📚 Year 8 Cambridge Maths: Case Study Practice | Year 8 剑桥数学：案例分析实战演练
In Year 8 Cambridge Mathematics, applying mathematical concepts to real-life situations is essential for developing problem-solving skills. Case study practice helps you link classroom topics – ratios, percentages, algebra, geometry, statistics, and probability – to everyday scenarios, from scaling recipes to comparing mobile phone plans. This article provides a series of guided case studies, each with step-by-step solutions in both English and Chinese, to strengthen your understanding and boost your confidence.
在Year 8剑桥数学课程中，将数学概念应用到实际生活中是培养解决问题能力的关键。案例分析练习能帮助你把课堂上学到的比例、百分数、代数、几何、统计和概率等知识与日常情景联系起来，例如食谱缩放和手机套餐比较。本文提供一系列引导式案例，每个案例都配有中英文分步解答，以加深你的理解并增强信心。
1. Ratio and Recipe Scaling | 比例与食谱缩放
Problem: A recipe designed for 4 people requires 200 g of flour. How much flour is needed if you want to make the same recipe for 6 people?
问题： 一份供4人食用的食谱需要200克面粉。如果要做给6人食用，需要多少面粉？
Solution – Method 1 (unitary method): First find the amount of flour per person.
解答 – 方法1（归一法）： 先求出每人需要的面粉量。
Step 1: Flour per person = 200 g ÷ 4 = 50 g per person.
步骤1：每人面粉量 = 200克 ÷ 4 = 50克/人。
Step 2: Multiply by the new number of people: 50 g × 6 = 300 g.
步骤2：乘以新人数：50克 × 6 = 300克。
Solution – Method 2 (scaling factor): The number of people increases by a factor of 6/4 = 1.5, so the flour also scales by 1.5.
解答 – 方法2（比例因子）： 人数增加了6/4 = 1.5倍，因此面粉量也按1.5倍缩放。
200 g × (6/4) = 200 g × 1.5 = 300 g.
200克 × (6/4) = 200克 × 1.5 = 300克。
Both methods show that 300 g of flour is needed for 6 people.
两种方法都表明，供6人食用需要300克面粉。
2. Discount and Sale Prices | 折扣与销售价格
Problem: A bicycle has an original price of £180. The shop offers a 15% discount. If a customer has a membership card giving an extra 10% off the already discounted price, what is the final price to pay?
问题： 一辆自行车原价180英镑，商店提供15%的折扣。如果顾客有会员卡，可在折扣价基础上再享受10%的优惠，最终需要支付多少钱？
Solution:
解答：
Step 1: Calculate the price after the first 15% discount.
步骤1：计算首次15%折扣后的价格。
Discount amount = 15% of £180 = 0.15 × 180 = £27.
折扣金额 = 180的15% = 0.15 × 180 = 27英镑。
Price after first discount = £180 − £27 = £153. (Alternatively, multiply by 0.85 directly: 180 × 0.85 = 153)
首次折扣后价格 = 180 − 27 = 153英镑。（或者直接乘以0.85：180 × 0.85 = 153）
Step 2: Apply the extra 10% member discount on the reduced price of £153.
步骤2：在153英镑的降价基础上再享受10%的会员折扣。
Extra discount = 10% of £153 = 0.10 × 153 = £15.30.
额外折扣 = 153的10% = 0.10 × 153 = 15.30英镑。
Final price = £153 − £15.30 = £137.70. (Or 153 × 0.90 = 137.70)
最终价格 = 153 − 15.30 = 137.70英镑。（或153 × 0.90 = 137.70）
The final price to pay is £137.70.
最终需要支付137.70英镑。
3. Area and Perimeter in Gardening | 花园中的面积与周长
Problem: A rectangular garden measures 8 m by 5.5 m. A path of width 1 m is to be built around the outside of the garden. Find the area of the path.
问题： 一个长方形花园长8米，宽5.5米。在花园外围要修一条1米宽的小路。求小路的面积。
Solution:
解答：
the outer rectangle (garden + path) has length = 8 + 2 = 10 m and width = 5.5 + 2 = 7.5 m.
步骤1：想象花园和小路。小路在每边增加1米，因此外矩形（花园+小路）的长 = 8 + 2 = 10米，宽 = 5.5 + 2 = 7.5米。
Step 2: Calculate the area of the outer rectangle: Aouter = 10 m × 7.5 m = 75 m².
步骤2：计算外矩形面积：A_外 = 10米 × 7.5米 = 75平方米。
Step 3: Calculate the area of the garden only: Agarden = 8 m × 5.5 m = 44 m².
步骤3：仅计算花园面积：A_花园 = 8米 × 5.5米 = 44平方米。
Step 4: Area of path = Aouter − Agarden = 75 m² − 44 m² = 31 m².
步骤4：小路面积 = 外面积 − 花园面积 = 75 − 44 = 31平方米。
The path covers an area of 31 m².
小路占地31平方米。
4. Volume and Capacity in a Fish Tank | 鱼缸的体积与容量
Problem: e tank at a rate of 5 litres per minute. How long will it take to fill the tank completely? (Remember: 1 litre = 1000 cm³)
问题： 一个长方形鱼缸内部尺寸为：长50厘米，宽30厘米，高40厘米。水泵以每分钟5升的速度注水。将鱼缸完全注满需要多长时间？（记住：1升 = 1000立方厘米）
Solution:
解答：
Step 1: Find the volume in cm³. Volume = length × width × height = 50 cm × 30 cm × 40 cm.
