📚 Year 9 Cambridge Mathematics: Unit Test Mock Paper Walkthrough | 剑桥9年级数学：单元测试模拟卷解析
This article provides a step‑by‑step walkthrough of a mock unit test designed for Year 9 Cambridge Mathematics. Each question targets a core topic from the curriculum, helping you strengthen essential skills in algebra, geometry, number, and statistics. Read the English explanation first, then reinforce your understanding by reading the parallel Chinese explanation immediately after.
本文为您逐题解析一份针对剑桥9年级数学设计的单元测试模拟卷。每道题都紧扣课程核心知识点，帮助您巩固代数、几何、数运算和统计等关键技能。每个要点都采用英文说明紧接中文对应翻译的方式，便于双语学习。
1. Simplifying Algebraic Expressions | 化简代数表达式
Simplify 3x + 4y – x + 2y. To simplify an algebraic expression, identify and combine like terms – terms that have exactly the same variable parts. Here, the like terms are the x‑terms (3x and –x) and the y‑terms (4y and 2y).
化简 3x + 4y – x + 2y。化简代数表达式时，先找出同类项并合并——同类项是指所含字母及指数完全相同的项。本题中，同类项为 x 项（3x 和 –x）和 y 项（4y 和 2y）。
Add the x‑terms: 3x – x = 2x. Add the y‑terms: 4y + 2y = 6y. The simplified expression is 2x + 6y.
合并 x 项：3x – x = 2x。合并 y 项：4y + 2y = 6y。化简后的表达式为 2x + 6y。
2. Solving Linear Equations | 解线性方程
Solve 2(x – 3) = 10. First, remove the parentheses by using the distributive law: 2(x – 3) becomes 2x – 6. The equation is now 2x – 6 = 10.
解方程 2(x – 3) = 10。首先用乘法分配律去掉括号：2(x – 3) 变为 2x – 6。方程变为 2x – 6 = 10。
h sides to isolate the term containing x: 2x = 16. Finally, divide both sides by 2 to find x = 8.
两边同时加 6，使含 x 的项单独在一边：2x = 16。最后两边同除以 2，得到 x = 8。
3. Ratio and Proportion | 比率与比例
Divide £ arts. Then work out the value of one part by dividing the total amount by the total parts: £96 ÷ 8 = £12 per part.
按比例 3 : 5 分配 96 英镑。按给定比例分配时，先求出总份数：3 + 5 = 8 份。再用总金额除以总份数得出每份的值：96 ÷ 8 = 12 英镑。
Multiply the value of one part by each ratio number. The first share is 3 × £12 = £36; the second share is 5 × £12 = £60. Check: £36 + £60 = £96.
用每份价值分别乘以比例中的各项。第一份为 3 × 12 = 36 英镑；第二份为 5 × 12 = 60 英镑。检验：36 + 60 = 96。
4. Angles in Parallel Lines | 平行线中的角度
In the diagram, two parallel lines are cut by a transversal. One of the alternate interior angles measures 70°. Find the measure of the other alternate angle, and explain why.
图中两条平行线被一条截线所截。已知一组内错角中的一个为 70°。求另一个内错角的度数并解释理由。
When parallel lines are cut by a transversal, alternate interior angles are equal. Therefore, the other alternate angle is also 70°. This property holds because parallel lines keep the same angle of intersection with a transversal.
两条平行线被截线所截时，内错角相等。因此另一个内错角也是 70°。这个性质成立是因为平行线与截线的夹角保持不变。
5. Area of Composite Shapes | 组合图形的面积
A composite shape is made from a rectangle of width 5 cm and length 8 cm, and a right‑angled triangle on top with a base of 5 cm and height of 4 cm. Find the total area.
一个组合图形由一个宽 5 cm、长 8 cm 的矩形和其上方一个底为 5 cm、高为 4 cm 的直角三角形组成。求总面积。
te the area of the triangle: (base × height) ÷ 2 = (5 × 4) ÷ 2 = 10 cm². The total area is 40 + 10 = 50 cm².
计算矩形面积：长 × 宽 = 8 × 5 = 40 cm²。计算三角形面积：(底 × 高) ÷ 2 = (5 × 4) ÷ 2 = 10 cm²。总面积为 40 + 10 = 50 cm²。
6. Basic Probability | 基础概率
A bag contains 3 red marbles and 5 blue marbles. One marble is picked at random. Find the probability that it is red.
一个袋子中有 3 个红色弹珠和 5 个蓝色弹珠。随机取出一个弹珠，求它是红色的概率。
er of favourable outcomes (red) is 3. Probability = favourable outcomes / total outcomes = 3/8.
