📚 Year 9 WJEC Mathematics: Mock Unit Test Analysis and Solutions | Year 9 WJEC 数学：单元测试模拟卷解析
This article provides a thorough walkthrough of a typical Year 9 WJEC Mathematics unit test mock paper. Each section presents a representative question, followed by step-by-step solutions, key concepts, and common pitfalls. The aim is to help you understand the core topics and build confidence for your real assessment. Use this revision guide to strengthen your problem-solving skills across Number, Algebra, Geometry, Statistics, and Ratio.
本文对一份典型的 Year 9 WJEC 数学单元测试模拟卷进行了详细解析。每个部分都包含一道代表性题目，并展示了分步解题过程、关键概念和常见错误。目的是帮助你理解核心知识点，为真正的考试树立信心。用这份复习指南来提升你在数、代数、几何、统计和比与比例方面的解题能力。
1. Number Skills and BIDMAS | 数字计算与运算顺序
Question: Work out 20 − 4 × 3 + 12 ÷ (2 + 1).
题目：计算 20 − 4 × 3 + 12 ÷ (2 + 1)。
Step 1: Apply BIDMAS – Brackets first. Calculate (2 + 1) = 3. The expression becomes 20 − 4 × 3 + 12 ÷ 3.
步骤 1：运用 BIDMAS 法则——先算括号。计算 (2 + 1) = 3。表达式变为 20 − 4 × 3 + 12 ÷ 3。
Step 2: There are no Indices, so move to Division and Multiplication, working left to right. First, 4 × 3 = 12. Next, 12 ÷ 3 = 4. The expression now reads 20 − 12 + 4.
步骤 2：没有指数，接下来从左到右计算除法和乘法。先算 4 × 3 = 12，再算 12 ÷ 3 = 4。现在表达式为 20 − 12 + 4。
Step 3: Finally, perform Addition and Subtraction from left to right. 20 − 12 = 8, then 8 + 4 = 12.
步骤 3：最后，从左到右进行加法和减法运算。20 − 12 = 8，然后 8 + 4 = 12。
Common mistake: Some students incorrectly do 20 − (4 × 3) + (12 ÷ 3) but then work 20 − 12 + 4 as 20 − 16 = 4. Remember that addition and subtraction have equal priority and must be performed left to right.
常见错误：有的学生虽然正确分组为 20 − 12 + 4，却错误地先算了 12 + 4 = 16，再用 20 − 16 = 4。请记住，加法和减法优先级相同，必须从左到右计算。
2. Fractions, Decimals and Percentages | 分数、小数与百分比
Question: Write 3/8 as a decimal and as a percentage. Then decide which is larger: 3/8 or 0.35.
题目：将 3/8 写成小数和百分比。然后判断 3/8 和 0.35 哪个更大。
Step 1: To convert a fraction to a decimal, divide the numerator by the denominator. 3 ÷ 8 = 0.375. A quick shortcut is to know that 1/8 = 0.125, so 3/8 = 0.375.
步骤 1：要将分数转换为小数，用分子除以分母。3 ÷ 8 = 0.375。一个快捷方式是记住 1/8 = 0.125，所以 3/8 = 0.375。
Step 2: To express as a percentage, multiply the decimal by 100. 0.375 × 100 = 37.5%.
步骤 2：要表示为百分比，将小数乘以 100。0.375 × 100 = 37.5%。
Step 3: Compare 3/8 and 0.35. Since 3/8 = 0.375, and 0.375 > 0.35, so 3/8 is the larger number.
步骤 3：比较 3/8 和 0.35。因为 3/8 = 0.375，且 0.375 > 0.35，所以 3/8 更大。
Key skill: Being able to fluently convert between fractions, decimals and percentages is essential for topics like probability and data interpretation. Practise recognising common equivalents such as 1/4 = 0.25 = 25% and 1/5 = 0.2 = 20%.
关键技能：能够熟练地在分数、小数和百分比之间进行转换，对概率和数据解读等主题至关重要。练习识别常见的等价形式，如 1/4 = 0.25 = 25%，1/5 = 0.2 = 20%。
3. Simplifying Algebraic Expressions | 化简代数表达式
Question: Simplify 5a − 3b + 2a + 7b − a.
