Year 8 Cambridge Maths: Mock Unit Test Walkthrough | 剑桥八年级数学：单元模拟卷全解

📚 Year 8 Cambridge Maths: Mock Unit Test Walkthrough | 剑桥八年级数学：单元模拟卷全解

Welcome to our detailed walkthrough of a Year 8 Cambridge Mathematics mock unit test. This article breaks down each question, explains the key concepts, and provides step-by-step solutions to help you prepare confidently for your assessment.

欢迎来到剑桥八年级数学单元模拟卷的详细解析。本文逐一拆解每道题目，解释核心概念，并提供分步解题思路，助你自信备考。

1. Multiplying Fractions | 分数乘法

Question: Calculate 2/3 × 3/4. Give your answer in its simplest form.

题目：计算 2/3 × 3/4，并以最简形式给出答案。

Step 1: Multiply the numerators (top numbers): 2 × 3 = 6.

步骤1：分子相乘（上方数字）：2 × 3 = 6。

Step 2: Multiply the denominators (bottom numbers): 3 × 4 = 12. This gives 6/12.

步骤2：分母相乘（下方数字）：3 × 4 = 12，得到 6/12。

or and denominator by their highest common factor (6). 6 ÷ 6 = 1, 12 ÷ 6 = 2. Answer: 1/2.

步骤3：分子分母同时除以最大公约数（6）进行化简。6 ÷ 6 = 1，12 ÷ 6 = 2。答案：1/2。

nd 4 share a factor 2, leaving 1/1 × 1/2 = 1/2. Always check for cancellation to save time!

小窍门：你也可以在相乘之前约分。在 2/3 × 3/4 中，3 可以约掉，2 和 4 有公因数 2，剩下 1/1 × 1/2 = 1/2。提前检查可否约分能节省时间！

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2. Decimal to Fraction | 小数化分数

Question: Convert 0.75 to a fraction in its simplest form.

题目：将 0.75 化为最简分数。

0.75 means 75 hundredths, so write it as 75/100.

0.75 表示 75 个百分之一，因此写为 75/100。

5. Divide top and bottom by 25: 75 ÷ 25 = 3, 100 ÷ 25 = 4. The simplified fraction is 3/4.

找出 75 和 100 的最大公约数，为 25。分子分母同时除以 25：75 ÷ 25 = 3，100 ÷ 25 = 4。最简分数为 3/4。

Key idea: The number of decimal places tells you the denominator – two decimal places means hundredths, three would be thousandths.

核心概念：小数位数决定了分母——两位小数对应百分位，三位则对应千分位。

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3. Simplifying Algebraic Expressions | 代数式化简

Question: Simplify 3a + 5b – 2a + 7b.

题目：化简 3a + 5b – 2a + 7b。

Identify like terms: terms with 'a' and terms with 'b'.

找出同类项：含有 'a' 的项和含有 'b' 的项。

Combine the a terms: 3a – 2a = 1a (which we write simply as a).

合并 a 项：3a – 2a = 1a（简写为 a）。

Combine the b terms: 5b + 7b = 12b.

合并 b 项：5b + 7b = 12b。

The simplified expression is a + 12b.

化简后的式子为 a + 12b。

Remember: Only like terms (same letter and same power) can be added or subtracted. The sign belongs to the term that follows it.

记住：只有同类项（相同的字母和相同的指数）才能相加减。符号属于它后面的那一项。

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4. Solving a Linear Equation | 解一次方程

Question: Solve 4x – 7 = 13 to find the value of x.

题目：解方程 4x – 7 = 13，求出 x 的值。

Our aim is to isolate x. First, undo the subtraction by adding 7 to both sides:

目标是使 x 单独在一边。首先，两边加 7 来消除减法：

4x – 7 + 7 = 13 + 7 → 4x = 20.

现在，两边除以 4 来得到 x：

Now, divide both sides by 4 to get x: 4x ÷ 4 = 20 ÷ 4 → x = 5.

现在，两边除以 4 来得到 x：4x ÷ 4 = 20 ÷ 4 → x = 5。

Always verify your answer: put x = 5 back into the original equation: 4(5) – 7 = 20 – 7 = 13. It works!

一定要验证答案：将 x = 5 代回原方程：4(5) – 7 = 20 – 7 = 13，成立！

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5. Percentage Increase | 百分数增加

Question: Increase £80 by 15%.

题目：将 £80 增加 15%。

Method 1: Find 15% of £80. 15% /100, so 15/100 × 80 = (15 × 80) / 100 = 1200/100 = 12. This is the increase. New amount = £80 + £12 = £92.

方法一：求 £80 的 15%。15% 即 15/100，所以 15/100 × 80 = (15 × 80) / 100 = 1200/100 = 12。这就是增加的量。新金额 = £80 + £12 = £92。

Method 2: Use a multiplier. An increase of 15% is the same as multiplying by 1.15. £80 × 1.15 = £92. This is faster once you are confident.

方法二：使用乘数。增加 15% 相当于乘以 1.15。£80 × 1.15 = £92。熟练后这个方法更快。

Key point: To increase by p%, multiply by (1 + p/100). To decrease, multiply by (1 – p/100).

要点：增加 p% 时，乘以 (1 + p/100)；减少时乘以 (1 – p/100)。

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6. Angles in a Triangle | 三角形内角

Question: In a triangle, two angles are 48° and 73°. Calculate the third angle.

题目：在一个三角形中，两个角分别是 48° 和 73°。求第三个角的大小。

The sum of the interior angles in any triangle is always 180°.

任何三角形的内角和恒为 180°。

Add the known angles: 48° + 73° = 121°.

