📚 Essential Maths Book 9i Answers | Year 9 数学 Essential Maths Book 9i 答案与解析
For many Year 9 students, the Essential Maths Book 9i is a core practice book that builds confidence across number, algebra, geometry and statistics. Using the answer pages well is not about copying; it is about checking your method, locating the exact step where you went wrong, and learning to self-correct before your next assessment.
对于许多 Year 9 学生来说，Essential Maths Book 9i 是一本核心练习册，帮助大家在数、代数、几何和统计方面建立信心。合理使用答案页面不是为了抄答案，而是检查你的解题方法，找到出错的具体步骤，并在下一次评估前学会自我纠正。
1. Understanding the Book 9i Structure | 了解 Book 9i 的结构
Book 9i is typically organised into units that mix fluency, reasoning and problem-solving questions. Each chapter ends with a review exercise, and the answers section at the back is usually split by exercise number. Before checking answers, write down the page and question number so you can revisit weak topics later.
Book 9i 通常按单元编排，混合了流利度、推理和问题解决题型。每一章末尾有复习练习，书后的答案部分通常按练习编号划分。在核对答案之前，写下页码和题号，这样你之后可以重新复习薄弱知识点。
- Always attempt the question fully before looking at the answer. / 在查看答案之前，一定要先完整地尝试题目。
- Mark your own work with a tick or cross, then correct errors in a different colour. / 用勾或叉批改自己的作业，然后用不同颜色订正错误。
- If your answer differs, compare the method, not just the final number. / 如果答案不同，要比较方法，而不仅仅是最终数字。
2. Integers, Decimals and Negative Numbers | 整数、小数与负数
a − b. Multiplying two negative numbers gives a positive product, while a negative times a positive gives a negative product.
本单元涵盖整数和小数的四则运算，以及有向数。常见题目是计算如 −7 + 12 或 (−3) × 4 这样的表达式。记住加一个负数等于减去它的相反数：a + (−b) = a − b。两个负数相乘得正，负数乘正数得负。
ate 8 − (−5). Since subtracting a negative is the same as adding, 8 − (−5) = 8 + 5 = 13. / 示例：计算 8 − (−5)。因为减去负数等于加上相反数，所以 8 − (−5) = 8 + 5 = 13。
For decimals, keep place value aligned when adding or subtracting, and count decimal places when multiplying. / 对于小数，加减时要对齐数位，乘法时要数小数位数。
3. Fractions, Decimals and Percentages | 分数、小数与百分比
to convert freely between fractions, decimals and percentages. For example, ¾ = 0.75 = 75%. To find a percentage of an amount, convert the percentage to a decimal and multiply: 15% = 0.15 × 240 = 36. For fraction addition and subtraction, find a common denominator first.
Book 9i 要求能够在分数、小数和百分比之间自由转换。例如，¾ = 0.75 = 75%。要求一个数量的百分比，先把百分比转为小数再相乘：240 的 15% = 0.15 × 240 = 36。分数加减时，先找公分母。
Worked example: Convert 0.35 to a fraction in its simplest form. 0.35 = 35/100 = 7/20. / 解题示例：将 0.35 化为最简分数。0.35 = 35/100 = 7/20。
When comparing quantities, change them all to the same form, such as decimals, to avoid mistakes. / 在比较数量时，把它们都化为同一种形式，例如小数，以避免出错。
4. Ratio and Proportion | 比率与比例
oth sides by the highest common factor, 6, giving 2:3. For sharing, add the parts: sharing £60 in the ratio 3:2 gives 3 + 2 = 5 parts, so one part is £12, and the shares are £36 and £24.
