📚 Homework Book Answers 8C: Final Worked Solutions | 8C 作业本答案：最终版精讲
This article provides a structured walkthrough of common questions from the 8C homework book, covering algebra, geometry, statistics and probability. Each section gives a worked example, the final answer, and the key method needed for exam success.
本文系统地讲解 8C 作业本中的常见题型，覆盖代数、几何、统计与概率。每一节给出例题、最终答案以及考试成功所需的关键方法。
1. Solving Linear Equations | 解一元一次方程
In the 8C homework book, linear equations often appear in the form ax + b = c. Use inverse operations in the correct order: first remove the constant by addition or subtraction, then divide by the coefficient of x.
在 8C 作业本中，一次方程常以 ax + b = c 的形式出现。按正确顺序使用逆运算：先通过加或减消去常数，再除以 x 的系数。
3x + 5 = 20 → 3x = 15 → x = 5
Always check by substituting x = 5 back into the original equation: 3(5) + 5 = 15 + 5 = 20, so the answer is correct.
始终将 x = 5 代回原方程检验：3(5) + 5 = 15 + 5 = 20，因此答案正确。
| Equation | Solution |
|---|---|
| 2x + 3 = 11 | x = 4 |
| 5x – 7 = 18 | x = 5 |
| x/4 + 2 = 6 | x = 16 |
2. Inequalities on a Number Line | 数轴上的不等式
For inequalities, the direction of the sign changes when multiplying or dividing both sides by a negative number. A number line helps to show the solution set visually.
对于不等式，当两边同时乘以或除以一个负数时，不等号方向改变。数轴有助于直观表示解集。
-2x ≤ 6 → x ≥ -3
Use an open circle for < or >, and a closed circle for ≤ or ≥. Shade the side that satisfies the inequality.
对于 < 或 > 使用空心圆，对于 ≤ 或 ≥ 使用实心圆。涂色表示满足不等式的区域。
- Open circle: x > 2 or x < -1
- Closed circle: x ≥ 0 or x ≤ 5
空心圆：x > 2 或 x < -1。实心圆：x ≥ 0 或 x ≤ 5。
3. Ratio and Proportion | 比例与比例关系
To split a quantity in a given ratio, add the parts to find the total number of shares, then divide the quantity by that total.
按给定比例分配数量时，先相加各部分得到总份数，再用总数量除以总份数。
One share = Total ÷ (a + b)
Divide £60 in the ratio 2:3. Total shares = 2 + 3 = 5; one share = 60 ÷ 5 = 12; amounts = 2 × 12 = £24 and 3 × 12 = £36.
将 £60 按 2:3 分配。总份数 = 2 + 3 = 5；一份 = 60 ÷ 5 = 12；两部分为 £24 与 £36。
4. Percentage Increase and Decrease | 百分数增减
Use the multiplier method for efficient percentage changes. For an increase of r%, multiply by (1 + r/100); for a decrease, multiply by (1 – r/100).
使用乘数法高效处理百分数变化。增加 r% 时乘以 (1 + r/100)；减少 r% 时乘以 (1 – r/100)。
Increase £80 by 15% → 80 × 1.15 = £92
Decrease £80 by 15%: multiplier = 0.85; new amount = 80 × 0.85 = £68.
减少 £80 的 15%：乘数 = 0.85；新金额 = 80 × 0.85 = £68。
5. Area and Circumference of Circles | 圆的面积与周长
The circumference of a circle is C = 2πr and the area is A = πr². In many 8C questions, π is taken as 3.14 or 22/7.
圆的周长公式为 C = 2πr，面积公式为 A = πr²。在 8C 的许多题目中，π 取 3.14 或 22/7。
For r = 7 cm: C = 2 × π × 7 ≈ 44 cm; A = π × 7² = 49π ≈ 154 cm²
Always include the correct units: circumference is a length, so use cm or m; area is a surface measure, so use cm² or m².
