📚 Year 9 Cambridge Maths: Formula & Theorem Quick Reference Handbook | 九年级剑桥数学：公式定理速查手册
This quick reference handbook summarises the essential formulas, theorems, and key concepts for Year 9 Cambridge Mathematics. It is designed to help you review and consolidate the topics covered in the Cambridge Lower Secondary or IGCSE foundation course. Each section presents the core ideas in a clear, bilingual format for easy memorisation and application.
本速查手册总结了九年级剑桥数学的基本公式、定理和核心概念，旨在帮助学生复习和巩固剑桥初中或 IGCSE 基础课程中的内容。每个小节都以清晰的双语形式呈现核心要点，方便记忆和运用。
Read it as a checklist: cover the right-hand column, work through the left, and make sure you can recall each result without looking. The formulas here are the ones that appear most often in Cambridge checkpoint tests, end-of-year papers, and the early IGCSE syllabus.
建议把它当作一份清单使用：遮住右侧，先看左侧回忆，确认每个结论都能脱口而出。这里收录的公式，是剑桥 checkpoint 测评、学年末考试以及 IGCSE 入门阶段出现频率最高的一批。
1. Number and Arithmetic | 数与算术
When two fractions share the same denominator, simply add or subtract the numerators and keep the denominator unchanged.
当两个分数分母相同时，直接对分子进行加减，分母保持不变。
a/d + b/d = (a + b)/d a/d − b/d = (a − b)/d
If the denominators differ, rewrite both fractions over the lowest common denominator (LCD) first, then combine. The general result is:
若分母不同，先把两个分数通分到最小公分母（LCD），再合并。一般公式为：
a/b + c/d = (ad + bc)/bd
Multiplication of fractions is direct, while division means multiplying by the reciprocal of the second fraction.
分数乘法直接相乘；分数除法即乘以第二个分数的倒数。
a/b × c/d = ac/bd a/b ÷ c/d = a/b × d/c = ad/bc
Always cancel common factors before multiplying, and always simplify the final answer to its lowest terms. For mixed numbers, convert to improper fractions before doing any arithmetic.
相乘之前先约分，最后结果一定要化简到最简分数。带分数要先化成假分数再运算。
Order of operations follows BIDMAS: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). Remember that multiplying or dividing two negative numbers gives a positive result.
运算顺序遵循 BIDMAS：括号、乘方、乘除（从左到右）、加减（从左到右）。注意负数与负数相乘或相除，结果为正。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three ways of writing the same number, and you must be able to move between them fluently.
分数、小数和百分数是同一个数的三种写法，必须能够熟练互化。
To change a decimal into a percentage, multiply by 100. To change a percentage into a decimal, divide by 100.
小数化百分数：乘以 100。百分数化小数：除以 100。
0.25 = 1/4 = 25% 0.6 = 3/5 = 60% 1/8 = 0.125 = 12.5%
Percentage increase or decrease: the new value equals the original value multiplied by one plus or minus the rate. If r is given as a percentage, use the first form; if r is already a decimal, use the second.
百分比增减：新值 = 原值 × (1 ± 比率)。若 r 以百分数形式给出，用前一种写法；若 r 已是小数，用后一种写法。
= Original × (1 ± r/100) or New = Original × (1 ± r)
Original = New ÷ (1 ± r/100). A common error is to apply the percentage to the wrong base.
反向百分数问题给出新值、求原值，此时应做除法而不是乘法：原值 = 新值 ÷ (1 ± r/100)。最常见的错误是把百分率用在了错误的基准量上。
For repeated percentage change over n periods (for example simple and compound interest), use:
对于连续 n 期的百分比变化（如单利与复利），使用：
A = P(1 + r/100)ⁿ
3. Indices, Roots and Standard Form | 指数、根式与科学记数法
The laws of indices let you combine powers of the same base quickly and safely.
指数律可以让你快速而安全地合并同底数的幂。
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ
Any non-zero number raised to the power zero equals 1, and a negative index means the reciprocal of the positive power.
任何非零数的零次幂等于 1；负指数表示正指数幂的倒数。
a⁰ = 1 a⁻ⁿ = 1/aⁿ a1/n = ⁿ√a
Fractional indices connect powers and roots: the denominator of the index is the root, and the numerator is the power. So am/n means the n-th root of a, all raised to the power m.
分数指数把幂与根号连起来：指数的分母是根号次数，分子是幂次。因此 am/n 表示先把 a 开 n 次方，再取 m 次幂。
Surds follow similar rules and can often be simplified by looking for square factors.
