📚 Essential Maths Book 9S Answers Compressed | Year 9 数学精华答案解析
This compressed revision guide works alongside the Year 9 Essential Maths Book 9S course. The answers and worked solutions below are parallel examples rather than a page-by-page copy of the book, so you can check the same skills, compare methods and correct your own working before a test.
本精华复习指南配合 Year 9 Essential Maths Book 9S 课程使用。下列答案与详解为平行例题，并非原书逐页复制，目的是让你核查同类技能、对比解法，并在测验前纠正自己的解题过程。
The guide covers the main 9S strands: number and fractions, ratio and proportion, indices and standard form, algebra, sequences and graphs, geometry and measure, Pythagoras and introductory trigonometry, plus statistics and probability. Every section gives the key rules, a worked example and the errors that cost the most marks.
本指南覆盖 9S 的主要板块：数与分数、比与比例、指数与标准形式、代数、数列与图象、几何与测量、毕达哥拉斯定理与三角入门，以及统计与概率。每一节都给出关键法则、一道详解例题，以及最容易丢分的常见错误。
1. Using the 9S Answers as a Revision Tool | 如何将 9S 答案用作复习工具
Always attempt the exercise before you look at the answer. Write down your full working, not just the final number, because method marks are awarded in school tests and later in IGCSE/GCSE papers.
在查看答案之前，一定要先独立完成练习。把完整过程写下来，而不只是最终答案，因为在学校测验以及之后的 IGCSE/GCSE 考试中，步骤分同样会被给出。
When you check an answer, mark the question right or wrong first, then diagnose the error. If you made a sign mistake, redo the last three lines of your working. If you misunderstood a word problem, underline the key information and try a similar question straight away.
核对答案时，先把题目判为对或错，然后诊断错误原因。如果是符号错误，就重做过程的最后三行；如果是没读懂文字题，就圈出关键信息，并立刻再做一道同类题。
Use the table below to turn every wrong answer into a specific action rather than a vague feeling of "I need to revise more".
用下面的表格，把每一个错题变成具体的行动，而不是笼统地觉得“我需要多复习”。
| Error type | 错误类型 | Symptom | 表现 | Fix | 纠正方法 |
| Sign slip | 符号错误 | −3x moved across without changing sign | 移项时 −3x 没有变号 | Redo the last three lines slowly | 放慢速度重做最后三行 |
| Misread question | 读题失误 | Answered "how many left" instead of "how many used" | 问“剩下多少”却答成“用了多少” | Underline the command words | 圈出题干中的指令词 |
| Missing working | 过程缺失 | Correct answer, zero method marks | 答案对但过程分为零 | Write one line per operation | 每一步运算各写一行 |
Keep a one-page error log: date, topic, question number, and the exact reason you lost the mark. Reread that log the night before a test.
准备一页错题日志：日期、知识点、题号，以及丢分的具体原因。测验前一晚重读这份日志即可。
2. Number Skills: Fractions and Decimals | 数字技能：分数与小数
Key skills include adding and subtracting fractions with different denominators, multiplying and dividing mixed numbers, and converting terminating decimals into fractions.
核心技能包括异分母分数的加减、带分数的乘除，以及有限小数化成分数。
Worked example: calculate 2/3 + 1/4, giving your answer as a fraction.
例题：计算 2/3 + 1/4，答案写成分数。
The lowest common denominator of 3 and 4 is 12, so rewrite both fractions over 12 before adding.
3 和 4 的最小公分母是 12，所以先把两个分数都化成以 12 为分母再相加。
2/3 + 1/4 = 8/12 + 3/12 = 11/12
Subtraction follows exactly the same steps; only the final operation changes. Always simplify the answer, and turn improper fractions such as 15/12 back into mixed numbers when the question asks for a mixed number.
