📚 Year 7 Cambridge Maths: Quick Reference Formulas & Theorems | 剑桥7年级数学：公式定理速查手册
Welcome to your handy revision guide for Year 7 Cambridge Mathematics. This resource compiles all the essential formulas, theorems, and rules you need to excel. Each concept is presented in both English and Chinese, with clear examples and a structured layout for quick reference.
欢迎使用剑桥7年级数学速查手册。本资料汇总了所有关键公式、定理和规则，助你取得优异成绩。每个概念均以中英双语呈现，并附有简明示例与结构化的布局，方便快速查阅。
1. Number and Place Value | 数与位值
Understand place value: each digit in a number has a value based on its position (ones, tens, hundreds, etc.). In decimals, places after the decimal point are tenths, hundredths, thousandths, and so on.
理解位值：数字中的每一位根据其位置拥有对应的值（个位、十位、百位等）。在小数中，小数点后的数位分别是十分位、百分位、千分位等。
Multiplying by 10, 100, 1000 moves the digits to the left; dividing moves digits to the right. Example: 3.4 × 10 = 34, 5.67 × 100 = 567.
乘以10、100、1000会使数字向左移动；除以10、100、1000会使数字向右移动。例如：3.4 × 10 = 34，5.67 × 100 = 567。
Rounding: look at the digit to the right of the required place value. If it is 5 or more, round up; otherwise round down.
四舍五入：看需保留数位右边的一位数字。如果大于或等于5，则进一；否则舍去。
2. Integers and Order of Operations | 整数与运算顺序
Adding integers: same signs, add and keep the sign; different signs, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.
整数加法：同号两数相加，取相同的符号，并把绝对值相加；异号两数相加，取绝对值较大数的符号，并用较大的绝对值减去较小的绝对值。
Subtracting an integer is the same as adding its opposite: a − b = a + (−b).
减去一个整数等于加上它的相反数：a − b = a + (−b)。
Multiplying and dividing integers: same signs → positive product/quotient; different signs → negative product/quotient.
整数乘除法：同号得正，异号得负。
Order of operations (BODMAS/BIDMAS): Brackets, Orders (powers/roots), Division and Multiplication (left to right), Addition and Subtraction (left to right). Example: 3 + 6 × 2 = 3 + 12 = 15.
运算顺序（BODMAS/BIDMAS）：括号、乘方/开方、除法和乘法（从左到右）、加法和减法（从左到右）。例如：3 + 6 × 2 = 3 + 12 = 15。
3. Fractions, Decimals, Percentages | 分数、小数和百分数
Equivalent fractions: a/b = (a×n)/(b×n). Simplify by dividing both numerator and denominator by their highest common factor.
等值分数：a/b = (a×n)/(b×n)。通过将分子分母同时除以最大公因数来化简分数。
Adding/subtracting fractions: find a common denominator. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
分数加减：找公分母。例：1/3 + 1/4 = 4/12 + 3/12 = 7/12。
Multiplying fractions: multiply numerators and multiply denominators. Dividing fractions: multiply by the reciprocal (KFC rule: Keep, Flip, Change).
分数乘法：分子乘分子，分母乘分母。分数除法：乘以除数的倒数（KFC法则：保持第一个分数不变，将除号改为乘号，将第二个分数分子分母互换）。
Converting between forms:
形式互化：
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/3 | 0.333… | 33⅓% |
| 1/10 | 0.1 | 10% |
4. Ratio and Proportion | 比和比例
Ratio compares two or more quantities. Simplifying a ratio is like simplifying a fraction by dividing all parts by their greatest common factor. The order of numbers must match the order of comparison.
比用来比较两个或多个数量。化简比的方法与化简分数类似，将所有组成部分除以它们的最大公因数。数字的顺序必须与比较的顺序一致。
Proportion: two ratios that are equal. Direct proportion: as one quantity increases, the other increases at the same rate; y is proportional to x (y = kx). Solving proportion problems often uses the unitary method.