始终使用正确单位：周长是长度，因此使用 cm 或 m；面积是表面度量，因此使用 cm² 或 m²。
6. Volume of Prisms | 棱柱体积
The volume of any prism is found by multiplying the area of its cross-section by its length. This applies to triangular, trapezoidal and cylindrical prisms.
任何棱柱的体积等于横截面积乘以长度。这适用于三棱柱、梯形棱柱和圆柱。
V = cross-sectional area × length
A triangular prism has cross-sectional area 12 cm² and length 10 cm. V = 12 × 10 = 120 cm³.
一个三棱柱的横截面积为 12 cm²，长度为 10 cm。V = 12 × 10 = 120 cm³。
7. Pythagoras’ Theorem | 勾股定理
For right-angled triangles, use a² + b² = c², where c is the hypotenuse. Only apply this theorem when the triangle contains a right angle.
对于直角三角形，使用 a² + b² = c²，其中 c 是斜边。只有当三角形包含直角时才能使用该定理。
6² + 8² = 36 + 64 = 100 → c = √100 = 10
The hypotenuse is always the longest side, opposite the right angle. If you need a shorter side, rearrange to a² = c² – b².
斜边始终是最长边，对着直角。如果求较短直角边，可变形为 a² = c² – b²。
8. Statistical Averages and Range | 统计平均数与极差
Mean is the total of all values divided by the number of values. Median is the middle value when data are ordered. Mode is the most frequent value. Range is maximum minus minimum.
平均数是所有数值之和除以数据个数。中位数是数据排序后的中间值。众数是出现次数最多的值。极差是最大值减去最小值。
| Data set | Mean | Median | Mode | Range |
|---|---|---|---|---|
| 4, 7, 7, 9, 12 | 7.8 | 7 | 7 | 8 |
When there are two middle values, the median is their mean. The range measures spread but is affected by extreme values.
当有两个中间值时，中位数取两者的平均数。极差衡量离散程度，但易受极端值影响。
9. Probability of Single Events | 单一事件概率
Probability is always between 0 and 1. For equally likely outcomes, P(event) = number of favourable outcomes ÷ total number of outcomes.
概率始终介于 0 和 1 之间。对于等可能结果，P(事件) = 有利结果数 ÷ 总结果数。
Rolling a fair die: P(odd) = 3/6 = 1/2
Simplify fractions where possible, and write probabilities as fractions, decimals or percentages as required.
尽可能化简分数，并根据题目要求将概率写成分数、小数或百分数。
- P(impossible event) = 0
- P(certain event) = 1
P(不可能事件) = 0；P(必然事件) = 1。
10. Sequences and nth Term | 数列与第 n 项
For an arithmetic sequence, the nth term is given by a + (n-1)d, where a is the first term and d is the common difference.
对于等差数列，第 n 项为 a + (n-1)d，其中 a 是首项，d 是公差。
Sequence 5, 8, 11, 14, … → nth term = 3n + 2
First term a = 5, common difference d = 3, so nth term = 5 + (n-1)×3 = 3n + 2.
首项 a = 5，公差 d = 3，所以第 n 项 = 5 + (n-1)×3 = 3n + 2。
11. Simplifying Algebraic Expressions | 化简代数式
Collect like terms by adding or subtracting the coefficients of the same variable. Then expand brackets where necessary.
通过加减相同变量的系数来合并同类项。必要时先去括号。
4a + 3b – a + 2b = 3a + 5b
Expand 3(x + 4) = 3x + 12. Then combine any like terms to give the simplest form.
展开 3(x + 4) = 3x + 12。然后合并同类项得到最简形式。
12. Directed Numbers | 有向数运算
Rules for directed numbers: adding a negative is the same as subtracting; subtracting a negative is the same as adding. Multiplication and division: same signs give positive, different signs give negative.
有向数规则：加上负数等于减去正数；减去负数等于加上正数。乘除法中，同号得正，异号得负。
(-2) × (-5) = 10; (-2) × 5 = -10; 7 – (-3) = 7 + 3 = 10
These rules are essential for simplifying expressions and solving equations that involve negative numbers.
这些规则对于化简含有负数的代数式和解方程至关重要。
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