📚 Year 7 WJEC Further Mathematics: Unit Test Mock Paper Walkthrough | 7年级WJEC进阶数学：单元测试模拟卷解析
This article presents a detailed walkthrough of a mock unit test designed for Year 7 WJEC Further Mathematics. Each question is solved step by step, with clear explanations in both English and Chinese to help you master the techniques and build your confidence for real assessments.
本文为7年级WJEC进阶数学模拟单元测试提供完整解析。每道题都给出了分步求解过程，并用中英双语详细讲解，帮助你掌握解题方法，为真正的考试建立信心。
1. Prime Factorisation | 质因数分解
Question: Write 72 as a product of its prime factors.
题目：将72写成质因数乘积的形式。
To find the prime factors, start by dividing 72 by the smallest prime number, 2.
为找出质因数，先用最小的质数2去除72。
72 ÷ 2 = 36. Since 36 is even, divide by 2 again: 36 ÷ 2 = 18.
72 ÷ 2 = 36。因为36是偶数，继续除以2：36 ÷ 2 = 18。
18 ÷ 2 = 9. Now 9 is not divisible by 2, so we move to the next prime number, 3.
18 ÷ 2 = 9。现在9不能被2整除，所以我们用下一个质数3。
9 ÷ 3 = 3, and 3 ÷ 3 = 1. We stop when we reach 1.
9 ÷ 3 = 3，再 3 ÷ 3 = 1。当得到1时停止。
. Therefore, 72 = 2 × 2 × 2 × 3 × 3, which can be written concisely using powers: 2³ × 3².
我们使用的质因数有2, 2, 2, 3, 3。因此，72 = 2 × 2 × 2 × 3 × 3，可以简洁地写成乘方形式：2³ × 3²。
Always check by multiplying: 2³ = 8, 3² = 9, 8 × 9 = 72, correct.
一定要检验：2³ = 8, 3² = 9, 8 × 9 = 72，正确。
2. Simplifying and Substituting | 化简与代入求值
Simplify the expression 5a + 3b - 2a + 7b, and then find its value when a = 3 and b = -2.
题目：化简表达式 5a + 3b - 2a + 7b，然后求当 a = 3 且 b = -2 时的值。
First, identify like terms. The terms containing a are 5a and -2a. The terms containing b are 3b and 7b.
首先，找出同类项。含有a的项是5a和-2a。含有b的项是3b和7b。
Combine the a terms: 5a - 2a = 3a.
合并a项：5a - 2a = 3a。
Combine the b terms: 3b + 7b = 10b.
合并b项：3b + 7b = 10b。
The simplified expression is 3a + 10b.
化简后的表达式为 3a + 10b。
Now substitute a = 3 and b = -2 into the simplified expression: 3(3) + 10(-2).
现在将 a = 3 和 b = -2 代入化简后的表达式：3(3) + 10(-2)。
Calculate each part: 3 × 3 = 9, and 10 × (-2) = -20.
计算各部分：3 × 3 = 9，10 × (-2) = -20。
Add them together: 9 + (-20) = -11.
将它们相加：9 + (-20) = -11。
The value of the expression is -11.
表达式的值为 -11。
3. Solving a Linear Equation | 解一元一次方程
Question: Solve the equation 3(y - 4) = 15.
题目：解方程 3(y - 4) = 15。
Begin by expanding the left-hand side: 3 × y and 3 × (-4).
首先展开左边：3 × y 和 3 × (-4)。
This gives 3y - 12 = 15.
得到 3y - 12 = 15。
To isolate the term with y, add 12 to both sides of the equation.
为了分离含y的项，在等式两边同时加12。
3y - 12 + 12 = 15 + 12, which simplifies to 3y = 27.
3y - 12 + 12 = 15 + 12，简化为 3y = 27。
Now divide both sides by 3: y = 27 ÷ 3.
现在两边同除以3：y = 27 ÷ 3。
Thus, y = 9.
因此，y = 9。
back into the original equation: 3(9 - 4) = 3 × 5 = 15, which matches the right-hand side.
把9代回原方程检验：3(9 - 4) = 3 × 5 = 15，与右边相等。
4. Angles in a Triangle | 三角形内角
Question: The angles of a triangle are (2x + 10)°, (3x)° and (x + 50)°. Find the value of x and the size of each angle.
