📚 Cambridge Lower Secondary Mathematics Workbook 8 Answers | 剑桥初中数学练习册 8 答案与解析
This article provides a structured guide to using the Cambridge Lower Secondary Mathematics Workbook 8 answers for effective revision. It covers worked solutions, common mistakes, and exam-style practice across the main topics of the CIE Lower Secondary Mathematics Stage 8 curriculum.
本文提供结构化指南，帮助有效利用《Cambridge Lower Secondary Mathematics Workbook 8》答案进行复习。内容涵盖 CIE 初中数学 Stage 8 课程主要专题的例题答案、常见错误与考试风格练习。
1. Understanding the Workbook Structure | 了解练习册结构
The Workbook 8 is organised into units covering number, algebra, geometry, measures, statistics and probability. Each unit ends with mixed exercises or review questions. Answers are usually listed by exercise number, so always note the unit, exercise and question number before checking.
练习册 8 按单元编排，涵盖数、代数、几何、测量、统计与概率。每个单元通常以混合练习或复习题结尾。答案一般按练习编号整理，因此检查前务必记录单元、练习和题号。
- Number and Place Value | 数与位值
- Fractions, Decimals and Percentages | 分数、小数与百分比
- Algebra and Equations | 代数与方程
- Geometry and Measures | 几何与测量
- Statistics and Probability | 统计与概率
- Ratio, Proportion and Rates of Change | 比、比例与变化率
Many answer keys in the workbook show only the final answer. For method-based questions, you should still write full working, because CIE assessments often award method marks even if the final answer is wrong.
练习册中的许多答案只显示最终结果。对于过程题，仍应写出完整步骤，因为 CIE 测评通常即使最终答案错误也会给步骤分。
2. How to Use the Answers Effectively | 如何高效使用答案
Do not copy answers directly. Attempt the question first, then compare your working with the answer. Answers should be used as a checking tool to identify gaps in understanding, not as a shortcut.
不要直接抄答案。先独立作答，再对照答案。答案应作为检查工具，找出理解漏洞，而不是走捷径。
For any incorrect answer, cover the solution and rework the problem without looking. Then check again. Keep an error log with the topic, question type and the mistake you made. This turns the answer key into a personalised revision plan.
对于错题，遮住答案重新计算，再检查一次。建立错题记录，标明主题、题型和错误类型。这能让答案变成个性化复习计划。
Always write your answer in the same form as the question requests. If the question asks for a decimal, give a decimal; if it asks for a fraction in simplest form, cancel fully.
始终按照题目要求的格式作答。如果题目要求小数，就写小数；如果要求最简分数，就完全约分。
3. Number and Place Value: Worked Solutions | 数与位值：例题答案
Example 1: Evaluate 64 ÷ 8 + 3 × 2.
例1：计算 64 ÷ 8 + 3 × 2。
plication are done before addition, so first calculate 64 ÷ 8 = 8 and 3 × 2 = 6, then add.
解：按运算顺序，先算除法和乘法再算加法。先求 64 ÷ 8 = 8，3 × 2 = 6，然后相加。
64 ÷ 8 + 3 × 2 = 8 + 6 = 14
Example 2: Write 2³ × 2² as a single power of 2.
例2：将 2³ × 2² 写成 2 的幂的形式。
Solution: When multiplying powers with the same base, add the exponents: 3 + 2 = 5.
解：同底数幂相乘，指数相加：3 + 2 = 5。
2³ × 2² = 2⁵
Example 3: Round 345.678 to 2 significant figures.
例3：将 345.678 保留 2 位有效数字。
Solution: The first two significant figures are 3 and 4. The next digit is 5, so round up the 4 to 5. The answer is 350, because the zero is needed to keep the place value.
解：前两位有效数字是 3 和 4。下一位是 5，所以将 4 进位到 5。答案是 350，因为需要 0 来保持数位。
345.678 ≈ 350 (2 s.f.)
