📚 Year 9 AQA Mathematics: Formula & Theorem Quick Reference Handbook | Year 9 AQA 数学：公式定理速查手册
This handbook brings together the essential formulas and theorems you will rely on during Year 9 AQA Mathematics. Use it to review, practise and build confidence for both classwork and assessments.
本手册汇集了你在Year 9 AQA数学学习中需要依赖的核心公式和定理，可用作复习、练习和增强课堂与考试信心的速查工具。
1. Algebra Basics and Formula Rearrangement | 代数基础与公式变换
In algebra, letters represent unknown numbers. Expressions can be simplified by collecting like terms, and equations are balanced pairs that can be solved through inverse operations.
代数用字母表示未知数。可以通过合并同类项简化表达式，而方程是通过逆运算求解的平衡式。
A formula shows the relationship between variables. To rearrange a formula and change its subject, perform the same inverse operation on each side of the equals sign.
公式展示变量间的关系。要重排公式、换个主项，只需在等号两边同时进行相同的逆运算。
s = v × t → v = s ÷ t
The subject of the formula is the variable that stands alone on one side. Rearranging helps you find any missing value without re-learning different rules.
公式的主项是单独位于一边的变量。重排公式能让你求出任一缺失值，无需重新学习不同规则。
2. Linear Equations and Inequalities | 线性方程与不等式
A linear equation has no powers higher than one. It can be solved by isolating the variable using addition, subtraction, multiplication or division.
线性方程的最高次幂为一，可以通过加减乘除隔离变量来求解。
The golden rule for equations: whatever you do to one side, you must do to the other. This keeps the equation balanced.
方程的黄金法则：对一边做什么，另一边也必须做同样的事，这才能使方程保持平衡。
Inequalities use signs such as <, >, ≤ and ≥. They are solved in the same way as equations, but remember: multiplying or dividing by a negative number reverses the inequality sign.
不等式使用 <, >, ≤, ≥ 等符号，求解方式与方程相同。但务必记住：乘以或除以负数时，不等号方向必须反转。
3x + 5 = 17 → 3x = 12 → x = 4
3. Laws of Indices | 指数定律
Indices are the small raised numbers that show repeated multiplication. The laws of indices make it easy to manipulate expressions with powers.
指数是表示重复相乘的小上标数字。指数定律让幂运算变得简单。
When multiplying powers with the same base, add the indices:
同底数幂相乘时，指数相加：
aᵐ × aⁿ = aᵐ⁺ⁿ
When dividing, subtract the indices:
相除时，指数相减：
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
To raise a power to another power, multiply the indices:
幂的乘方，指数相乘：
(aᵐ)ⁿ = aᵐⁿ
o the power of zero equals one: a⁰ = 1. A negative index means a reciprocal: a⁻ⁿ = 1 / aⁿ.
任何非零数的零次方等于 1：a⁰ = 1。负指数表示倒数：a⁻ⁿ = 1 / aⁿ。
4. Fractions, Decimals and Percentages | 分数、小数和百分比
To convert a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100%.
分数转小数，用分子除以分母；小数转百分数，乘以 100%。
Percentage change is calculated by dividing the difference by the original value and multiplying by 100%.
百分比变化的计算方式为：(变化量 ÷ 原量) × 100%。
Percentage increase = (increase ÷ original) × 100%
To find a percentage of an amount, write the percentage as a fraction or decimal and multiply.
求一个量的百分之几，将百分数写成分数或小数再相乘。
Repeated percentage changes use a multiplier. An increase of 8% uses a multiplier of 1.08; a decrease of 12% uses 0.88.
连续的百分比变化使用乘数：增长 8% 用乘数 1.08；减少 12% 用乘数 0.88。
5. Ratio and Proportion | 比率与比例
Ratio compares two or more quantities in the same units. Ratios can be simplified by dividing all parts by a common factor.
比率用来比较两个或多个相同单位的量，可以通过除以公因数来化简。
To share an amount in a given ratio, add the parts to find the total number of shares, then work out the value of one share.
按给定比率分配数量时，先求出总份数，得到每一份的值。
y = kx. Inverse proportion means as one quantity increases, the other decreases: y = k/x.
正比例指两个量以相同速率增减，可写成 y = kx。反比例指一个量增加时另一个减少： y = k/x。
Use the unitary method or scaling to solve proportion problems. If 4 pens cost £3, then 10 pens cost £3 × (10÷4) = £7.50.
用单位法或比例缩放解决比例问题。如果4支笔3英镑，那么10支笔：£3 × (10÷4) = £7.50。
6. Perimeter, Area and Volume | 周长、面积与体积
Perimeter is the distance around a 2D shape. Area measures the surface covered. Volume measures the space a 3D solid occupies.
周长是二维图形边界的长度，面积度量其覆盖的表面大小，体积度量三维立体占据的空间。
Key formulas for common shapes are shown in the table below. Learn them and practise substituting numbers correctly.
下面表格列出常见图形的重要公式，请熟记并练习正确代入数值。
| Shape (English) | Area / Volume formula | 中文 |
|---|---|---|
| Rectangle | A = l × w | 矩形：长 × 宽 |
| Triangle | A = ½ × b × h | 三角形：½ × 底 × 高 |
| Parallelogram | A = b × h | 平行四边形：底 × 高 |
| Trapezium | A = ½ (a + b)h | 梯形：½(上底 + 下底) × 高 |
| Cuboid (volume) | V = l × w × h | 长方体：长 × 宽 × 高 |
| Prism (volume) | V = area of cross-section × length | 棱柱：横截面积 × 长度 |
of a rectangle is P = 2(l + w). The circumference of a circle appears in the next section.
周长即各边长度相加。长方形周长为 P = 2(长+宽)。圆的周长将在下一节介绍。
7. Circles | 圆
A circle' on its radius r and diameter d (where d = 2r). The constant π (pi) is approximately 3.142.
圆的周长和面积取决于半径 r 和直径 d（d = 2r）。常数 π 约等于 3.142。
Circumference: C = πd = 2πr
Area: A = πr²
When an answer asks for a value in terms of π, leave π in the expression. Otherwise, use the π button on your calculator for accuracy.
若题目要求用 π 表示结果，则保留 π。否则，用计算器的 π 键获取精确值。
Half a circle (semicircle) has an arc length of πr and area ½πr², but do not forget to add the diameter when finding the perimeter of a semicircle.
半圆的弧长为 πr，面积为 ½πr
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