题目：化简 5a − 3b + 2a + 7b − a。
Step 1: Identify like terms. Terms containing ‘a’ are 5a, +2a, and −a. Terms containing ‘b’ are −3b and +7b.
步骤 1：识别同类项。含 a 的项有 5a、+2a 和 −a。含 b 的项有 −3b 和 +7b。
Step 2: Combine the ‘a’ terms by adding their coefficients: 5 + 2 − 1 = 6, giving 6a. Combine the ‘b’ terms: −3 + 7 = 4, giving 4b.
步骤 2：合并 a 项，将它们的系数相加：5 + 2 − 1 = 6，得到 6a。合并 b 项：−3 + 7 = 4，得到 4b。
Step 3: Write the simplified expression as 6a + 4b. You can factorise further to 2(3a + 2b), but either form is acceptable unless the question specifies fully factorised form.
步骤 3：写出化简后的表达式 6a + 4b。你还可以进一步因式分解为 2(3a + 2b)，但除非题目要求写成完全因式分解的形式，否则两种写法都是可以的。
Crucial reminder: The sign in front of a term belongs to that term. So, −a means the coefficient is −1. Losing track of negative signs is one of the most frequent errors in algebra.
重要提醒：项前面的符号属于该项。所以 −a 表示系数为 −1。搞丢负号是代数中最常见的错误之一。
4. Solving Linear Equations | 解线性方程
Question: Solve 4(x + 3) = 28.
题目：解方程 4(x + 3) = 28。
Method 1 – Expand first: Multiply out the bracket: 4x + 12 = 28. Then subtract 12 from both sides: 4x = 16. Finally, divide by 4: x = 4.
方法 1——先去括号：将括号展开：4x + 12 = 28。然后等式两边同时减去 12：4x = 16。最后除以 4：x = 4。
Method 2 – Divide first: Notice that 4 is a factor of 28. Divide both sides by 4: (x + 3) = 7. Then subtract 3: x = 4. This method is more efficient when the coefficient divides the constant neatly.
方法 2——先除以系数：注意到 4 是 28 的因数。两边同时除以 4：(x + 3) = 7。然后减去 3：x = 4。当系数能整除常数时，这种方法更高效。
Always check your solution by substituting x = 4 back into the original equation: 4(4 + 3) = 4 × 7 = 28, which is correct.
始终将 x = 4 代回原方程检验：4(4 + 3) = 4 × 7 = 28，结果正确。
Common exam error: When expanding, some students incorrectly write 4x + 3 instead of 4x + 12. Remember that the 4 multiplies every term inside the bracket.
常见考试错误：去括号时，一些学生错误地写成 4x + 3 而不是 4x + 12。请记住，系数 4 要与括号内的每一项相乘。
5. Angles in Polygons and Parallel Lines | 多边形与平行线中的角度
Question: In a triangle, two of the angles are 48° and 67°. Find the size of the third angle. Then, given that a straight line intersects a pair of parallel lines, one of the alternate interior angles is 55°, find the other.
题目：在一个三角形中，有两个角分别是 48° 和 67°。计算第三个角的大小。然后，已知一条直线与一组平行线相交，其中一对内错角之一为 55°，求另一个角。
Triangle part: The sum of angles in any triangle is 180°. Add the two given angles: 48° + 67° = 115°. Subtract from 180°: 180° − 115° = 65°. The third angle is 65°.
三角形部分：任何三角形的内角和都是 180°。将两个已知角相加：48° + 67° = 115°。用 180° 减去这个和：180° − 115° = 65°。第三个角是 65°。
Parallel lines part: When a transversal cuts two parallel lines, alternate interior angles are equal. Therefore, if one alternate angle is 55°, the other alternate angle is also 55°.
平行线部分：当一条截线与两条平行线相交时，内错角相等。因此，如果一个内错角是 55°，另一个内错角也是 55°。
It is vital to know the angle properties: vertically opposite angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°. Always state the reason clearly in WJEC-style ‘give a reason’ questions.
