Year 8 Cambridge Maths: Quick Reference Formulas and Theorems | Year 8 剑桥数学：公式定理速查手册

📚 Year 8 Cambridge Maths: Quick Reference Formulas and Theorems | Year 8 剑桥数学：公式定理速查手册

This quick-reference handbook covers the essential formulas, theorems, and key rules for Year 8 Cambridge Mathematics. Keep it handy for revision and homework – every formula is presented with a clear explanation and a matching Chinese translation.

本速查手册涵盖 Year 8 剑桥数学的核心公式、定理与重要法则，便于复习与完成作业时查阅。每条公式均配有清晰解释与对应的中文翻译。

1. Operations with Integers and Decimals | 整数与小数运算

BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) determines the order of operations. Always work from left to right for division and multiplication.

BIDMAS（括号、指数、乘除、加减）决定运算顺序。乘除同级时，从左到右依次计算。

Example: 10 − 2 × 3 = 10 − 6 = 4, not 8 × 3 = 24.

例如：10 − 2 × 3 = 10 − 6 = 4，而不是 8 × 3 = 24。

For adding and subtracting decimals, align the decimal points before carrying out the operation. Fill empty spaces with zeros if needed.

进行小数加减时，先对齐小数点再做计算，必要时用零补齐空位。

Multiplying decimals: first multiply as whole numbers, then count the total decimal places in the factors to place the decimal point in the product.

小数乘法：先按整数相乘，再根据因数中小数位数之和确定积的小数点位置。

Dividing by a decimal: multiply both the dividend and divisor by 10, 100, etc., until the divisor is a whole number.

小数除法：将被除数和除数同时乘以 10、100 等，使除数变为整数再除。

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

n to a decimal, divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375.

分数转小数：用分子除以分母。例如 3/8 = 3 ÷ 8 = 0.375。

To change a percentage to a decimal, divide by 100 – move the decimal point two places left. 45% = 0.45.

百分数转小数：除以 100，小数点左移两位。45% = 0.45。

Adding or subtracting fractions: find a common denominator first, then add or subtract the numerators. The denominator stays the same.

分数加减法：先通分，再对分子进行加减，分母不变。

Multiplying fractions: multiply the numerators together and the denominators together. Simplify if possible.

分数乘法：分子相乘作新分子，分母相乘作新分母，最后化简。

a/b × c/d = (a×c) / (b×d)

iding by a fraction is equivalent to multiplying by its reciprocal. a/b ÷ c/d = a/b × d/c.

除以一个分数等于乘以它的倒数：a/b ÷ c/d = a/b × d/c。

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3. Algebraic Expressions and Simplification | 代数表达式与化简

Like terms have exactly the same variable raised to the same power. They can be combined by adding or subtracting the coefficients.

同类项：变量和指数完全相同的项，可以通过系数相加减合并。

Distributive law: a(b + c) = ab + ac. This is used to expand brackets.

分配律：a(b + c) = ab + ac，用于展开括号。

When expanding two binomials, use the FOIL method: (a + b)(c + d) = ac + ad + bc + bd.

展开两个二项式时，可使用 FOIL 法则：(a + b)(c + d) = ac + ad + bc + bd。

ng is the reverse of expanding; look for a common factor. For example, 6x + 9 = 3(2x + 3).

因式分解是展开的逆运算；找出公因数即可。例如 6x + 9 = 3(2x + 3)。

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4. Solving Linear Equations | 解一元一次方程

To solve a linear equation, perform the same inverse operation on both sides to keep the equation balanced.

解方程时，两边同时进行相同的逆运算，以保持等式平衡。

If you add, subtract, multiply or divide both sides by the same non-zero number, the equation remains true.

等式两边同加、同减、同乘或同除以同一个非零数，等式依然成立。

Example: 2x + 5 = 13 → subtract 5: 2x = 8 → divide by 2: x = 4.

例如：2x + 5 = 13 → 两边减 5 得 2x = 8 → 两边除以 2 得 x = 4。

riable appears on both sides, collect like terms first. 5x − 3 = 2x + 9 → 3x = 12 → x = 4.

若未知数在两边，先移项合并同类项。5x − 3 = 2x + 9 → 3x = 12 → x = 4。

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5. Powers and Index Laws | 幂与指数定律

An exponent tells how many times a base is multiplied by itself: aⁿ means a × a × ... (n times).

指数表示底数自乘的次数：aⁿ 表示 a 自乘 n 次。

Multiplication law: a^m × aⁿ = a^m+n.

同底数幂相乘，指数相加：a^m × aⁿ = a^m+n。

Division law: a^m ÷ aⁿ = a^m−n.

同底数幂相除，指数相减：a^m ÷ aⁿ = a^m−n。

Power of a power: (a^m)ⁿ = a^m×n.

幂的乘方，指数相乘：(a^m)ⁿ = a^mn。

Zero exponent: a⁰ = 1 (provided a ≠ 0).

零指数：a⁰ = 1（a ≠ 0）。

Negative exponent: a⁻ⁿ = 1 / aⁿ.

负指数定义：a⁻ⁿ = 1 / aⁿ。

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6. Ratio and Proportional Reasoning | 比例与比例推理

A ratio compares two or more quantities of the same kind. It can be simplified by dividing each term by a common factor.

比用来比较同类量，可以通过每一项同除以一个公因数化简。

If a ratio is given as a : b, the total number of parts is a + b.

若比为 a : b，总份数为 a + b。

To share a quantity according to a ratio, multiply the quantity by each part’s fraction of the whole. For a ratio 3 : 5, the fractions are 3/8 and 5/8.

按比例分配时，将总量分别乘以各份数占总份数的分数。例如 3 : 5，则分数为 3/8 与 5/8。

two quantities increase at the same rate. If y is directly proportional to x, then y = kx.

正比例：两个量按相同速率增加。若 y 与 x 成正比，则 y = kx。

Inverse proportion: one quantity increases as the other decreases, so xy = k.

反比例：一个量增加，另一个量减少，满足 xy = k。

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7. Angles and Parallel Lines | 角与平行线

Angles on a straight line sum to 180°.

平角（直线上相邻的角）之和为 180°。

Angles around a point add up to 360°.

周角（一点周围的所有角）之和为 360°。

Vertically opposite angles are equal.

对顶角相等。

When a transversal crosses parallel lines:

当一条截线与平行线相交时：

– Corresponding angles are equal.

– 同位角相等。

– Alternate interior angles are equal.

– 内错角相等。

– Interior angles on the same side of the transversal are supplementary (sum to 180°).

– 同旁内角互补（和为 180°）。

The sum of interior angles in a

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