📚 Essential Maths Book 9C Answers Compressed: Year 9 Key Topic Guide | Year 9 数学 Essential Maths 9C 答案压缩精讲
This guide breaks down the main topics covered in Essential Maths Book 9C so that checking answers becomes a genuine revision task, not just copying numbers.
本指南拆解 Essential Maths Book 9C 的主要知识模块，让核对答案成为真正的复习过程，而不是简单抄写数字。
1. Understanding the 9C Syllabus and Answer Book | 了解 9C 大纲与答案册
Essential Maths Book 9C is designed for Year 9 students who are working at the higher end of Key Stage 3 and preparing for the transition to GCSE.
Essential Maths Book 9C 面向 Year 9 中较高水平的学生，旨在帮助他们从 Key Stage 3 平稳过渡到 GCSE。
The answer section usually gives final values only, so you should always show the steps that lead to those values in your exercise book.
答案部分通常只给出最终结果，因此你应当在练习本中完整写出得到这些结果的过程。
A compressed answer file simply collects these results in a shorter form; it is useful for quick checking but should never replace worked solutions.
压缩答案文件只是把这些结果以更简短的形式汇总，适合快速核对，但不能代替完整解题步骤。
2. Number: Fractions, Decimals and Percentages | 数：分数、小数与百分数
n to a decimal, divide the numerator by the denominator; for example, 3/8 = 3 ÷ 8 = 0.375.
把分数化为小数，用分子除以分母即可；例如 3/8 = 3 ÷ 8 = 0.375。
To convert a decimal to a percentage, multiply by 100; so 0.375 becomes 37.5%.
把小数化为百分数，乘以 100；因此 0.375 变为 37.5%。
For percentage increase, multiply the original amount by 1 plus the rate: a 15% increase gives new value = original × 1.15.
计算百分数增加时，用原数乘以 1 加百分率：增加 15% 后，新值 = 原数 × 1.15。
For percentage decrease, multiply by 1 minus the rate: a 20% decrease gives new value = original × 0.80.
计算百分数减少时，用原数乘以 1 减百分率：减少 20% 后，新值 = 原数 × 0.80。
Fraction → Decimal → Percentage: 3/8 = 0.375 = 37.5%
百分数增减公式：新值 = 原值 × (1 ± 百分率)
3. Ratio and Proportion | 比与比例
of a whole; for example, a ratio 2:3 means the whole is split into 2 + 3 = 5 equal parts.
比用来表示整体中各部分的关系；例如 2:3 表示整体被分成 2 + 3 = 5 等份。
To share £60 in the ratio 2:3, one part is 60 ÷ 5 = £12, so the shares are 2 × 12 = £24 and 3 × 12 = £36.
按 2:3 分配 £60，一份是 60 ÷ 5 = £12，因此两部分分别为 2 × 12 = £24 和 3 × 12 = £36。
Direct proportion problems often use the unitary method: find the value of one unit first, then multiply by the required number of units.
正比例问题常用单位法：先求出一个单位的值，再乘以所需单位数。
For example, if 5 pens cost £3.50, one pen costs £0.70, so 8 pens cost 8 × £0.70 = £5.60.
例如，5 支笔售价 £3.50，一支笔为 £0.70，因此 8 支笔为 8 × £0.70 = £5.60。
4. Algebra: Expressions and Equations | 代数：表达式与方程
Simplify expressions by collecting like terms: 3x + 2y + 5x - y = 8x + y.
化简代数式要合并同类项：3x + 2y + 5x - y = 8x + y。
on both sides; for 3x + 5 = 20, subtract 5 to get 3x = 15, then divide by 3 to get x = 5.
解方程时，两边进行相同运算：对于 3x + 5 = 20，先减 5 得 3x = 15，再除以 3 得 x = 5。
brackets by multiplying each term inside: (x + 3)(x - 2) = x² - 2x + 3x - 6 = x² + x - 6.
展开括号要把括号内每一项分别相乘：(x + 3)(x - 2) = x² - 2x + 3x - 6 = x² + x - 6。
Factorising reverses expanding; for x² + 5x + 6, find two numbers that multiply to 6 and add to 5, giving (x + 2)(x + 3).
因式分解是展开的逆运算；对于 x² + 5x + 6，找到乘积为 6、和为 5 的两个数，得到 (x + 2)(
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