务必掌握各种角的性质：对顶角相等，同位角相等，同旁内角互补（和为 180°）。在 WJEC 常考的“说明理由”类题目中，要清晰地陈述理由。
6. Perimeter, Area and Volume | 周长、面积与体积
Question: A rectangle has a length of 8 cm and a width of 5 cm. Calculate its perimeter and area. A cuboid has the same length and width, and a height of 3 cm. Find its volume.
题目：一个长方形的长是 8 cm，宽是 5 cm。计算它的周长和面积。一个长方体有同样的长和宽，高是 3 cm。求它的体积。
Perimeter of rectangle: P = 2(length + width) = 2(8 + 5) = 2 × 13 = 26 cm.
长方形周长：P = 2 × (长 + 宽) = 2 × (8 + 5) = 2 × 13 = 26 cm。
Area of rectangle: A = length × width = 8 × 5 = 40 cm². Note that area units are always squared.
长方形面积：A = 长 × 宽 = 8 × 5 = 40 cm²。注意面积单位总是带平方。
Volume of cuboid: V = length × width × height = 8 × 5 × 3 = 120 cm³. Volume units are cubed.
长方体体积：V = 长 × 宽 × 高 = 8 × 5 × 3 = 120 cm³。体积单位是立方。
Many marks are lost by forgetting to include the correct units. Always check whether the answer is a length, area or volume, and write cm, cm² or cm³ accordingly. If the question mixes units (e.g. mm and cm), convert all measurements to the same unit first.
很多分数因为忘记写正确的单位而丢失。始终检查答案是长度、面积还是体积，并相应地写上 cm、cm² 或 cm³。如果题目混合了不同单位（如 mm 和 cm），要先把所有测量值转换成相同的单位。
7. Sequences and nth Term | 数列与第 n 项
Question: The nth term of a sequence is given by the rule 3n + 2. Write down the first three terms and find the 20th term. Is 101 a term in this sequence? Explain.
题目：某个数列的第 n 项公式为 3n + 2。写出该数列的前三项，并求出第 20 项。101 是这个数列中的一项吗？请解释。
First term (n = 1): 3(1) + 2 = 5.
第一项 (n = 1)：3(1) + 2 = 5。
Second term (n = 2): 3(2) + 2 = 8. Third term (n = 3): 3(3) + 2 = 11. The sequence begins 5, 8, 11, …
第二项 (n = 2)：3(2) + 2 = 8。第三项 (n = 3)：3(3) + 2 = 11。数列开始为 5, 8, 11, ……
20th term: Substitute n = 20 into the rule: 3(20) + 2 = 60 + 2 = 62.
第 20 项：将 n = 20 代入公式：3(20) + 2 = 60 + 2 = 62。
To check if 101 is in the sequence, set 3n + 2 = 101. Solve: 3n = 99, n = 33. Because n is a positive whole number, 101 is the 33rd term. If the result were not an integer, the number would not be a term.
要检查 101 是否在数列中，令 3n + 2 = 101。求解：3n = 99，n = 33。因为 n 是一个正整数，所以 101 是第 33 项。如果结果不是整数，该数就不是数列中的项。
Understanding how to generate terms, find a specific term, and check term membership is fundamental to linear sequences. As the sequence grows, the constant difference is the coefficient of n (here 3).
理解如何生成项、找到特定项以及检查某项是否在数列中，是掌握线性数列的基础。随着数列逐项增加，各项的差是常数且等于 n 的系数（这里是 3）。
8. Averages and Range | 平均数与极差
Question: Here are the marks of 5 students: 5, 8, 5, 12, 10. Find the mean, median, mode and range.
题目：以下是 5 名学生的分数：5, 8, 5, 12, 10。求平均数、中位数、众数和极差。
Mean: Add all values: 5 + 8 + 5 + 12 + 10 = 40. Divide by the number of values: 40 ÷ 5 = 8. The mean is 8.
平均数：将所有数值相加：5 + 8 + 5 + 12 + 10 = 40。除以数值的个数：40 ÷ 5 = 8。平均数是 8。
Median: Arrange in order: 5, 5, 8, 10, 12. The middle value is the third number, which is 8. Median = 8.