根式遵循类似规则，通常可以通过寻找平方因子来化简。
√a × √b = √ab √a ÷ √b = √a/b
Standard form writes a number as a value between 1 and 10 multiplied by a power of ten, which is essential for very large or very small quantities.
科学记数法把一个数写成 1 到 10 之间的数乘以 10 的幂，这在处理极大或极小的量时非常必要。
A × 10ⁿ where 1 ≤ A < 10 and n is an integer
4. Ratio, Proportion and Rates | 比、比例与变化率
A ratio compares quantities of the same kind and should always be written in its simplest whole-number form. To divide an amount in the ratio a : b, first find the value of one part.
比用于比较同类量，必须写成最简整数比。要把一个总量按 a : b 分配，先求出一份的值。
one part = total ÷ (a + b)
Two quantities are in direct proportion when their ratio stays constant, and in inverse proportion when their product stays constant.
两个量成正比例时，比值保持不变；成反比例时，乘积保持不变。
Direct: y = kx Inverse: y = k/x
Rate problems in Year 9 are usually built on three formulas that must be memorised together with their units.
九年级的变化率问题通常建立在三个公式之上，记忆时必须连同单位一起记。
distance ÷ time density = mass ÷ volume pressure = force ÷ area
When a rate changes, use the unitary method: find the value for one unit first, then scale up or down. Always check that the units are consistent before substituting.
当比率发生变化时，使用归一法：先求单位量的值，再按比例放大或缩小。代入前务必检查单位是否统一。
5. Algebraic Expressions and Factorising | 代数式与因式分解
Expanding brackets uses the distributive law, and the same idea extends to double brackets.
展开括号使用分配律，同样的思路可推广到双括号相乘。
a(b + c) = ab + ac (x + a)(x + b) = x² + (a + b)x + ab
Three special products appear again and again and are worth memorising as patterns.
有三个特殊乘法公式反复出现，值得作为固定模式记下来。
a² + 2ab + b² (a − b)² = a² − 2ab + b² (a + b)(a − b) = a² − b²
Factorising reverses expansion. Check for a common factor first, then look for the difference of two squares, and finally try to split a quadratic trinomial into two brackets.
因式分解是展开的逆过程。先提取公因式，再看是否为平方差，最后尝试把二次三项式拆成两个括号。
x² + (a + b)x + ab = (x + a)(x + b)
Algebraic fractions are simplified by factorising the numerator and denominator first and then cancelling common factors. You may only cancel factors, never individual terms.
化简代数分式时，先对分子分母分解因式，再约去公因式。只能约去因式，绝不能约去单独的项。
6. Equations, Inequalities and Simultaneous Equations | 方程、不等式与方程组
A linear equation is solved by doing the same operation to both sides until the unknown stands alone.
解一次方程的原则是：两边同时进行相同的运算，直到未知数单独留在一边。
ax + b = c ⟹ x = (c − b)/a
Inequalities are solved in exactly the same way, with one crucial exception: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
解不等式的方法完全相同，但有一个关键例外：两边同时乘以或除以负数时，不等号方向要翻转。
−2x > 6 ⟹ x < −3
Simultaneous equations in two unknowns can be solved by elimination (add or subtract the equations to remove one variable) or by substitution (express one variable in terms of the other).
二元一次方程组可用消元法（两式相加减消去一个未知数）或代入法（用一个未知数表示另一个）求解。
Quadratic equations are solved by factorising, by completing the square, or by the quadratic formula, which works in every case.
二次方程可用因式分解、配方法或求根公式求解，其中求根公式在任何情况下都适用。
x = (−b ± √b² − 4ac) / 2a
The discriminant b² − 4ac tells you how many real roots exist: positive gives two, zero gives one repeated root, and negative gives none.
判别式 b² − 4ac 决定了实数根的个数：大于 0 有两个根，等于 0 有一个重根，小于 0 无实根。
7. Sequences, Functions and Graphs | 数列、函数与图像
In a linear sequence the difference between consecutive terms is constant, so the n-th term takes the form an + b where a is the common difference.
等差数列中相邻两项之差为常数，因此第 n 项形如 an + b，其中 a 为公差。
n-th term = a + (n − 1)d
In a quadratic sequence the first differences are not constant, but the second differences are. This tells you the n-th term contains an n² term.
二次数列的一阶差不是常数，但二阶差是常数，这说明第 n 项中含有 n² 项。
The equation of a straight line is written in gradient-intercept form, and the gradient measures how steeply the line rises.