先求所有可能结果的总数：3 + 5 = 8。有利结果（红色）的个数为 3。概率 = 有利结果数 / 总结果数 = 3/8。
7. Gradient of a Straight Line | 直线的斜率（梯度）
s (1, 2) and (4, 8). The gradient m is the change in y divided by the change in x: m = (y₂ – y₁) ÷ (x₂ – x₁).
求经过点 (1, 2) 和 (4, 8) 的直线的斜率。斜率 m 等于 y 的变化量除以 x 的变化量：m = (y₂ – y₁) ÷ (x₂ – x₁)。
Substitute the coordinates: y₂ – y₁ = 8 – 2 = 6, x₂ – x₁ = 4 – 1 = 3. Therefore, m = 6 ÷ 3 = 2. A gradient of 2 means that for every 1 unit increase in x, y increases by 2.
代入坐标：y₂ – y₁ = 8 – 2 = 6，x₂ – x₁ = 4 – 1 = 3。因此 m = 6 ÷ 3 = 2。斜率为 2 表示 x 每增加 1 个单位，y 增加 2 个单位。
8. Percentage Increase and Decrease | 百分比增减
A shirt originally costs £40. The price is increased by 15%. What is the new price? To find the increase, calculate 15% of £40: (15 ÷ 100) × 40 = 6. Then add the increase to the original price: £40 + £6 = £46.
一件衬衫原价 40 英镑，价格上涨了 15%。新价格是多少？先求增加额：40 的 15% 是 (15 ÷ 100) × 40 = 6 英镑。再把增加额加到原价上：40 + 6 = 46 英镑。
The multiplier method offers a faster route. An increase of 15% means the new price is 115% of the original, so multiply by 1.15: £40 × 1.15 = £46. Both methods give the same result.
使用倍数法更快捷。上涨 15% 意味着新价格是原价的 115%，所以乘以 1.15：40 × 1.15 = 46 英镑。两种方法结果一致。
9. Pythagoras’ Theorem | 勾股定理
In a right‑ d the length of the hypotenuse. Pythagoras’ theorem states: hypotenuse² = side₁² + side₂².
一个直角三角形的两条直角边分别为 6 cm 和 8 cm。求斜边长。勾股定理指出：斜边² = 直角边₁² + 直角边₂²。
hypotenuse² = 6² + 8² = 36 + 64 = 100. Take the square root to find the hypotenuse: √100 = 10 cm. The hypotenuse is 10 cm long.
代入已知边长：斜边² = 6² + 8² = 36 + 64 = 100。开平方得斜边长：√100 = 10 cm。斜边长为 10 cm。
10. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots each student’s height on the x‑axis and their weight on the y‑axis. The points show a general upward trend from left to right. Describe the correlation and explain what it means.
一张散点图将每位学生的身高标在 x 轴上，体重标在 y 轴上。各点整体呈从左到右上升的趋势。请描述其相关性并解释含义。
This pattern indicates a positive correlation: as height increases, weight tends to increase as well. However, the points are not perfectly aligned, so the correlation is not perfect; individual variation still exists.
这种模式表明 正相关：身高增加时，体重也往往增加。但各点并未完全在一条线上，因此不是完全正相关，个体差异仍然存在。
11. Index Laws (Laws of Exponents) | 指数律
². When multiplying the same base, add the exponents; when raising a power to another power, multiply the exponents; when dividing, subtract the exponents.
利用指数律化简下列各式：(a) a³ × a⁴，(b) (b²)³，(c) c⁵ ÷ c²。同底数幂相乘，指数相加；幂的乘方，指数相乘；同底数幂相除，指数相减。
c) c⁵ ÷ c² = c⁵⁻² = c³. These rules work because exponents count repeated multiplications.
(a) a³ × a⁴ = a³⁺⁴ = a⁷。(b) (b²)³ = b²×³ = b⁶。(c) c⁵ ÷ c² = c⁵⁻² = c³。这些法则之所以成立，是因为指数本质上是重复乘法的计数。
12. Word Problem: Forming and Solving Equations | 应用题：列方程求解
ber be n. Then the next consecutive number is n + 1. Set up an equation: n + (n + 1) = 43.
两个连续整数的和为 43。设较小的数为 n，则下一个连续整数为 n + 1。列方程：n + (n + 1) = 43。
2n + 1 = 43. Subtract 1 from both sides: 2n = 42. Divide by 2: n = 21. The two numbers are 21 and 22. Always check: 21 + 22 = 43.
化简左边：2n + 1 = 43。两边减 1：2n = 42。两边除以 2：n = 21。这两个数为 21 和 22。务必检验：21 + 22 = 43。
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