Year 9 SQA Maths: Formula and Theorem Quick-Reference Handbook | SQA 九年级数学：公式定理速查手册

📚 Year 9 SQA Maths: Formula and Theorem Quick-Reference Handbook | SQA 九年级数学：公式定理速查手册

This handbook collects the essential formulas, rules, and theorems you will meet in the Year 9 SQA Mathematics curriculum. Keep it handy for quick checks, homework support, and last-minute revision. Each section pairs clear English explanations with Chinese translations so that bilingual learners can reinforce understanding from both sides.

本手册汇集了 SQA 九年级数学课程中必备的公式、法则和定理。你可以用它快速查阅、辅助作业和考前冲刺。每个板块都采用英文与中文配对讲解，帮助双语学习者从两种语言的角度巩固理解。

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1. Number Facts and Factors | 数字性质与因数

A prime number has exactly two distinct factors: 1 and itself. Composite numbers have more than two factors. The number 1 is neither prime nor composite.

质数只有两个因数：1 和它本身。合数有两个以上因数。1 既不是质数也不是合数。

To find the Highest Common Factor (HCF) of two numbers, list all prime factors and multiply the common ones. The Lowest Common Multiple (LCM) is found by multiplying the highest powers of all prime factors that appear.

求两个数的最大公因数（HCF）时，列出所有质因数，然后把公有的质因数相乘。最小公倍数（LCM）取所有出现的质因数的最高次幂相乘。

When multiplying powers with the same base, add the exponents.

aᵐ × aⁿ = aᵐ⁺ⁿ

同底数幂相乘，底数不变，指数相加。

When dividing powers with the same base, subtract the exponents.

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

同底数幂相除，底数不变，指数相减。

To raise a power to another power, multiply the exponents.

(aᵐ)ⁿ = aᵐⁿ

幂的乘方，底数不变，指数相乘。

Any non‑zero number raised to the power of zero equals 1.

a⁰ = 1 (a ≠ 0)

任何非零数的零次方都等于1。

A negative exponent means the reciprocal of the positive power.

a⁻ⁿ = 1 / aⁿ

负指数表示取正指数幂的倒数。

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2. Fractions, Decimals and Percentages | 分数、小数与百分比

To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100 and add the % symbol.

分数转小数：用分子除以分母。小数转百分数：乘以100再加上 %。

To increase a quantity by p%, multiply by (1 + p/100). To decrease it, multiply by (1 – p/100).

New amount = original × (1 ± p/100)

增加 p%，用原数乘以 (1 + p/100)；减少 p%，乘以 (1 – p/100)。

To add or subtract fractions, rewrite them with a common denominator first.

分数加减法必须先通分，化成同分母再计算。

Multiply fractions by multiplying numerators together and denominators together. Division is performed by multiplying by the reciprocal.

a/b × c/d = ac/bd

a/b ÷ c/d = a/b × d/c = ad/bc

分数相乘：分子乘分子，分母乘分母。分数相除：除以一个数等于乘它的倒数。

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3. Algebraic Expressions | 代数表达式

The distributive law allows you to expand products: multiply each term inside the bracket by the term outside.

a(b + c) = ab + ac

分配律：用括号外的项乘括号内的每一项。

Expanding two binomials follows the FOIL pattern: multiply First, Outer, Inner, Last terms and simplify.

(x + a)(x + b) = x² + (a + b)x + ab

展开两个二项式遵循首、外、内、尾的乘法模式，然后合并同类项。

Factorising by a common factor reverses the distributive law.

ab + ac = a(b + c)

提取公因式就是分配律的逆运算。

The difference of two squares factorises into a product of sum and difference.

a² – b² = (a – b)(a + b)

平方差公式可分解为和与差的乘积。

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4. Linear Equations and Inequalities | 线性方程与不等式

To solve a linear equation, isolate the variable by performing inverse operations on both sides. Always keep the equation balanced.

解一元一次方程时，通过逆运算把未知数移到一边，同时保持等式两边平衡。

If an equation contains brackets, expand first, then collect like terms before solving.

若方程中含括号，先展开，再合并同类项，最后求解。

When solving an inequality, multiplying or dividing both sides by a negative number reverses the inequality sign.

解不等式时，若两边同时乘或除以一个负数，不等号方向要改变。

If a < b then –a > –b

例如，若 a < b，则 –a > –b。

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5. Ratio and Proportion | 比率与比例

Ratios compare quantities of the same kind. Simplify a ratio by dividing all parts by their highest common factor.

比率用来比较同类量。化简比率时用最大公因数约分。

When sharing a quantity in a given ratio, divide the quantity by the total number of parts, then multiply by each part of the ratio.

按比率分配数量时，先除以总份数，再乘以每一份的数值。

nt. The relationship can be written as y = kx, where k is the constant of proportionality.

y = kx

两个量成正比例时，比值保持不变，关系式为 y = kx，k 是比例常数。

In inverse proportion, the product of the two quantities is constant: y = k/x.

y = k / x

成反比例时，两个量的乘积为定值，关系式为 y = k/x。

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6. Angles and Polygons | 角与多边形

When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and co‑interior angles sum to 180°.

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