📚 Year 9 Higher Homework Compressed: Core Mathematics Revision | 九年级高阶作业浓缩：核心数学复习
This revision guide compresses the key topics from a typical Year 9 Higher homework pack into one focused resource. You will revisit algebra, equations, fractions, ratio, graphs, indices, geometry and probability, with straightforward methods and examples.
本复习指南将九年级高阶作业包中的核心主题压缩为一份重点资源。你将复习代数、方程、分数、比、图像、指数、几何和概率，并配有简洁的方法和示例。
1. Algebraic Simplification | 代数化简
variable and power can be added or subtracted. For example, 5x + 3y − 2x + 7y = 3x + 10y.
通过合并同类项进行化简。具有相同变量和次数的项可以相加或相减。例如 5x + 3y − 2x + 7y = 3x + 10y。
Always handle signs carefully. Multiplying terms uses index laws: x × x² = x³.
始终小心处理符号。项与项相乘使用指数法则：x × x² = x³。
5x + 3y − 2x + 7y = 3x + 10y
2. Solving Linear Equations | 解一元一次方程
do the same operation to both sides. For 3x − 5 = 16, add 5 then divide by 3 to get x = 7.
解方程时，对方程两边进行相同运算。对于 3x − 5 = 16，先加 5 再除以 3，得到 x = 7。
ect them first. For 5x + 2 = 2x + 11, subtract 2x and subtract 2 to find 3x = 9, so x = 3.
如果变量出现在两边，先合并变量。对于 5x + 2 = 2x + 11，减去 2x 并减去 2，得到 3x = 9，所以 x = 3。
3x − 5 = 16 → x = 7
3. Expanding and Factorising | 展开与因式分解
Expand brackets by multiplying each term inside by the factor outside: 3(x + 4) = 3x + 12.
展开括号时，将括号内的每一项乘以外面的因数：3(x + 4) = 3x + 12。
Factorising reverses the process. Look for the highest common factor: 6x + 9 = 3(2x + 3).
因式分解是展开的逆过程。找出最高公因数：6x + 9 = 3(2x + 3)。
Double brackets expand to three terms: (x + 2)(x + 5) = x² + 7x + 10.
双括号展开得到三项：(x + 2)(x + 5) = x² + 7x + 10。
(x + 2)(x + 5) = x² + 7x + 10
4. Fractions, Decimals and Percentages | 分数、小数与百分比
Convert between forms by remembering key equivalences: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%.
通过记住关键等值关系在形式之间转换：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%。
To add fractions, use a common denominator: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
分数相加时使用公分母：1/3 + 1/4 = 4/12 + 3/12 = 7/12。
Find a percentage of a quantity by multiplying by the decimal or fraction equivalent: 15% of 60 = 0.15 × 60 = 9.
求一个量的百分比，乘以对应的小数或分数：60 的 15% = 0.15 × 60 = 9。
1/3 + 1/4 = 4/12 + 3/12 = 7/12
5. Ratio and Proportion | 比与比例
Simplify ratios by dividing all parts by a common factor, e.g. 12:18 = 2:3.
通过将各部分除以公因数来化简比，例如 12:18 = 2:3。
Share a quantity in a ratio by finding the total number of parts. For £50 in a 2:3 ratio, total parts = 5, so each part = £10, giving £20 and £30.
按比例分配数量时，先求总份数。将 50 英镑按 2:3 分配，总份数 = 5，每份 = 10 英镑，因此得到 20 英镑和 30 英镑。
Direct proportion means when one quantity doubles, the other doubles too. Use a multiplier method to solve problems.
正比例意味着一个量翻倍，另一个量也翻倍。使用倍率法解决问题。
12:18 = 2:3
6. Linear Graphs and Gradient | 直线图像与斜率
of a straight line is often y = mx + c, where m is the gradient and c is the y-intercept.
直线方程通常为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
asures steepness: gradient = change in y ÷ change in x. A positive gradient slopes upward.
斜率衡量倾斜程度：斜率 = y 的变化量 ÷ x 的变化量。正斜率向上倾斜。
Parallel lines have the same gradient, for example y = 2x + 1 and y = 2x − 4.
平行线具有相同的斜率，例如 y = 2x + 1 和 y = 2x − 4。
y = mx + c
7. Indices and Standard Form | 指数与标准形式
Index laws simplify products and powers: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ.
指数法则可简化乘积和幂：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。
number to the power 0 is 1: 5⁰ = 1. A negative index gives a reciprocal: 2⁻³ = 1/2³ = 1/8.
任何非零数的 0 次方为 1：5⁰ = 1。负指数表示倒数：2⁻³ = 1/2³ = 1/8。
Standard form writes numbers as a × 10ⁿ where 1 ≤ a < 10. For example, 4500 = 4.5 × 10³.
标准形式将数字写作 a × 10ⁿ，其中 1 ≤ a < 10。例如 4500 = 4.5 × 10³。
aᵐ × aⁿ = aᵐ⁺ⁿ
8. Angles in Polygons | 多边形内角
e sum of interior angles of an n-sided polygon is (n − 2) × 180°. A pentagon has sum 540°.
n 边形内角和为 (n − 2) × 180°。五边形内角和为 540°。
The sum of exterior angles of any polygon is always 360°. Each exterior angle of a regular polygon is 360° ÷ n.
任何多边形外角和始终为 360°。正多边形每个外角为 360° ÷ n。
Angles on a straight line add to 180°, and angles around a point add to 360°.
直线上的角相加为 180°，围绕一点的角相加为 360°。
Sum of interior angles = (n − 2) × 180°
9. Area, Perimeter and Volume | 面积、周长与体积
tangle is length × width; triangle is 1/2 × base × height; trapezium is 1/2 × (a + b) × h.
矩形面积 = 长 × 宽；三角形面积 = 1/2 × 底 × 高；梯形面积 = 1/2 × (a + b) × 高。
cube is side³; cuboid is length × width × height; prism is area of cross-section × length.
正方体体积 = 边长³；长方体体积 = 长 × 宽 × 高；棱柱体积 = 横截面积 × 长度。
rcumference of a circle = 2πr and area = πr². Use π ≈ 3.14 or leave answers in terms of π.
圆的周长 = 2πr，面积 = πr²。可使用 π ≈ 3.14 或用 π 表示答案。
A = πr²
10. Probability of Combined Events | 组合事件概率
of favourable outcomes ÷ total number of outcomes. Probabilities always range from 0 to 1.
事件的概率 = 有利结果数 ÷ 总结果数。概率总是在 0 到 1 之间。
probabilities: P(A and B) = P(A) × P(B). For mutually exclusive events, add probabilities.
对于独立事件，概率相乘：P(A 与 B) = P(A) × P(B)。对于互斥事件，概率相加。
A tree diagram helps list outcomes for two or more events. The sum of probabilities on branches from a point is 1.
树状图有助于列出两个或多个事件的结果。从一点出发的各分支概率之和为 1。
P(A and B) = P(A) × P(B)
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