📚 8H Answers: Key Stage 3 Mathematics Practice Solutions | 8H 答案：关键阶段3数学练习解析
This article provides a complete set of worked answers for common 8H mathematics exercises. The 8H block covers number operations, algebra, geometry, ratio, probability and statistics. Each section below gives a typical question, the answer, and a short explanation to help you check your work and improve your method.
本文提供常见 8H 数学练习的完整答案解析。8H 单元涵盖数的运算、代数、几何、比、概率和统计。下面每一节给出典型题目、答案和简要解释，帮助你检查作业并改进解题方法。
1. Number and Place Value | 数与位值
Example 1: Write 0.000608 in standard form.
示例 1：将 0.000608 写成标准形式。
Answer: 6.08 × 10⁻⁴
答案：6.08 × 10⁻⁴
Explanation: Move the decimal point four places to the right to get 6.08, then multiply by 10⁻⁴ because the original number is small.
解释：将小数点向右移动四位得到 6.08，然后乘以 10⁻⁴，因为原数很小。
Example 2: Round 37.295 to 2 decimal places.
示例 2：将 37.295 四舍五入到两位小数。
Answer: 37.30
答案：37.30
Explanation: The third decimal digit is 5, so round the second decimal digit up from 9 to 10, carrying 1 to give 37.30.
解释：第三位小数是 5，因此将第二位小数从 9 进位到 10，向前进 1，得到 37.30。
Example 3: Work out 15% of 240.
示例 3：计算 240 的 15%。
Answer: 36
答案：36
Explanation: 10% of 240 is 24, 5% is 12, so 15% is 24 + 12 = 36.
解释：240 的 10% 是 24，5% 是 12，所以 15% 是 24 + 12 = 36。
2. Fractions, Decimals and Percentages | 分数、小数和百分数
Example 1: Convert 7/20 to a percentage.
示例 1：将 7/20 转换成百分数。
Answer: 35%
答案：35%
Explanation: Multiply the fraction by 100: 7/20 × 100 = 700/20 = 35.
解释：将分数乘以 100：7/20 × 100 = 700/20 = 35。
Example 2: Calculate 2/3 + 1/4.
示例 2：计算 2/3 + 1/4。
Answer: 11/12
答案：11/12
Use a common denominator of 12, so 2/3 = 8/12 and 1/4 = 3/12, giving 8/12 + 3/12 = 11/12.
解释：用公分母 12，所以 2/3 = 8/12，1/4 = 3/12，得到 8/12 + 3/12 = 11/12。
Example 3: Find 0.6 × 0.25.
示例 3：计算 0.6 × 0.25。
Answer: 0.15
答案：0.15
en place the decimal point 3 places from the right because there are three decimal digits in total.
解释：先算 6 × 25 = 150，由于两个数共有三位小数，从右边数三位点小数点，得到 0.15。
3. Ratio and Proportion | 比与比例
Example 1: Divide £80 in the ratio 3:5.
示例 1：按 3:5 的比例分配 80 英镑。
Answer: £30 and £50
答案：30 英镑和 50 英镑
Explanation: Total parts = 3 + 5 = 8. One part = £80 ÷ 8 = £10. The shares are 3 × £10 = £30 and 5 × £10 = £50.
解释：总份数 = 3 + 5 = 8。一份 = 80 ÷ 8 = 10 英镑。两份分别为 3 × 10 = 30 英镑和 5 × 10 = 50 英镑。
Example 2: A recipe uses 200 g flour for 4 people. How much for 10 people?
示例 2：一份食谱为 4 人提供 200 克面粉。10 人需要多少？
Answer: 500 g
答案：500 克
Explanation: Flour per person = 200 g ÷ 4 = 50 g. For 10 people: 50 g × 10 = 500 g.
解释：每人面粉 = 200 ÷ 4 = 50 克。10 人需要：50 × 10 = 500 克。
Example 3: Simplify the ratio 18:24.
示例 3：化简比 18:24。
Answer: 3:4
答案：3:4
on: Divide both sides by the highest common factor, which is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4.
解释：两边同时除以最大公因数 6。18 ÷ 6 = 3，24 ÷ 6 = 4。
4. Algebraic Expressions | 代数表达式
Example 1: Simplify 5a + 3b - 2a + b.