4. Fractions, Decimals and Percentages: Worked Solutions | 分数、小数与百分比：例题答案
Example 1: Convert 3/8 to a decimal and a percentage.
例1：将 3/8 转换为小数和百分比。
by the denominator: 3 ÷ 8 = 0.375. To convert a decimal to a percentage, multiply by 100.
解：分子除以分母：3 ÷ 8 = 0.375。把小数转换为百分比，乘以 100。
3/8 = 0.375 = 37.5%
Example 2: Find 15% of 240.
例2：求 240 的 15%。
Solution: Write 15% as 0.15 and multiply by 240. Alternatively, find 10% first, then 5%, and add the results.
解：把 15% 写成 0.15，再乘以 240。也可以先求 10%，再求 5%，然后把结果相加。
0.15 × 240 = 36
Example 3: Work out 2/3 + 1/4. Give your answer as a mixed number.
例3：计算 2/3 + 1/4，结果写成带分数。
nd 4 is 12, so convert both fractions: 2/3 = 8/12 and 1/4 = 3/12. Then add the numerators.
解：先找公分母。3 和 4 的最小公倍数是 12，因此转换：2/3 = 8/12，1/4 = 3/12。再把分子相加。
8/12 + 3/12 = 11/12
5. Algebra and Equations: Worked Solutions | 代数与方程：例题答案
Example 1: Solve 5x + 3 = 18.
例1：解方程 5x + 3 = 18。
3 from both sides to isolate the term containing x: 5x = 15. Then divide both sides by 5.
解：两边同时减去 3，得到 5x = 15。然后两边同时除以 5。
5x + 3 = 18 → 5x = 15 → x = 3
Example 2: Expand and simplify 3(x + 4) − 2(x − 1).
例2：展开并化简 3(x + 4) − 2(x − 1)。
Solution: Multiply out the brackets. Remember that −2 multiplied by −1 gives +2. Then collect like terms.
解：先展开括号。注意 −2 乘以 −1 得到 +2。然后合并同类项。
3x + 12 − 2x + 2 = x + 14
Example 3: Simplify the expression 4a + 3b − 2a + 5b.
例3：化简表达式 4a + 3b − 2a + 5b。
Solution: Collect the a terms and the b terms separately: 4a − 2a = 2a and 3b + 5b = 8b.
解：分别合并 a 项和 b 项：4a − 2a = 2a，3b + 5b = 8b。
4a + 3b − 2a + 5b = 2a + 8b
6. Geometry and Measures: Worked Solutions | 几何与测量：例题答案
Example 1: Find the area of a triangle with base 8 cm and height 5 cm.
例1：求底为 8 cm、高为 5 cm 的三角形的面积。
formula for the area of a triangle: Area = ½ × base × height. Substitute the given values.
解：使用三角形面积公式：面积 = ½ × 底 × 高。代入已知数值。
Area = ½ × 8 × 5 = 20 cm²
Example 2: A triangle has angles 65° and 48°. Find the third angle.
例2：三角形的两个角分别为 65° 和 48°。求第三个角。
Solution: The sum of angles in a triangle is 180°. Add the two known angles, then subtract from 180°.
解：三角形内角和为 180°。先加两个已知角，再用 180° 减去。
180° − 65° − 48° = 67°
Example 3: Find the volume of a cuboid with length 6 cm, width 4 cm and height 3 cm.
例3：求长 6 cm、宽 4 cm、高 3 cm 的长方体的体积。
Solution: Use the formula Volume = length × width × height. Multiply the three dimensions.
解：使用体积公式：体积 = 长 × 宽 × 高。将三个尺寸相乘。
Volume = 6 × 4 × 3 = 72 cm³
7. Statistics and Probability: Worked Solutions | 统计与概率：例题答案
Example 1: Find the mean of 12, 15, 18, 21 and 24.
