📚 Year 7 CAIE Advanced Mathematics: Formula & Theorem Quick Reference Handbook | Year 7 CAIE 进阶数学：公式定理速查手册
This handbook provides a concise overview of essential formulas and theorems for Year 7 CAIE Advanced Mathematics. It is designed as a quick reference to support your revision and problem-solving.
本手册为 Year 7 CAIE 进阶数学必备公式与定理提供简明概览，可作为复习与解题的速查助手。
1. Integers & Number Properties | 整数与数的性质
A factor is a number that divides exactly into another number. A multiple is the product of a number and any integer.
因数是可以整除另一个数的数。倍数是某个数与任意整数的乘积。
Prime numbers have exactly two distinct factors: 1 and the number itself. The number 1 is not prime.
质数恰好有两个不同的因数：1 和它本身。数字 1 不是质数。
The Highest Common Factor (HCF) is the largest factor shared by two or more numbers. The Lowest Common Multiple (LCM) is the smallest multiple they share.
最大公因数 (HCF) 是两个或多个数共有的最大因数。最小公倍数 (LCM) 是它们共有的最小倍数。
For any two numbers a and b: a × b = HCF(a, b) × LCM(a, b)
² = 25. A cube number is the product of three equal factors, e.g. 3³ = 27. The square root √n reverses squaring, and the cube root ³√n reverses cubing.
平方数是整数自乘的结果，例如 5² = 25。立方数是三个相同因数的乘积，例如 3³ = 27。平方根 √n 是平方的逆运算，立方根 ³√n 是立方的逆运算。
and a × b = b × a; the associative law groups numbers differently without changing the result; the distributive law connects multiplication and addition: a(b + c) = ab + ac.
遵循运算顺序 BODMAS（括号、指数、除/乘、加/减）。交换律指 a + b = b + a 和 a × b = b × a；结合律改变分组而不改变结果；分配律连接乘法与加法：a(b + c) = ab + ac。
2. Fractions, Decimals & Percentages | 分数、小数与百分数
Equivalent fractions represent the same part of a whole. Simplify a fraction by dividing numerator and denominator by their HCF.
等值分数表示相同的部分。化简分数时分子分母同时除以它们的最大公因数。
To add or subtract fractions, write them with a common denominator first. Then add or subtract the numerators and keep the denominator unchanged.
要加减分数，先化为同分母，然后只把分子相加或相减，分母保持不变。
a/b + c/b = (a + c)/b
Multiply fractions by multiplying the numerators together and the denominators together. Divide by a fraction by multiplying by its reciprocal.
分数相乘时分子乘分子、分母乘分母。除以一个分数等于乘以它的倒数。
a/b ÷ c/d = a/b × d/c
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To find a percentage of an amount, multiply the amount by the percentage written as a decimal.
分数化小数用分子除以分母。小数化百分数乘以 100。求一个数的百分之几，用这个数乘以百分数写成的小数。
Percentage increase = original × (1 + percentage/100); percentage decrease = original × (1 – percentage/100).
增长后的数值 = 原值 × (1 + 百分数/100)；减少后的数值 = 原值 × (1 – 百分数/100)。
3. Negative Numbers | 负数
Negative numbers are less than zero. On a number line they lie to the left of zero. Adding a negative number is the same as subtracting its positive value.
负数小于零，在数轴上位于零的左侧。加上一个负数相当于减去它的相反数。
a + (–b) = a – b
Subtracting a negative number is equivalent to adding the positive: a – (–b) = a + b.
减去一个负数等于加上它的正数：a – (–b) = a + b。
When multiplying or dividing two numbers, if the signs are the same the answer is positive; if the signs are different the answer is negative.
两数相乘或相除时，同号得正，异号得负。
(–a) × (–b) = +ab ; (–a) × b = –ab
Use number-line reasoning or the rules above when dealing with chains of operations. Always handle brackets and BODMAS carefully.
处理连续运算时，可运用数轴推理或上述规则，并细心处理括号与运算顺序。
4. Algebraic Expressions | 代数表达式
An algebraic expression contains variables (letters), numbers and operation signs. The number in front of a variable is the coefficient. Terms with exactly the same variable parts are like terms and can be combined by adding or subtracting their coefficients.
代数表达式包含变量（字母）、数字和运算符号。变量前面的数字是系数。含有完全相同变量部分的项称为同类项，可以加减它们的系数来合并。
3x + 2y – x = 2x + 2y
To expand brackets, multiply each term inside the bracket by the term outside. This uses the distributive law.