示例 1：化简 5a + 3b - 2a + b。
Answer: 3a + 4b
答案：3a + 4b
Explanation: Collect like terms: 5a - 2a = 3a and 3b + b = 4b.
解释：合并同类项：5a - 2a = 3a，3b + b = 4b。
Example 2: Expand 3(x - 4).
示例 2：展开 3(x - 4)。
Answer: 3x - 12
答案：3x - 12
Explanation: Multiply both terms inside the bracket by 3: 3 × x = 3x and 3 × (-4) = -12.
解释：将括号内两项分别乘以 3：3 × x = 3x，3 × (-4) = -12。
Example 3: Factorise 6x + 9.
示例 3：因式分解 6x + 9。
Answer: 3(2x + 3)
答案：3(2x + 3)
Explanation: The highest common factor of 6 and 9 is 3. Take 3 outside the bracket: 3(2x + 3).
解释：6 和 9 的最大公因数是 3。把 3 提到括号外：3(2x + 3)。
5. Solving Linear Equations | 解一元一次方程
Example 1: Solve 5x - 3 = 2x + 9.
示例 1：解方程 5x - 3 = 2x + 9。
Answer: x = 4
答案：x = 4
Subtract 2x from both sides: 3x - 3 = 9. Add 3 to both sides: 3x = 12. Divide by 3: x = 4.
解释：两边同时减去 2x：3x - 3 = 9。两边同时加 3：3x = 12。两边同时除以 3：x = 4。
Example 2: Solve 4(x + 2) = 28.
示例 2：解方程 4(x + 2) = 28。
Answer: x = 5
答案：x = 5
Explanation: Divide both sides by 4: x + 2 = 7. Subtract 2 from both sides: x = 5.
解释：两边同时除以 4：x + 2 = 7。两边同时减去 2：x = 5。
Example 3: Solve 2x/3 = 8.
示例 3：解方程 2x/3 = 8。
Answer: x = 12
答案：x = 12
Explanation: Multiply both sides by 3: 2x = 24. Divide both sides by 2: x = 12.
解释：两边同时乘以 3：2x = 24。两边同时除以 2：x = 12。
6. Sequences and nth Term | 数列与第 n 项
Example 1: Find the nth term of the sequence 5, 8, 11, 14, ...
示例 1：求数列 5, 8, 11, 14, ... 的第 n 项。
Answer: 3n + 2
答案：3n + 2
nth term starts with 3n. When n = 1, 3 × 1 = 3, but the first term is 5, so add 2: 3n + 2.
解释：公差是 3，因此第 n 项以 3n 开头。当 n = 1 时，3 × 1 = 3，但首项是 5，所以加上 2：3n + 2。
Example 2: What is the 20th term of the sequence with nth term 2n - 1?
示例 2：第 n 项为 2n - 1 的数列的第 20 项是多少？
Answer: 39
答案：39
Explanation: Substitute n = 20: 2 × 20 - 1 = 40 - 1 = 39.
解释：将 n = 20 代入：2 × 20 - 1 = 40 - 1 = 39。
Example 3: Is 50 in the sequence with nth term 3n + 2?
示例 3：50 是否在 nth 项为 3n + 2 的数列中？
Answer: Yes, when n = 16
答案：是，当 n = 16 时
2 = 50. Subtract 2: 3n = 48. Divide by 3: n = 16, a whole number, so 50 is the 16th term.
解释：令 3n + 2 = 50。减去 2：3n = 48。除以 3：n = 16，为整数，因此 50 是该数列的第 16 项。
7. Angles and Shapes | 角与图形
Example 1: In a triangle, two angles are 55° and 70°. Find the third angle.
示例 1：三角形中两个角分别是 55° 和 70°，求第三个角。
Answer: 55°
答案：55°
Explanation: Angles in a triangle sum to 180°. Third angle = 180° - 55° - 70° = 55°.
解释：三角形内角和为 180°。第三个角 = 180° - 55° - 70° = 55°。
Example 2: Find the size of one interior angle of a regular hexagon.
示例 2：求正六边形一个内角的大小。
Answer: 120°
答案：120°
Explanation: Sum of interior angles = (6 - 2) × 180° = 720°. Each angle = 720° ÷ 6 = 120°.
解释：内角和 = (6 - 2) × 180° = 720°。每个内角 = 720° ÷ 6 = 120°。
Example 3: An exterior angle of a regular polygon is 20°. How many sides does it have?