中位数：按顺序排列：5, 5, 8, 10, 12。中间的数是第三个，即 8。中位数为 8。
Mode: The number that appears most often is 5 (it occurs twice). Mode = 5.
众数：出现次数最多的数是 5（出现了两次）。众数为 5。
Range: Subtract the smallest from the largest: 12 − 5 = 7. The range is 7.
极差：最大值减最小值：12 − 5 = 7。极差为 7。
In WJEC exams, you may need to compare two data sets using mean and range. A larger mean generally indicates higher values on average, while a smaller range shows more consistency. Always label your answers clearly.
在 WJEC 考试中，你可能需要利用平均数和极差来比较两组数据。较大的平均数通常表示平均数值更高，而较小的极差表明数据更稳定。始终清晰地标注答案。
9. Probability Scale and Simple Events | 概率尺度与简单事件
Question: On a probability scale from 0 to 1, mark the probability of rolling an even number on a standard fair 6-sided dice. Then, a bag contains 3 red, 5 blue and 2 green counters. What is the probability of picking a blue counter?
题目：在 0 到 1 的概率尺度上，标出掷一枚标准公平六面骰子得到偶数的概率。然后，一个袋子里有 3 个红色、5 个蓝色和 2 个绿色筹码。抽到蓝色筹码的概率是多少？
Dice probability: There are 6 equally likely outcomes. Even numbers are 2, 4, 6 – that is 3 favourable outcomes. Probability = 3/6 = 1/2 = 0.5. On the probability scale, mark the midpoint between 0 and 1, and label it ‘0.5’ or ‘evens’.
骰子的概率：共有 6 种等可能的结果。偶数是 2、4、6——共 3 种有利结果。概率 = 3/6 = 1/2 = 0.5。在概率尺度上，在 0 和 1 的中点做标记，并标注 ‘0.5’ 或 ‘evens’。
Counter probability: Total number of counters = 3 + 5 + 2 = 10. Number of blue counters = 5. Probability(blue) = 5/10 = 1/2 = 0.5 or 50%.
筹码的概率：筹码总数 = 3 + 5 + 2 = 10。蓝色筹码数 = 5。概率（蓝色）= 5/10 = 1/2 = 0.5 或 50%。
Probability can be expressed as a fraction, decimal or percentage, but the question often specifies the required form. Always simplify fractions where possible. Events with probability 0.5 are just as likely to happen as not.
概率可以用分数、小数或百分比表示，但题目通常会指明需要的形式。分数要尽可能化简。概率为 0.5 的事件表示发生与不发生的可能性相同。
10. Ratio and Proportion | 比与比例
Question: Divide £60 between two friends in the ratio 3:2. How much does each friend receive?
题目：将 60 英镑按 3:2 的比例分给两个朋友。每人各得多少？
Step 1: Find the total number of parts. The ratio 3:2 means there are 3 + 2 = 5 equal parts in total.
步骤 1：求出总份数。比例 3:2 意味着总共有 3 + 2 = 5 等份。
Step 2: Calculate the value of one part by dividing the total amount by the total parts: £60 ÷ 5 = £12 per part.
步骤 2：用总额除以总份数，计算一份的价值：£60 ÷ 5 = £12/份。
Step 3: Multiply to find each share. First friend gets 3 parts: 3 × £12 = £36. Second friend gets 2 parts: 2 × £12 = £24. Always check that the sum is £36 + £24 = £60.
步骤 3：相乘求出每人所得。第一个朋友得到 3 份：3 × £12 = £36。第二个朋友得到 2 份：2 × £12 = £24。务必检查总额：£36 + £24 = £60。
This unitary method is powerful for any ratio problem. WJEC often includes ratio in recipes, maps and scale drawing contexts. Keep your working organized by clearly showing the value of one part.
这种“归一法”对任何比例问题都非常有效。WJEC 经常在食谱配方、地图和比例尺绘图等情境中考查比例。解题时清晰展示一份的价值，可使过程条理分明。
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