比例：两个相等的比。正比例：一个量增加，另一个量以相同速率增加；y 与 x 成正比（y = kx）。解决比例问题常使用归一法（先求单一量）。
Unitary method: find the value of one unit first, then multiply to find the required amount. E.g., if 5 pens cost £2, 1 pen costs £0.40, so 8 pens cost 8 × £0.40 = £3.20.
归一法：先求出单一量，再乘以所需数量。例如，若5支笔2英镑，则1支笔0.40英镑，8支笔共 8 × 0.40 = 3.20 英镑。
5. Algebra Basics | 初等代数
Variables and expressions: letters stand for unknown numbers. Terms are separated by + or – signs. Like terms (same variable raised to the same power) can be combined by adding or subtracting their coefficients.
变量与表达式：字母代表未知数。项由加号或减号分隔。同类项（相同变量的相同次幂）可以通过加减其系数来合并。
Simplifying: 3a + 2b + 5a − b = 8a + b. 4x² + 2x − x² + 3x = 3x² + 5x.
化简：3a + 2b + 5a − b = 8a + b。4x² + 2x − x² + 3x = 3x² + 5x。
Substitution: replace the variable with its given number. If x = 4, then 2x + 7 = 15.
代入法：用给定的数值替换变量并计算。若 x = 4，则 2x + 7 = 15。
Expanding brackets (distributive property): a(b + c) = ab + ac. Example: 3(x + 4) = 3x + 12.
展开括号（分配律）：a(b + c) = ab + ac。例：3(x + 4) = 3x + 12。
Factorising (reverse of expanding): find the highest common factor. 6x + 9 = 3(2x + 3).
因式分解（展开的逆运算）：找出最大公因数。6x + 9 = 3(2x + 3)。
6. Linear Equations | 线性方程
Solving an equation means finding the value of the unknown that makes the equality true. Always do the same operation to both sides to keep the equation balanced.
解方程意味着找出使等式成立的未知数的值。始终要在等式两边进行相同的运算，以保持方程平衡。
One-step equations: use inverse operations. x + 5 = 12 → x = 7; 3x = 15 → x = 5; x/4 = 2 → x = 8.
一步方程：使用逆运算。x + 5 = 12 → x = 7；3x = 15 → x = 5；x/4 = 2 → x = 8。
Two-step equations: undo addition/subtraction first, then multiplication/division. 2x + 3 = 11 → 2x = 8 → x = 4.
两步方程：先做加减逆运算，再做乘除逆运算。2x + 3 = 11 → 2x = 8 → x = 4。
Equations with brackets: expand first, then solve. 2(x + 4) = 14 → 2x + 8 = 14 → x = 3.
含括号的方程：先展开，再解方程。2(x + 4) = 14 → 2x + 8 = 14 → x = 3。
7. Geometry: Angles and Lines | 几何：角与线
Angle types: acute (0° < angle < 90°), right angle (90°), obtuse (90° < angle < 180°), straight line (180°), and reflex (180° < angle < 360°).
角的种类：锐角（0°<角<90°），直角（90°），钝角（90°<角<180°），平角（180°），优角（180°<角<360°）。
Angles on a straight line sum to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的角之和为180°。绕一点一周的角之和为360°。对顶角相等。
Parallel lines: corresponding angles are equal; alternate angles are equal; co-interior (allied) angles sum to 180°.
平行线：同位角相等；内错角相等；同旁内角（邻补角）之和为180°。
Triangles: interior angles sum to 180°. The exterior angle of a triangle equals the sum of the two opposite interior angles. Equilateral triangle: all angles 60°; isosceles: two equal base angles.
三角形：内角和为180°。三角形的一个外角等于与它不相邻的两个内角之和。等边三角形每个角60°；等腰三角形两底角相等。
8. Perimeter, Area, Volume | 周长、面积
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