将已知两角相加：48° + 73° = 121°。

Subtract from 180° to find the missing angle: 180° – 121° = 59°.

用 180° 减去这个和，得到未知角：180° – 121° = 59°。

So the third angle is 59°. Always double-check that your three angles add up to 180°.

因此第三个角是 59°。务必检查三个角之和是否为 180°。

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7. Area of a Trapezium | 梯形面积

Question: Find the area of a trapezium with parallel sides 8 cm and 12 cm, and height 5 cm.

题目：一个梯形的平行边长分别为 8 cm 和 12 cm，高为 5 cm，求它的面积。

The formula for the area of a trapezium is: Area = 1/2 × (sum of parallel sides) × height.

梯形面积公式为：面积 = 1/2 ×（平行边之和）× 高。

Sum of parallel sides = 8 + 12 = 20 cm.

平行边之和 = 8 + 12 = 20 cm。

Area = 1/2 × 20 × 5 = 10 × 5 = 50 cm².

面积 = 1/2 × 20 × 5 = 10 × 5 = 50 cm²。

Tip: Remember that the height must be the perpendicular distance between the parallel sides, not the length of a slanted side.

提示：切记高必须是平行边之间的垂直距离，而不是斜边的长度。

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8. Calculating the Mean | 求平均值

Question: Find the mean of the data set: 5, 9, 12, 7, 11.

题目：计算这组数据的平均值：5, 9, 12, 7, 11。

Step 1: Add all the values together. 5 + 9 + 12 + 7 + 11 = 44.

步骤1：将所有数值相加。5 + 9 + 12 + 7 + 11 = 44。

Step 2: Count how many values there are. Here we have 5 numbers.

步骤2：数一数共有多少个数值。这里有 5 个数。

Step 3: Divide the total by the number of values: Mean = 44 ÷ 5 = 8.8.

步骤3：总和除以数值的个数：平均值 = 44 ÷ 5 = 8.8。

The mean is 8.8. The mean can be a decimal even if all the data values are integers.

平均值为 8.8。即使所有数据都是整数，平均值也可能是小数。

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9. Ratio and Proportion | 比例问题

Question: The ratio of sugar to flour in a recipe is 3:5. If you use 150 g of sugar, how much flour do you need?

题目：一个食谱中糖和面粉的比例是 3:5。如果你用了 150 克糖，需要多少面粉？

The ratio 3:5 means for every 3 parts of sugar, there are 5 parts of flour. We can find the value of one part.

比例 3:5 表示每 3 份糖对应 5 份面粉。我们可以先求出一份的量。

Sugar's 3 parts correspond to 150 g. So 1 part = 150 ÷ 3 = 50 g.

糖的 3 份等于 150 克。因此 1 份 = 150 ÷ 3 = 50 克。

Flour is 5 parts, so flour needed = 5 × 50 = 250 g.

面粉占 5 份，所以需要面粉 = 5 × 50 = 250 克。

ctions: 3/5 = 150/x. Cross-multiply: 3x = 5 × 150 → 3x = 750 → x = 250. Both methods work.

也可以用等值分数来解：3/5 = 150/x，交叉相乘：3x = 5 × 150 → 3x = 750 → x = 250。两种方法均可。

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10. nth Term of a Sequence | 序列的第 n 项

Question: A sequence begins 3, 7, 11, 15, … Find an expression for the nth term.

题目：某序列的前几项为 3, 7, 11, 15, … 求第 n 项的表达式。

Check the difference between consecutive terms: 7 – 3 = 4, 11 – 7 = 4, 15 – 11 = 4. The common difference is 4, so this is an arithmetic sequence.

检查相邻两项的差：7 – 3 = 4，11 – 7 = 4，15 – 11 = 4。公差为 4，因此这是一个等差数列。

neral rule for an arithmetic sequence is: nth term = (common difference) × n + adjustment.

等差数列的通项公式为：第 n 项 = 公差 × n + 调整值。

→ 11, n=4 → 15. Notice that 4 × 1 = 4, but we need 3, so subtract 1. So the nth term = 4n – 1.

给各项标上位置编号：n=1 → 3，n=2 → 7，n=3 → 11，n=4 → 15。注意到 4 × 1 = 4，但我们需要 3，所以要减 1。因此第 n 项 = 4n – 1。

Test it: when n=1, 4(1)–1 = 3; n=2, 4(2)–1 = 7. It works perfectly.

检验：当 n=1 时，4(1)–1 = 3；n=2 时，4(2)–1 = 7，完全吻合。

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11. Common Mistakes and Tips | 常见错误与应对技巧

Mistake: Forgetting to simplify fractions completely. Always check if both numerator and denominator can be divided by the same number again.

错误：分数没有化简到最简。始终检查分子分母是否还能继续除以同一个数。

+ 2b = 5ab, which is incorrect. You can only combine terms with identical variable parts.

错误：代数中将不同类项相加，如 3a + 2b = 5ab，这是错误的。只能合并变量部分完全一样的项。

Mistake: Confusing area and perimeter formulas. Always refer to the formula sheet if allowed, and label units correctly (cm² for area, cm for perimeter).

错误：混淆面积和周长的公式。如果允许，随时查阅公式表，并正确标注单位（面积用 cm²，周长用 cm）。

Tip: Show all your working. Even if the final answer is wrong, you can earn marks for correct methods.

技巧：展示完整的解题步骤。即使最后答案错了，正确的解法也能获得步骤分。

Tip: Manage your time. Start with the questions you find easiest, then return to harder ones. This builds confidence.

技巧：管理好时间。先做你觉得最简单的题，再回头做难题，这样可以建立信心。

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