去括号时，将括号外的项乘以括号内的每一项，这运用了分配律。
a(b + c) = ab + ac
mon factor of the terms and write it outside the bracket. For example, 6x + 9 = 3(2x + 3).
因式分解是去括号的逆过程：找出各项的最大公因数并放在括号外，如 6x + 9 = 3(2x + 3)。
Laws of indices for the same base: when multiplying add the exponents, when dividing subtract the exponents, when raising a power to a power multiply the exponents. Any non-zero number raised to the power 0 equals 1.
同底数幂的指数法则：相乘时指数相加，相除时指数相减，幂的乘方时指数相乘。任何非零数的 0 次方均为 1。
aᵐ × aⁿ = aᵐ⁺ⁿ ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ ; (aᵐ)ⁿ = aᵐˣⁿ ; a⁰ = 1 (a ≠ 0)
5. Linear Equations | 一次方程
A linear equation contains a variable raised only to the power 1. To solve it, we use inverse operations to isolate the variable while keeping the equation balanced.
一次方程中的变量指数仅为 1。解方程时使用逆运算，在保持等式平衡的同时把变量单独解出。
om both sides. For two-step equations like ax + b = c, first subtract b, then divide by a.
对于 x + a = b 这类一步方程，两边同时减 a。对于 ax + b = c 这种两步方程，先减 b，再除以 a。
If 3x + 5 = 20 then 3x = 15 and x = 5
If the equation contains brackets, expand them first. If variables appear on both sides, collect all variable terms on one side and constants on the other.
若方程含有括号，先去括号。若变量在等式两边都出现，将所有含变量的项移到一边，常数项移到另一边。
Always check your solution by substituting it back into the original equation.
得到解后应代入原方程检验。
6. Ratio & Proportion | 比与比例
A ratio compares two or more quantities of the same kind. Ratios can be simplified by dividing all parts by their HCF.
比用来比较两个或多个同类量。可将比的各项除以它们的最大公因数来化简。
12:8 simplifies to 3:2
To share a quantity in a given ratio, find the total number of parts, divide the quantity by that total, and multiply by each part of the ratio.
按给定比例分配一个量时，先求总份数，用总量除以总份数，再乘以比的每一项。
m stays constant. This is expressed as y = kx, where k is the constant of proportionality.
若两个量的比值保持不变，则它们成正比，可表示为 y = kx，其中 k 为比例常数。
Use a unitary method or a ratio table to solve proportion problems: find the value of one unit first.
用归一法或比例表解决比例问题：先算出每份的量。
7. Angles & Lines | 角与直线
(<90°), right (=90°), obtuse (between 90° and 180°), straight (=180°) and reflex (>180°).
角以度 (°) 为单位。类型有锐角 (<90°)、直角 (=90°)、钝角 (90° 到 180°)、平角 (=180°) 和优角 (>180°)。
Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的邻角之和为 180°。绕一点的周角之和为 360°。对顶角相等。
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and co-interior (allied) angles sum to 180°.
当两条平行线被一条横线所截时，同位角相等，内错角相等，同旁内角之和为 180°。
The sum of the interior angles of a triangle is 180°. The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和为 180°。三角形的一个外角等于与它不相邻的两个内角之和。
Sum of interior angles of an n-sided polygon = (n – 2) × 180°
8. Perimeter, Area & Volume | 周长、面积与体积
meter is the distance around a shape. For a rectangle, P = 2(l + w); for a square, P = 4s.
周长是图形边界的总长。矩形周长 P = 2(l + w)；正方形周长 P = 4s。
e surface enclosed. Rectangle: A = l × w; Square: A = s²; Triangle: A = ½ × base × height.
面积度量封闭区域。矩形面积 A = l × w；正方形 A = s²；三角形面积 A = ½ × 底 × 高。
Parallelogram: A = b × h
Trapezium: A = ½ (a + b) h, where a and b are parallel sides
For a circle with radius r and diameter d = 2r: circumference C = 2πr or πd; area A = πr².
对于半径为 r、直径为 d = 2r 的圆：周长 C = 2πr 或 πd；面积 A = πr²。
h; Cube: V = s³. Surface area of a cuboid is the total area of all faces: 2(lw + lh + wh).
体积度量立体图形内部空间。长方体体积 V = l ×
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