📚 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。
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)。
gers: same signs → positive product/quotient; different signs → negative product/quotient.
整数乘除法：同号得正，异号得负。
eft 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
(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。
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。
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）。
e 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。
panding 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。
p 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。
do 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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