直线方程写成斜截式，其中斜率表示直线的倾斜程度。
y = mx + c m = (y₂ − y₁) / (x₂ − x₁)
Two lines are parallel when their gradients are equal, and perpendicular when the product of their gradients is −1.
两直线平行时斜率相等；垂直时斜率之积为 −1。
The midpoint and the distance between two points can be found directly from their coordinates.
两点的中点坐标与距离可直接由坐标求得。
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) Distance = √((x₂ − x₁)² + (y₂ − y₁)²)
8. Angles, Polygons and Circle Facts | 角、多边形与圆的性质
Angle facts on straight lines and around points form the basis of almost every angle question in Year 9.
直线上的角与绕一点的角，构成了九年级几乎所有角度问题的基础。
Angles on a straight line add up to 180°. | 直线上的角之和为 180°。
Angles around a point add up to 360°. | 绕一点的角之和为 360°。
Vertically opposite angles are equal. | 对顶角相等。
Angles in a triangle add up to 180°, and the exterior angle equals the sum of the two opposite interior angles. | 三角形内角和为 180°，外角等于不相邻两内角之和。
For parallel lines cut by a transversal, alternate angles are equal, corresponding angles are equal, and co-interior angles add up to 180°.
当平行线被一条截线所截时：内错角相等，同位角相等，同旁内角之和为 180°。
Polygon angle formulas follow directly from splitting the shape into triangles.
多边形的角度公式可由把图形分割成三角形直接推出。
Interior angle sum = (n − 2) × 180° Exterior angle sum = 360°
For a regular polygon, each exterior angle is 360°/n and each interior angle is 180° − 360°/n.
正多边形的每个外角为 360°/n，每个内角为 180° − 360°/n。
Circle vocabulary also matters: the circumference and area formulas appear throughout the mensuration topics.
圆的术语同样重要：周长与面积公式贯穿整个测量专题。
Circumference = 2πr = πd Area = πr²
9. Pythagoras and Trigonometry | 勾股定理与三角学
Pythagoras' theorem applies only to right-angled triangles and connects the three sides.
勾股定理只适用于直角三角形，它把三条边联系起来。
a² + b² = c² (c is the hypotenuse)
st whether a triangle is right-angled: if a² + b² = c² holds, the angle opposite c is 90°.
反过来，可用它判断三角形是否为直角三角形：若 a² + b² = c² 成立，则 c 所对的角为 90°。
Trigonometry relates angles to side lengths through three ratios, remembered as SOHCAHTOA.
三角函数通过三个比值把角与边长联系起来，记忆口诀为 SOHCAHTOA。
hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent
Some exact values should be recalled without a calculator, since they appear in non-calculator papers.
部分特殊角的三角函数值必须不依赖计算器记忆，因为它们在不可使用计算器的试卷中经常出现。
| θ | sin θ | cos θ | tan θ |
| 30° | ½ | (√3)/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | (√3)/2 | ½ | √3 |
In three-dimensional problems, find a right-angled triangle in the plane you are working with, and use Pythagoras twice if necessary. Angles of elevation are measured upward from the horizontal, and angles of depression downward.
在立体几何问题中，先在你所处理的平面内找出一个直角三角形，必要时两次使用勾股定理。仰角从水平线向上量，俯角从水平线向下量。
10. Perimeter, Area and Volume | 周长、面积与体积
Area formulas for plane shapes should be memorised together with the shapes they belong to.
平面图形的面积公式应与图形一一对应地记忆。
Triangle = ½ × b × h Parallelogram = b × h Trapezium = ½(a + b)h
The height of a triangle or parallelogram is always the perpendicular height, not the slant edge. For compound shapes, split the figure into rectangles, triangles and semicircles, then add the parts.
三角形或平行四边形的高始终是垂直高，而不是斜边。遇到组合图形，把它拆成矩形、三角形和半圆，再把各部分相加。
The volume of a prism equals the area of its uniform cross-section multiplied by its length, which covers cuboids and cylinders alike.
棱柱的体积等于均匀横截面积乘以长度，长方体和圆柱都适用。
V = cross-section area × length Cylinder: V = πr²h
Surface area is the total area of all faces. For a closed cylinder it is the two circular ends plus the curved rectangle that unrolls around the side.
表面积是所有面的面积之和。对于封闭圆柱，它是两个圆面加上侧面展开形成的矩形。
Cylinder surface area = 2πr² + 2πrh
rsions catch many students out, so learn the key relationships: 1 cm = 10 mm, 1 m = 100 cm
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