例1：求 12、15、18、21 和 24 的平均数。
Solution: Add all the values together, then divide by the number of values. The sum is 90 and there are 5 values.
解：把所有数相加，再除以数据的个数。总和为 90，共有 5 个数。
Mean = 90 ÷ 5 = 18
Example 2: A fair six-sided dice is rolled once. Find the probability of rolling a prime number.
例2：掷一次公平六面骰子。求掷出质数的概率。
Solution: The prime numbers on a dice are 2, 3 and 5, so there are 3 favourable outcomes out of 6 possible outcomes.
解：骰子上的质数是 2、3 和 5，因此有 3 个有利结果，总共有 6 个可能结果。
P(prime) = 3/6 = 1/2
Example 3: A bag contains 3 red, 5 blue and 2 green counters. One counter is taken at random. Find the probability that it is not red.
例3：袋中有 3 个红色、5 个蓝色和 2 个绿色筹码。随机取出一个，求它不是红色的概率。
number of counters: 3 + 5 + 2 = 10. The number of counters that are not red is 5 + 2 = 7.
解：先求筹码总数：3 + 5 + 2 = 10。不是红色的筹码数为 5 + 2 = 7。
P(not red) = 7/10
8. Ratio, Proportion and Rates of Change: Worked Solutions | 比、比例与变化率：例题答案
Example 1: Simplify the ratio 24 : 36.
例1：化简比 24 : 36。
Solution: Divide both sides by the highest common factor, which is 12.
解：两边同时除以最大公因数 12。
24 : 36 = 2 : 3
Example 2: If 5 pens cost $3.50, find the cost of 8 pens.
例2：如果 5 支笔花费 $3.50，求 8 支笔的费用。
find the cost of one pen: 3.50 ÷ 5 = 0.70. Then multiply by 8 to find the cost of 8 pens.
解：先求一支笔的费用：3.50 ÷ 5 = 0.70。再乘以 8，得到 8 支笔的费用。
8 × 0.70 = $5.60
Example 3: Convert 60 km/h to metres per second.
例3：将 60 km/h 转换为米每秒。
Solution: There are 1000 metres in a kilometre and 3600 seconds in an hour. Multiply by 1000 and divide by 3600.
解：1 公里 = 1000 米，1 小时 = 3600 秒。乘以 1000 再除以 3600。
60 × 1000 ÷ 3600 = 16.7 m/s (to 3 s.f.)
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
s. Many students add before multiplying, for example writing 3 + 4 × 5 = 35 instead of 23.
错误1：忽略运算顺序。许多学生先加后乘，例如把 3 + 4 × 5 写成 35，而正确答案是 23。
Mistake 2: Sign errors when expanding brackets. A negative outside the bracket changes the sign of every term inside.
错误2：展开括号时符号出错。括号外的负号会改变括号内每一项的符号。
Mistake 3: Confusing area and perimeter. Remember that area uses square units and perimeter is a length with linear units.
错误3：混淆面积与周长。面积使用平方单位，而周长是长度，使用线性单位。
Mistake 4: Forgetting to include units in the final answer. Always write the correct unit, such as cm, cm², cm³, km/h or $.
错误4：最终答案漏写单位。务必写出正确单位，如 cm、cm²、cm³、km/h 或 $。
10. Exam-Style Practice Answers | 考试风格练习答案
Below are some typical exam-style questions with answers. Attempt each question before checking the solution.
以下是一些典型的考试风格题目及答案。先尝试作答，再核对答案。
| Question | 题目 | Working | 解题过程 | Answer | 答案 |
|---|---|---|
| Evaluate 3² + 4 × 5 | 计算 3² + 4 × 5 | 3² = 9 and 4 × 5 = 20, then add | 29 |
| A car travels 180 km in 2 hours 30 minutes. Find its average speed in km/h | 一辆汽车 2 小时 30 分钟行驶 180 km，求平均速度（km/h） | 2
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