Year 7 CIE Maths: Formula & Theorem Quick Reference Guide | CIE 七年级数学公式定理速查手册

📚 Year 7 CIE Maths: Formula & Theorem Quick Reference Guide | CIE 七年级数学公式定理速查手册

This quick reference guide brings together all the essential formulas, theorems, and properties you need to master in Year 7 CIE Mathematics. From number operations to geometry, algebra, and statistics, each section provides clear rules and examples designed for fast revision before tests and exams. Keep this handbook handy to strengthen your problem‑solving confidence.

本速查手册汇集了 CIE 七年级数学中你必须掌握的全部核心公式、定理与性质。内容涵盖数的运算、几何、代数与统计，每个小节均配有简明法则与示例，专为考前高效复习而设计。随时翻阅这本手册，让你解题更有信心。

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1. Number Properties and Operations | 数的性质与运算

The commutative law for addition and multiplication tells us that the order of the numbers does not change the sum or product.

a + b = b + a and a × b = b × a

交换律：加法或乘法中，交换数字的顺序不会改变和或积。

The associative law shows that the way numbers are grouped does not affect the sum or product.

(a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)

结合律：加法或乘法中，数字的分组方式不影响结果。

The distributive law connects multiplication and addition; multiplying a sum by a number gives the same result as multiplying each addend separately and then adding.

a(b + c) = ab + ac

分配律：一个数乘以括号内的和，等于这个数分别乘以各项再相加。

The order of operations (BODMAS/BIDMAS) must be followed: Brackets first, then Orders (powers/indices), then Division and Multiplication (left to right), and finally Addition and Subtraction (left to right). For example, 3 + 4 × 2 = 3 + 8 = 11.

运算顺序必须遵守 BODMAS/BIDMAS 规则：先算括号，再算乘方（指数），接着算乘除（从左到右），最后算加减（从左到右）。例如 3 + 4 × 2 = 3 + 8 = 11。

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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.

a/b = a ÷ b

分数化小数：用分子除以分母。

To convert a decimal to a percentage, multiply by 100.

Decimal × 100 = Percentage %

小数化百分数：将小数乘以 100。

To add or subtract fractions with the same denominator, keep the denominator and add or subtract the numerators.

a/b + c/b = (a + c)/b

同分母分数相加减：分母不变，分子相加减。

To add or subtract fractions with different denominators, find a common denominator, then apply the rule for like denominators.

a/b + c/d = (ad + bc)/bd

异分母分数相加减：先通分，化为同分母后再计算。

To multiply fractions, multiply the numerators and multiply the denominators. Simplify if possible.

a/b × c/d = ac/bd

分数乘法：分子乘分子，分母乘分母，能约分的要约分。

To divide by a fraction, multiply by its reciprocal (flip the second fraction and then multiply).

(a/b) ÷ (c/d) = (a/b) × (d/c)

分数除法：除以一个分数等于乘以它的倒数。

To find a percentage of a quantity, write the percentage as a fraction over 100 and multiply. For percentage increase or decrease, find the change and divide by the original amount, then multiply by 100.

求一个数的百分之几：将百分数写成分母为 100 的分数，再与数量相乘。计算百分比增减：用变化量除以原量，再乘以 100。

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3. Algebraic Expressions and Substitution | 代数表达式与代入

In algebra, you can simplify expressions by collecting like terms – terms that have the same variable raised to the same power.

ax + bx = (a + b)x

代数表达式中，可通过合并同类项（含有相同变量且相同指数的项）进行化简。

Substitution means replacing each letter in an expression with the number it represents, then performing the arithmetic carefully.

代入法：将表达式中的每个字母替换为它所代表的数值，然后认真进行算术运算。

For example, if a = 2 and b = 3, then 3a + 2b = 3(2) + 2(3) = 6 + 6 = 12. Always follow BODMAS when substituting.

例如已知 a = 2, b = 3，则 3a + 2b = 3×2 + 2×3 = 6 + 6 = 12。代入时必须遵守运算顺序规则。

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4. Solving Linear Equations | 解一元一次方程

To solve a simple linear equation, perform the same operation on both sides to keep the equation balanced and isolate the variable.

x + a = b → x = b – a

解简单一元一次方程：对方程两边进行相同的运算以保持平衡，从而解出未知数。

ax = b → x = b / a (a ≠ 0)

如 ax = b，则 x = b ÷ a（a 不为零）。

ax + b = c → x = (c – b) / a

如 ax

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