📚 Year 9 Core Mathematics: Compressed Revision Guide | 九年级核心数学：压缩复习指南
This compressed revision guide covers the essential skills in the Year 9 Core Mathematics course. It is organised into focused sections that provide clear explanations, worked examples, and key formulas needed for success in end-of-year assessments. Number work, ratio, algebra, graphs, geometry, Pythagoras, statistics and probability are all included, so the whole course can be revised in a few sittings.
本压缩复习指南涵盖九年级核心数学课程的基本技能。全文分为若干重点小节，提供清晰讲解、例题和关键公式，帮助你在期末考试中取得成功。数的运算、比与比例、代数、图像、几何、勾股定理、统计与概率全部收录，几次复习就能过完整个课程。
1. Number Sense and Place Value | 数感与位值
f ten, so the number can be written as 3×1000 + 4×100 + 5×10 + 6×1 + 7×(1/10) + 8×(1/100).
位值是九年级所有数字计算的基础。数字 3 456.78 中各位数字代表不同的十的幂，可以写作 3×1000 + 4×100 + 5×10 + 6×1 + 7×(1/10) + 8×(1/100)。
Multiplying by 10, 100 or 1000 moves every digit one, two or three places to the left; dividing moves digits to the right. This is why the decimal point appears to 'move'.
乘以 10、100、1000 会把每一位数字分别向左移动一位、两位或三位；除以这些数则向右移动。这就是小数点看起来在 '移动' 的原因。
0⁻⁴. Large and small numbers in science questions are almost always expected in this form.
标准形式把一个数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。因此 3 456.78 可写作 3.45678 × 10³，0.00042 可写作 4.2 × 10⁻⁴。科学情境中的大数和小数通常都要求写成这种形式。
A × 10ⁿ, 1 ≤ A < 10
Rounding belongs here too: round to a given number of decimal places or significant figures, counting significant figures from the first non-zero digit. Always ask whether an answer is sensible in size before writing it down.
四舍五入也属于这一部分：按给定的小数位数或有效数字取近似值，有效数字从第一个非零数字开始数。写答案前先判断数量级是否合理。
2. Order of Operations and Negative Numbers | 运算顺序与负数
Always follow the order of operations: brackets first, then indices, then division and multiplication from left to right, then addition and subtraction from left to right. This is often remembered as BIDMAS or BODMAS.
始终遵守运算顺序：先算括号，再算指数，然后从左到右计算乘除，最后从左到右计算加减。这通常被记为 BIDMAS 或 BODMAS。
Brackets → Indices → Division/Multiplication → Addition/Subtraction
+ 12 = 17. A very common error is to add before multiplying, which gives the wrong answer.
例题：5 + 3 × (8 − 6)² = 5 + 3 × 2² = 5 + 3 × 4 = 5 + 12 = 17。非常常见的错误是先加后乘，那样会得到错误答案。
sitive; multiplying or dividing two negatives gives a positive. For example, (−3) × (−4) = 12 and (−5) + (−2) = −7.
负数遵循一致的规则：加上一个负数等同于减去一个正数；减去一个负数等同于加上一个正数；两个负数相乘或相除结果为正数。例如，(−3) × (−4) = 12，(−5) + (−2) = −7。
On a calculator, protect a negative with brackets: type (−3)², not −3², because the first gives 9 and the second gives −9. On paper, treat a long fraction bar as a bracket around the whole numerator and denominator.
使用计算器时要用括号保护负数：输入 (−3)²，而不是 −3²，因为前者得 9，后者得 −9。在纸上运算时，把长分数线看作给整个分子和分母都加上了括号。
3. Fractions, Decimals and Percentages | 分数、小数与百分数
by the denominator, then multiply by 100 for a percentage. For example, 3/8 = 0.375 = 37.5%, and 0.6 = 3/5 = 60%.
三者之间的互化只需一条规则：用分子除以分母，再乘以 100 就得到百分数。例如 3/8 = 0.375 = 37.5%，而 0.6 = 3/5 = 60%。
To add or subtract fractions, write them over a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal - keep the first fraction, flip the second, change to multiply.
分数加减要先通分；分数相乘则分子乘分子、分母乘分母；分数相除要乘以倒数——第一个分数不变，第二个分数上下颠倒，除号变乘号。
Percentage change = (change ÷ original) × 100%. To increase by 15%, multiply by 1.15; to decrease by 15%, multiply by 0.85. Reverse problems work backwards: if a price is £60 after a 20% rise, the original price is 60 ÷ 1.2 = £50.
百分数变化 =（变化量 ÷ 原值）× 100%。增加 15% 就乘以 1.15，减少 15% 就乘以 0.85。逆向问题要反推：若某商品涨价 20% 后为 60 英镑，原价为 60 ÷ 1.2 = 50 英镑。
a decimal and T in years. Compound interest applies the multiplier again each year, so after 3 years an amount grows by the factor 1.05³ for 5% per year.
单利公式为 I = P × R ×
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