题目：三角形三个内角的度数分别为 (2x + 10)°、(3x)° 和 (x + 50)°。求x的值及各角的度数。
Recall that the sum of interior angles in any triangle is 180°.
回忆一下，任何三角形的内角和都是180°。
Set up an equation: (2x + 10) + (3x) + (x + 50) = 180.
列出方程：(2x + 10) + (3x) + (x + 50) = 180。
Combine like terms on the left: 2x + 3x + x = 6x, and 10 + 50 = 60.
合并左边同类项：2x + 3x + x = 6x，10 + 50 = 60。
The equation becomes 6x + 60 = 180.
方程变为 6x + 60 = 180。
Subtract 60 from both sides: 6x = 120.
两边都减去60：6x = 120。
Divide both sides by 6: x = 120 ÷ 6 = 20.
两边除以6：x = 120 ÷ 6 = 20。
Now substitute x = 20 to find each angle: first angle = 2(20) + 10 = 40 + 10 = 50°.
现在代入 x = 20 求各角：第一个角 = 2(20) + 10 = 40 + 10 = 50°。
Second angle = 3(20) = 60°.
第二个角 = 3(20) = 60°。
Third angle = 20 + 50 = 70°.
第三个角 = 20 + 50 = 70°。
Check the sum: 50 + 60 + 70 = 180°, which confirms our solution.
检验总和：50 + 60 + 70 = 180°，确认解答正确。
5. Area of a Composite Shape | 组合图形面积
Question: A shape is made from a rectangle of length 8 cm and width 5 cm, with a right-angled triangle attached along one of the 8 cm sides. The triangle has a base of 8 cm and a height of 4 cm. Find the total area of the shape.
题目：一个图形由一个长8 cm、宽5 cm的矩形和一个直角三角形组成，三角形沿着矩形的一条8 cm边贴附。三角形底边8 cm，高4 cm。求该图形的总面积。
Start by calculating the area of the rectangle: length × width = 8 × 5 = 40 cm².
先计算矩形的面积：长 × 宽 = 8 × 5 = 40 cm²。
Next, calculate the area of the right-angled triangle: 1/2 × base × height = 1/2 × 8 × 4.
接着，计算直角三角形的面积：1/2 × 底 × 高 = 1/2 × 8 × 4。
1/2 of 8 is 4, and 4 × 4 = 16, so the triangle area is 16 cm².
8的一半是4，4 × 4 = 16，所以三角形面积为16 cm²。
The total area is the sum of the two areas: 40 + 16 = 56 cm².
总面积等于两者之和：40 + 16 = 56 cm²。
It is helpful to draw the shape and label sides to avoid using wrong measurements.
画出图形并标注边长有助于避免用错数据。
6. Ratio and Proportion | 比与比例分配
Question: Share £120 in the ratio 2 : 3 : 5.
题目：将120英镑按2 : 3 : 5的比例分配。
First, find the total number of parts in the ratio: 2 + 3 + 5 = 10 parts.
首先，求出比例的总份数：2 + 3 + 5 = 10份。
Each part is worth the total amount divided by the total parts: 120 ÷ 10 = £12 per part.
每一份的价值是总额除以总份数：120 ÷ 10 = 每份12英镑。
For the first share, multiply the number of parts by the value per part: 2 × 12 = £24.
第一份：份数乘以每份价值：2 × 12 = 24英镑。
Second share: 3 × 12 = £36.
第二份：3 × 12 = 36英镑。
Third share: 5 × 12 = £60.
第三份：5 × 12 = 60英镑。
Always verify the sum: 24 + 36 + 60 = £120, which matches the original amount.
务必验证总和：24 + 36 + 60 = 120英镑，与原始金额一致。
7. Interpreting Pie Charts | 饼图解读
Question: A pie chart represents students' favourite colours. The angles are: Red 90°, Blue 120°, Green 60°. The rest is Yellow. If there are 36 students in total, how many chose Yellow?
题目：一个饼图展示了学生们最喜欢的颜色。已知各部分的圆心角为：红色90°，蓝色120°，绿色60°，其余为黄色。如果总共有36名学生，选择黄色的有多少人？
has 360°. First, calculate the total angle for Red, Blue and Green: 90 + 120 + 60 = 270°.