减法的步骤完全相同，只是最后一步的运算变了。答案一定要化简；当题目要求带分数时，要把 15/12 这样的假分数化回带分数。
For multiplication, convert mixed numbers to improper fractions first, then multiply numerators and denominators. For division, multiply by the reciprocal of the second fraction.
做乘法时，先把带分数化成假分数，再分子乘分子、分母乘分母。做除法时，乘以第二个分数的倒数。
1 2/3 × 3/4 = 5/3 × 3/4 = 15/12 = 1 1/4
36 = 36/100 = 9/25, and 0.375 = 375/1000 = 3/8. Check by dividing back on your calculator.
有限小数化分数时利用数位：0.36 = 36/100 = 9/25，0.375 = 375/1000 = 3/8。用计算器除回去检验即可。
3. Percentages, Ratio and Proportion | 百分比、比与比例
Percentage questions are fastest with multipliers. Increasing by 15% means multiplying by 1.15; decreasing by 15% means multiplying by 0.85. Two successive changes can be combined by multiplying the multipliers together.
百分比问题用乘数最快。增加 15% 就是乘以 1.15；减少 15% 就是乘以 0.85。连续两次变化，只要把两个乘数相乘即可合并。
Reverse percentage questions are the ones students lose most often. If a price rises by 15% to ¥92, the ¥92 is 115% of the original, so divide rather than subtract.
反向百分比是最容易丢分的一类。如果一件商品涨价 15% 后是 92 元，那么 92 元是原价的 115%，因此要除，而不是减。
Original = 92 ÷ 1.15 = 80
tion use repeated multipliers with a power, for example 2 000 × 1.05³ for three years at 5% per year.
复利与折旧使用带指数的重复乘数，例如每年 5%、共三年的情况写成 2 000 × 1.05³。
For ratio questions, first reduce to the form 1 : n if the question asks for a unit rate, then find the value of one part before scaling up.
比的题目中，若要求单位比，先化简成 1 : n 的形式；然后先求出“一份”的值，再按比例放大。
Worked example: share ¥240 in the ratio 3 : 5.
例题：把 240 元按 3 : 5 分配。
rt = 240 ÷ 8 = 30. The shares are 3 × 30 = 90 and 5 × 30 = 150. Check that 90 + 150 = 240.
总份数 = 3 + 5 = 8，所以一份 = 240 ÷ 8 = 30。两份分别为 3 × 30 = 90 和 5 × 30 = 150。检验 90 + 150 = 240。
Direct proportion means both quantities grow together; inverse proportion means one grows as the other shrinks, so the product stays constant.
正比例指两个量同增同减；反比例指一个增大另一个减小，二者的乘积保持不变。
4. Indices, Roots and Standard Form | 指数、根式与标准形式
The index laws must be learned exactly, because mixing them up is the single most common algebra error in Year 9.
指数法则必须准确记住，因为把它们混淆是 Year 9 最常见的代数错误。
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1
Negative and fractional indices follow the same pattern: a⁻ⁿ = 1 / aⁿ, a1/2 = √a, and a1/3 = ∛a. So 5⁻² = 1/25 and 271/3 = 3.
负指数与分数指数遵循同一规律：a⁻ⁿ = 1 / aⁿ，a1/2 = √a，a1/3 = ∛a。因此 5⁻² = 1/25，271/3 = 3。
Surds can be simplified by splitting out the largest square factor: √50 = √25 × √2 = 5√2.
根式可通过拆出最大的平方因子来化简：√50 = √25 × √2 = 5√2。
here 1 ≤ A < 10. Multiply or divide the numbers first, then deal with the powers of 10.
标准形式写成 A × 10ⁿ，其中 1 ≤ A < 10。先算前面的数字，再处理 10 的幂。
3.2 × 10⁴ × 2.5 × 10⁻² = 8.0 × 10²
For addition and subtraction in standard form, rewrite both numbers with the same power of 10 first, otherwise the answer will be wrong.
标准形式的加减法，必须先把两个数化成
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