示例 3：一个正多边形的一个外角是 20°。它有多少条边？
Answer: 18 sides
答案：18 条边
Explanation: Exterior angles sum to 360°. Number of sides = 360° ÷ 20° = 18.
解释：外角和为 360°。边数 = 360° ÷ 20° = 18。
8. Area and Perimeter | 面积与周长
Example 1: Find the area of a circle with radius 7 cm. Use π = 3.14.
示例 1：求半径为 7 cm 的圆的面积。取 π = 3.14。
Answer: 153.86 cm²
答案：153.86 cm²
Explanation: Area = πr² = 3.14 × 7² = 3.14 × 49 = 153.86 cm².
解释：面积 = πr² = 3.14 × 7² = 3.14 × 49 = 153.86 cm²。
Example 2: Find the perimeter of a rectangle 8 cm by 5 cm.
示例 2：求长 8 cm、宽 5 cm 的长方形的周长。
Answer: 26 cm
答案：26 cm
Explanation: Perimeter = 2 × (length + width) = 2 × (8 + 5) = 2 × 13 = 26 cm.
解释：周长 = 2 × (长 + 宽) = 2 × (8 + 5) = 2 × 13 = 26 cm。
Example 3: Find the area of a trapezium with parallel sides 6 m and 10 m, height 4 m.
示例 3：求上底 6 m、下底 10 m、高 4 m 的梯形面积。
Answer: 32 m²
答案：32 m²
Explanation: Area = ½ × (a + b) × h = ½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 m².
解释：面积 = ½ × (上底 + 下底) × 高 = ½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 m²。
9. Coordinates and Graphs | 坐标与图像
Example 1: Find the midpoint of (2, 5) and (8, 11).
示例 1：求点 (2, 5) 和 (8, 11) 的中点。
Answer: (5, 8)
答案：(5, 8)
Explanation: Midpoint = ((2 + 8)/2, (5 + 11)/2) = (10/2, 16/2) = (5, 8).
解释：中点 = ((2 + 8)/2, (5 + 11)/2) = (10/2, 16/2) = (5, 8)。
Example 2: Find the gradient of the line passing through (1, 2) and (3, 8).
示例 2：求经过点 (1, 2) 和 (3, 8) 的直线的斜率。
Answer: 3
答案：3
Explanation: Gradient = (change in y) / (change in x) = (8 - 2) / (3 - 1) = 6 / 2 = 3.
解释：斜率 = (y 的变化量) / (x 的变化量) = (8 - 2) / (3 - 1) = 6 / 2 = 3。
Example 3: Find the equation of the straight line with gradient 2 and y-intercept 5.
示例 3：求斜率为 2、y 截距为 5 的直线方程。
Answer: y = 2x + 5
答案：y = 2x + 5
Use the slope-intercept form y = mx + c, where m is the gradient and c is the y-intercept.
解释：使用斜截式 y = mx + c，其中 m 是斜率，c 是 y 截距。
10. Statistics and Probability | 统计与概率
Example 1: Find the mean of 4, 7, 8, 11, 15.
示例 1：求 4, 7, 8, 11, 15 的平均数。
Answer: 9
答案：9
Explanation: Sum = 4 + 7 + 8 + 11 + 15 = 45. Mean = 45 ÷ 5 = 9.
解释：总和 = 4 + 7 + 8 + 11 + 15 = 45。平均数 = 45 ÷ 5 = 9。
Example 2: A bag has 3 red, 2 blue and 5 green marbles. What is the probability of picking a blue marble?
示例 2：一个袋子里有 3 个红球、2 个蓝球和 5 个绿球。摸到蓝球的概率是多少？
Answer: 1/5
答案：1/5
Explanation: Total marbles = 3 + 2 + 5 = 10. Probability = 2/10 = 1/5.
解释：总球数 = 3 + 2 + 5 = 10。概率 = 2/10 = 1/5。
Example 3: Find the median of 3, 5, 7, 9, 10, 12.
示例 3：求 3, 5, 7, 9, 10, 12 的中位数。
Answer: 8
答案：8
ere are six numbers, so the median is the mean of the 3rd and 4th values: (7 + 9) ÷ 2 = 8.
解释：共有六个数，中位数是第 3 和第 4 个数的平均数：(7 + 9) ÷ 2 = 8。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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