一个整圆的圆心角为360°。先计算红、蓝、绿色所占的角度和：90 + 120 + 60 = 270°。
The angle for Yellow is 360 - 270 = 90°.
黄色对应的圆心角为 360 - 270 = 90°。
The fraction of the circle for Yellow is 90/360, which simplifies to 1/4.
黄色所占圆心角的比例是 90/360，化简为 1/4。
Therefore, the number of students who chose Yellow is 1/4 of 36: 36 ÷ 4 = 9 students.
因此，选择黄色的学生人数是36人的1/4：36 ÷ 4 = 9人。
360 = 9, Blue 120° gives 36 × 120/360 = 12, Green 60° gives 6, and Yellow 9, totalling 36.
快速检验：红色90°对应 36 × 90/360 = 9人，蓝色120°对应12人，绿色60°对应6人，黄色9人，总计36人。
8. Finding the nth Term of a Sequence | 找出数列的第n项
Question: Find the nth term of the arithmetic sequence 5, 11, 17, 23, ...
题目：求等差数列 5, 11, 17, 23, ... 的第n项。
tween consecutive terms: 11 - 5 = 6, 17 - 11 = 6, 23 - 17 = 6. The common difference is 6.
观察相邻两项的差：11 - 5 = 6，17 - 11 = 6，23 - 17 = 6。公差为6。
an arithmetic sequence can be written as: term = first term + (n - 1) × common difference.
等差数列第n项的公式为：项 = 首项 + (n - 1) × 公差。
Substitute the values: 5 + (n - 1) × 6.
代入数值：5 + (n - 1) × 6。
Expand: 5 + 6n - 6 = 6n - 1.
展开：5 + 6n - 6 = 6n - 1。
So the nth term is given by 6n - 1.
因此，第n项为 6n - 1。
st for n = 1: 6(1) - 1 = 5 (first term correct); n = 2: 12 - 1 = 11 (second term correct).
检验：n=1时，6(1) - 1 = 5（首项正确）；n=2时，12 - 1 = 11（第二项正确）。
9. Operations with Negative Numbers | 负数运算
Question: Work out: (-4) × (-6) + (-3) × 5.
题目：计算：(-4) × (-6) + (-3) × 5。
Follow the order of operations: multiplication before addition.
遵循运算顺序：先乘除，后加减。
First, calculate (-4) × (-6). A negative times a negative gives a positive: 4 × 6 = 24.
首先计算 (-4) × (-6)。负数乘负数得正数：4 × 6 = 24。
(-3) × 5. A negative times a positive gives a negative: 3 × 5 = 15, so the product is -15.
接着计算 (-3) × 5。负数乘正数得负数：3 × 5 = 15，因此积为 -15。
Now add the results: 24 + (-15). This is the same as 24 - 15 = 9.
现在将结果相加：24 + (-15)。等同于 24 - 15 = 9。
The final answer is 9.
最终答案为 9。
Remember: if the signs are the same, the product is positive; if they are different, the product is negative.
记住：同号得正，异号得负。
10. Speed, Distance and Time | 速度、距离与时间
Question: A car travels 150 km in 2 hours 30 minutes. Find its average speed in km/h. If it continues at this speed, how long will it take to travel 100 km? Give your answer in hours and minutes.
题目：一辆汽车行驶150公里用时2小时30分钟。求其平均速度，单位 km/h。如果以此速度继续行驶，行驶100公里需要多长时间？用小时和分钟表示答案。
onvert the time to hours only. 30 minutes is 0.5 hours, so 2 hours 30 minutes = 2.5 hours.
首先，把时间统一用小时表示。30分钟是0.5小时，所以2小时30分 = 2.5小时。
Average speed = total distance ÷ total time = 150 ÷ 2.5.
平均速度 = 总距离 ÷ 总时间 = 150 ÷ 2.5。
ultiply numerator and denominator by 10: 1500 ÷ 25 = 60. So, the average speed is 60 km/h.
为简化除法，分子分母同乘10：1500 ÷ 25 = 60。因此，平均速度为60 km/h。
Now, for 100 km at 60 km/h, time = distance ÷ speed = 100 ÷ 60.
现在，以60 km/h的速度行驶100公里，时间 = 距离 ÷ 速度 = 100
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