步骤1：计算体积（立方厘米）。体积 = 长 × 宽 × 高 = 50 × 30 × 40。
50 × 30 = 1500; 1500 × 40 = 60,000 cm³.
50 × 30 = 1500；1500 × 40 = 60000立方厘米。
Step 2: Convert to litres: 60,000 ÷ 1000 = 60 litres.
步骤2：换算为升：60000 ÷ 1000 = 60升。
Step 3: Time = total volume ÷ filling rate = 60 litres ÷ 5 litres per minute = 12 minutes.
步骤3：时间 = 总体积 ÷ 注水速度 = 60升 ÷ 5升/分钟 = 12分钟。
It will take 12 minutes to fill the tank completely.
完全注满鱼缸需要12分钟。
5. Linear Equations and Mobile Phone Plans | 线性方程与手机套餐
Problem: A mobile phone company offers two plans. Plan A costs £10 per month plus 5p per minute of calls. Plan B costs £15 per month plus 3p per minute. For how many minutes of calls per month will the two plans cost exactly the same?
问题： 一家手机公司提供两种套餐。A套餐每月10英镑，加上每分钟通话费5便士。B套餐每月15英镑，加上每分钟通话费3便士。每月通话多少分钟时，两种套餐费用完全相同？
Solution:
解答：
Step 1: Write expressions for total cost. Work in pounds: 5p = £0.05, 3p = £0.03.
步骤1：写出总费用表达式。统一用英镑：5便士 = 0.05英镑，3便士 = 0.03英镑。
Cost of Plan A = 10 + 0.05m, where m = number of minutes.
A套餐费用 = 10 + 0.05m，其中m为通话分钟数。
Cost of Plan B = 15 + 0.03m.
B套餐费用 = 15 + 0.03m。
Step 2: Set them equal: 10 + 0.05m = 15 + 0.03m.
步骤2：两者相等：10 + 0.05m = 15 + 0.03m。
Step 3: Subtract 0.03m from both sides: 10 + 0.02m = 15.
步骤3：两边减去0.03m：10 + 0.02m = 15。
Step 4: Subtract 10 from both sides: 0.02m = 5.
步骤4：两边减去10：0.02m = 5。
Step 5: Divide by 0.02: m = 5 ÷ 0.02 = 250.
步骤5：除以0.02：m = 5 ÷ 0.02 = 250。
plans cost the same at 250 minutes of calls. At this point each plan costs 10 + 0.05×250 = £22.50.
当通话250分钟时，两种套餐费用相同。此时每个套餐费用为10 + 0.05 × 250 = 22.50英镑。
6. Interpreting Bar Charts and Pie Charts | 解释条形图与饼图
Problem: A survey asked 48 students their favourite sport. The results were: Football 18, Basketball 12, Swimming 8, Tennis 6, Other 4. Calculate the percentage and the angle for each sector if these data were shown in a pie chart. Which two sports together account for more than half of the students?
问题： 一项调查询问了48名学生最喜欢的运动。结果为：足球18人，篮球12人，游泳8人，网球6人，其他4人。计算若用饼图表示时每项的百分比和扇区角度。哪两项运动加起来超过学生总数的一半？
Solution:
解答：
Step 1: Total frequency = 48.
步骤1：总频数 = 48。
| Sport 运动 | Frequency 人数 | Percentage 百分比 | Angle 角度 |
|---|---|---|---|
| Football 足球 | 18 | (18/48)×100 = 37.5% | (18/48)×360° = 135° |
| Basketball 篮球 | 12 | (12/48)×100 = 25% | (12/48)×360° = 90° |
| Swimming 游泳 | 8 | (8/48)×100 ≈ 16.7% | (8/48)×360° = 60° |
| Tennis 网球 | 6 | (6/48)×100 = 12.5% | (6/48)×360° = 45° |
| Other 其他 | 4 | (4/48)×100 ≈ 8.3% | (4/48)×360° = 30° |
e two largest groups. Football (18) + Basketball (12) = 30 students, which is 30/48 = 62.5%, more than half. So Football and Basketball together exceed 50%.
步骤2：检查人数最多的两项。足球（18）+ 篮球（12）= 30人，占30/48 = 62.5%，超过一半。因此足球和篮球合计超过50%。
7. Probability and Game Strategy | 概率与游戏策略
Problem: A game spinner has four colours: red, blue, green, and yellow. The probability of landing on red is 1/2, blue is 1/4, green is 1/8, and yellow is 1/8. It costs £0.50 to play. The prizes are: red wins £1, blue wins £0.50, green wins £1.50, yellow wins nothing. Is the game fair? Calculate the expected profit per game for a player and suggest a change to make the game fair.
问题： 一个游戏转盘有四种颜色：红、蓝、绿、黄。转到红色的概率是1/2，蓝色1/4，绿色1/8，黄色1/8。每次玩花费0.50英镑。奖品为：红色赢1英镑，蓝色赢0.50英镑，绿色赢1.50英镑，黄色无奖。游戏公平吗？计算玩家每局游戏的期望利润，并提出一个让游戏公平的改动。
Solution:
解答：
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