Book 9i 中的比率题常要求化简、按比例分配或使用比例因子。化简 12:18 时，两边同时除以最大公因数 6，得到 2:3。分配时先把份数相加：按 3:2 分 60 英镑，总份数为 3 + 2 = 5 份，每份是 12 英镑，所以分得 36 英镑和 24 英镑。
- Check that the units are the same before simplifying a ratio. / 化简比率前，确保单位相同。
- For proportion, use a multiplier or the unitary method to find a missing value. / 对于比例，使用倍数法或单位法求未知值。
5. Algebraic Expressions and Equations | 代数表达式与方程
olve 2x + 5 = 13, subtract 5 from both sides to get 2x = 8, then divide by 2 to get x = 4.
Year 9 的代数学习包括去括号、合并同类项和解一元一次方程。去括号：3(x + 4) = 3x + 12。合并同类项：5a + 2b + 3a = 8a + 2b。解方程 2x + 5 = 13 时，两边同时减 5 得到 2x = 8，再除以 2 得到 x = 4。
olve 4(y − 2) = 12. Expand the bracket: 4y − 8 = 12, add 8: 4y = 20, divide by 4: y = 5. / 解题示例：解方程 4(y − 2) = 12。去括号：4y − 8 = 12，加 8：4y = 20，除以 4：y = 5。
Always write each step on a new line so that the examiner or your teacher can follow your logic. / 每一步换行书写，这样考官或老师能跟上你的逻辑。
6. Sequences and Functions | 数列与函数
achine, apply the operations in the correct order: input 3 into x → 2x + 1 gives output 7.
对于等差数列，找出公差并写成第 n 项。数列 5, 9, 13, 17, ... 的公差是 4，因此第 n 项为 4n + 1。对于函数机器，按正确顺序执行运算：输入 3 到 x → 2x + 1 得到输出 7。
To check a sequence formula, substitute n = 1, 2, 3 and compare with the given terms. / 检查数列公式时，代入 n = 1、2、3 并与给定项比较。
Some Book 9i questions ask you to find the rule from a table of values; draw a small table and look for the difference between consecutive outputs. / Book 9i 中的一些题目要求从数值表中找出规律；画一个小表，观察相邻输出值之间的差。
7. Geometry: Angles, Area and Volume | 几何：角、面积与体积
ume of a cuboid is V = l × w × h, and volume of a prism is area of cross-section × length.
关键角度事实包括：直线上的角之和为 180°，点周围的角之和为 360°，三角形内角和为 180°。面积方面，矩形 A = 长 × 宽，三角形 A = ½ × 底 × 高，圆 A = πr²。长方体体积 V = 长 × 宽 × 高，棱柱体积 = 截面面积 × 长度。
ple: Find the area of a triangle with base 8 cm and height 5 cm. A = ½ × 8 × 5 = 20 cm². / 解题示例：求底为 8 cm、高为 5 cm 的三角形面积。A = ½ × 8 × 5 = 20 cm²。
Remember to include units in your final answer and to square or cube the unit for area or volume. / 最终答案要带单位，面积用平方单位，体积用立方单位。
8. Statistics and Probability | 统计与概率
Statistics questions ask for the mean, median, mode and range. The mean is the sum of values divided by the number of values. The median is the middle value when data are ordered; if there are two middle numbers, take their mean. Probability is calculated as number of favourable outcomes ÷ total number of outcomes, and it always lies between 0 and 1.
统计题要求计算平均数、中位数、众数和极差。平均数等于数值之和除以数值个数。中位数是数据排序后的中间值；如果中间有两个数，取它们的平均数。概率计算为有利结果数 ÷ 总结果数，其值始终在 0 到 1 之间。
When data are grouped, read the question carefully to see whether you need to estimate the mean using midpoints. / 当数据是分组数据时，仔细读题，看是否需要使用组中点来估算平均数。
For probability, write answers as a fraction, decimal or percentage but simplify fractions where possible. / 概率可以写成分数、小数或百分比，但分数要尽量化简。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
One frequent error is dropping the negative sign when solving equations or working with directed numbers. Another is mixing up area and perimeter formulas. A third mistake is forgetting to simplify fractions or ratios in the final answer. To catch these
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