GCSE Maths: Last-Minute Revision Notes | GCSE 数学：考前冲刺笔记

📚 GCSE Maths: Last-Minute Revision Notes | GCSE 数学：考前冲刺笔记

GCSE Maths can feel overwhelming, but a sharp, focused set of revision notes can make all the difference. This guide distills the key topics you need to master, with essential formulas, concepts, and tips to help you feel confident on exam day.

GCSE 数学可能让人感到不知所措，但一份清晰、有重点的冲刺笔记可以带来很大的不同。本指南提炼了你需要掌握的关键主题，包含重要公式、概念和技巧，帮助你在考试当天充满信心。

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1. Number and Arithmetic | 数与算术

Understand the real number system: natural numbers (ℕ), integers (ℤ), rational and irrational numbers. Prime numbers have exactly two factors.

理解实数系统：自然数 (ℕ)、整数 (ℤ)、有理数和无理数。质数只有两个因数。

BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) determines order of operations.

BIDMAS（括号、指数、除/乘、加/减）决定运算顺序。

Rounding to decimal places: count digits after the decimal point; to significant figures: start from the first non-zero digit.

四舍五入到小数位：计算小数点后的位数；四舍五入到有效数字：从第一个非零数字开始数。

Error intervals: when a number is rounded, the true value lies within a half-unit interval. For 2.3 rounded to 1 d.p., true value is 2.25 ≤ x < 2.35.

误差区间：当数字被四舍五入后，真实值在半个单位区间内。例如 2.3 保留一位小数，真实值是 2.25 ≤ x < 2.35。

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

To add or subtract fractions, find a common denominator first.

分数加减：首先找到公分母。

Multiplying fractions: multiply numerators together and denominators together. Division: multiply by the reciprocal.

分数乘法：分子乘分子，分母乘分母。除法：乘以倒数。

Convert a fraction to a decimal by dividing numerator by denominator. To a percentage, multiply by 100 and add the % sign.

分数化小数：分子除以分母；化百分数：乘以 100 并加上 % 号。

Percentage change = (difference ÷ original) × 100. A 20% increase means you multiply by 1.2; a 15% decrease means multiply by 0.85.

百分比变化 = (差值 ÷ 原值) × 100。增加 20% 意味着乘以 1.2；减少 15% 意味着乘以 0.85。

Reverse percentages: if a price after a 15% decrease is £85, original = 85 ÷ 0.85.

逆推百分数：如果降价 15% 后价格为 £85，原价 = 85 ÷ 0.85。

Simple interest: I = P × r × t, where P = principal, r = rate per year (as decimal), t = time in years.

单利：I = P × r × t，P 为本金，r 为年利率（小数），t 为年数。

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3. Ratio and Proportion | 比与比例

A ratio a : b can be simplified like a fraction by dividing by the highest common factor. Always express ratios in their simplest integer form.

比 a : b 可以像分数一样除以最大公因数来化简。始终用最简整数比表示。

To share £60 in the ratio 2:3, add parts (2+3=5). One part = £12, so the shares are £24 and £36.

将 £60 按 2:3 比例分配，总份数 2+3=5，每份 £12，分得 £24 和 £36。

Direct proportion: y = kx, graph is a straight line through the origin. Inverse proportion: y = k/x, graph is a hyperbola.

正比例：y = kx，图像是过原点的直线。反比例：y = k/x，图像是双曲线。

For best buy problems, calculate the unit cost (price per gram, per ml, etc.) and compare.

解决最划算购买问题时，计算单位成本（每克、每毫升等的价格）并进行比较。

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4. Algebraic Expressions | 代数表达式

Collect like terms: 3x + 2y – x + 5y = 2x + 7y.

合并同类项：3x + 2y – x + 5y = 2x + 7y。

Expanding brackets: a(b + c) = ab + ac. Double brackets: (x + a)(x + b) = x² + (a+b)x + ab.

展开括号：a(b + c) = ab + ac。双括号：(x + a)(x + b) = x² + (a+b)x + ab。

Factorising: look for the highest common factor, e.g., 6x + 9 = 3(2x + 3). Quadratic factorising: x² + 5x + 6 = (x+2)(x+3).

因式分解：寻找最大公因数，如 6x + 9 = 3(2x + 3)。二次三项式因式分解：x² + 5x + 6 = (x+2)(x+3)。

Difference of two squares: a² – b² = (a+b)(a-b). Example: x² – 16 = (x+4)(x-4).

平方差公式：a² – b² = (a+b)(a-b)。例如 x² – 16 = (x+4)(x-4)。

a² – b² = (a + b)(a – b)

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5. Solving Equations & Inequalities | 解方程与不等式

Solve linear equations by using inverse operations to isolate x. Example: 2x + 3 = 11 → 2x = 8 → x = 4.

解一元一次方程：用逆运算孤立 x。例如 2x + 3 = 11 → 2x = 8 → x = 4。

With unknowns on both sides, collect all x-terms on one side first. 5x – 3 = 2x + 9 → 3x = 12 → x = 4.

方程两边都含未知数时，先将含 x 项移到一边。5x – 3 = 2x + 9 → 3x = 12 → x = 4。

For quadratic equations, if factorised to (ax+b)(cx+d) = 0, then solutions are x = -b/a and x = -d/c. Always check it factorises first.

二次方程：若能因式分解为 (ax+b)(cx+d) = 0，则解为 x = –

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