
Welcome to your handy quick reference guide for Year 8 CAIE Further Mathematics. This handbook brings together all the essential formulas, theorems, and rules you need, neatly arranged by topic. Keep it close when revising, completing homework, or checking your understanding – it will help you work faster and more accurately.
欢迎使用这份便捷的 Year 8 CAIE 进阶数学速查手册。本手册将你所需的所有关键公式、定理和规则按主题清晰整理。无论是复习、做作业还是自我检查，随时翻阅都能让你解题更快、更准确。
1. Number and Arithmetic | 数与运算
Number skills form the foundation of all further mathematics. You need to be confident with fractions, decimals, percentages, ratios, and basic financial calculations.
数的运算是进阶数学的基础。你需要熟练掌握分数、小数、百分数、比和基本的金融计算。
Percentage change = (Change ÷ Original amount) × 100%
A positive answer indicates a percentage increase; a negative answer shows a percentage decrease. Always divide the actual change by the original value before multiplying by 100.
正数表示增长百分比，负数表示减少百分比。务必先用实际变化量除以原值，再乘以100。
Simple Interest: I = P × r × t
essed as a decimal, and t is the time in years. The total amount accumulated is A = P + I.
P 为本金，r 为年利率（用小数表示），t 为年数。累积总额为 A = P + I。
Percentage profit = (Profit ÷ Cost price) × 100%
This shows the profit as a percentage of the amount originally paid. It is useful for comparing business deals or solving word problems in examinations.
它表示利润占成本价的百分数。常用于比较交易盈亏或解决考试中的应用题。
Finding the original amount after a percentage change: Original = New value ÷ Multiplier
ultiplier = 1 + (percentage/100) for an increase, and 1 - (percentage/100) for a decrease.
其中，增长乘数 = 1 + 百分数/100，减少乘数 = 1 - 百分数/100。
2. Algebra Basics | 代数基础
In algebra, letters stand for unknown numbers. You must be able to collect like terms, expand brackets, and factorise simple expressions.
代数中，字母代表未知数。你需要掌握合并同类项、展开括号以及简单的因式分解。
Distributive law: a(b + c) = ab + ac
e term outside the bracket by each term inside. Be careful with signs: a(b - c) = ab - ac.
将括号外的项乘以括号内的每一项。注意符号：a(b - c) = ab - ac。
Expanding two binomials: (a + b)(c + d) = ac + ad + bc + bd
Remember the FOIL method: First, Outer, Inner, Last. Simplify by combining like terms afterwards.
使用FOIL法则：首项、外项、内项、末项。之后合并同类项进行化简。
Factorisation: ab + ac = a(b + c)
Look for the highest common factor (HCF) of all terms and place it outside the bracket. This is the reverse of expansion.
找出所有项的最高公因式（HCF）并将其放在括号外。这是展开的逆运算。
3. Linear Equations and Inequalities | 线性方程与不等式
Solving equations means finding the value of the unknown that makes the statement true. Inequalities show a range of possible values.
解方程就是求出使等式成立的未知数的值。不等式则表示一个取值范围。
Basic rule: If x + a = b, then x = b – a
Perform the same operation on both sides to keep the equation balanced. Use inverse operations to isolate the variable.
在方程两边执行相同的运算以保持平衡。使用逆运算来分离变量。
Solving: ax + b = c → x = (c – b) ÷ a
Always undo addition or subtraction before multiplication or division. Check your solution by substituting it back into the original equation.
总是先做加减的逆运算，再做乘除的逆运算。将解代入原方程进行检验。
Inequality sign rules: multiplying or dividing by a negative number reverses the sign
ng by -2 gives x > -3. Remember to flip < to > and ≤ to ≥ when using negative multipliers.
例如，若 -2x < 6，两边除以 -2 得 x > -3。乘或除以负数时，小于号变为大于号，小于等于号变为大于等于号。
4. Sequences | 数列
A sequence is an ordered set of numbers that follows a rule. You need to find the nth term and recognise both arithmetic and geometric patterns.
数列是按一定规律排列的一组数。你需要会求第 n 项，并能识别等差数列和等比数列。
Arithmetic sequence nth term: aₙ = a₁ + (n – 1)d
a₁ is the first term, d is the common difference. This formula gives any term directly without listing the whole sequence.
a₁ 为首项，d 为公差。该公式可直接求出任意项，无需列出整个数列。
Geometric sequence nth term: aₙ = a₁ × rⁿ⁻¹
r is the common ratio (multiplier between consecutive terms). Remember that the exponent is n – 1, not n.
r 为公比（连续两项的乘数）。注意指数是 n – 1 而非 n。
Term-to-term rule: next term = current term ± d (or × r)
If the question asks for the next few terms, use the recurrence rule repeatedly instead of the nth term formula.
若题目要求写出后几项，使用递推关系逐次计算，而不必套用通项公式。
5. Ratio and Proportion | 比和比例
Ratios compare quantities of the same kind; proportion compares two amounts that change together. Direct and inverse proportion are key ideas in further maths.
比用于比较同类数量；比例描述两个量之间的变化关系。正比和反比是进阶数学的重要概念。
Simplifying a ratio: divide all parts by their HCF
12:18 = 2:3 after dividing by 6. Ratios have no units and should be in the simplest form.
例如，12:18 除以 6 后得到 2:3。比没有单位，应化为最简比。
Direct proportion: y = kx
k is the constant of proportionality. When x doubles, y also doubles. The graph is a straight line through the origin.
k 为比例常数。x 加倍时 y 也加倍。图像为过原点的直线。
Inverse proportion: y = k/x
As x increases, y decreases so that the product xy remains constant (k). The graph is a hyperbola.
x 增大时 y 减小，乘积 xy 保持为常数 k。图像为双曲线。
6. Geometry of Lines and Angles | 线与角的几何
This topic covers angle facts, parallel lines, and properties of polygons. Learn these rules to find missing angles in diagrams.
该主题涵盖角的性质、平行线及多边形内角关系。掌握这些规则以便在图形中求未知角。
Sum of angles on a straight line = 180°
Angles around a point total 360°. Vertically opposite angles are equal.
平角为 180°，周角为 360°。对顶角相等。
Angles in a triangle sum to 180°; quadrilateral sum to 360°
For any n-sided polygon, the sum of interior angles = (n – 2) × 180°. Exterior angles always sum to 360° for any convex polygon.
任意三角形内角和 180°，四边形内角和 360°。任意 n 边形的内角和 = (n – 2) × 180°。凸多边形的外角和总是 360°。
Parallel lines: corresponding angles are equal, alternate angles are equal, co‑interior (allied) angles sum to 180°
Identifying the F‑shape, Z‑shape
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