📚 Real-World Maths Case Studies: Practical Problem Solving | 现实世界数学案例分析：实战演练
Mathematics is not just about numbers in a textbook. Every day we use mathematical thinking to budget money, design spaces, adjust recipes and interpret data. This article presents a series of real-world case studies that require the skills you learn in Year 8 AQA Mathematics. Work through each scenario to build your confidence in problem solving, using fractions, percentages, algebra, geometry, statistics and probability. Each case is broken down into clear steps with practice questions to test your understanding.
数学不仅仅是课本上的数字。我们每天都在用数学思维做预算、设计空间、调整食谱和解读数据。本文通过一系列现实世界的案例分析，实践你在AQA Year 8数学中学到的技能。每个场景都会帮助你提升解决问题的信心，综合运用分数、百分数、代数、几何、统计和概率。每个案例都被拆解成清晰的步骤，并附有练习题来检验你的理解。
1. Budgeting a School Trip | 学校旅行的预算
A Year 8 class of 30 students is planning a visit to the science museum. The coach hire costs a fixed £120. The museum entry fee is £4.80 per student, but a group discount of 15% applies for groups of 20 or more. Work out the total cost of the trip and how much each student should pay if they share the cost equally. Then, if every student gives £10, find the change per student.
一个Year 8班级的30名学生计划去科学博物馆参观。租用大巴的固定费用是120英镑。博物馆门票为每名学生4.80英镑，但对20人及以上的团体提供15%的折扣。请计算旅行的总费用，以及如果平均分摊的话每名学生应付多少钱。再假设每名学生交10英镑，算出每人能找回多少零钱。
First, calculate the total museum entry without discount: 30 × £4.80 = £144. Apply the 15% discount: discount amount = 0.15 × 144 = £21.60. The discounted entry total = 144 - 21.60 = £122.40. Then add the coach hire: total cost = £120 + £122.40 = £242.40.
首先，计算无折扣时的门票总额：30 × 4.80 = 144英镑。15%的折扣金额为0.15 × 144 = 21.60英镑。折扣后的门票总额 = 144 − 21.60 = 122.40英镑。再加上大巴租金：总费用 = 120 + 122.40 = 242.40英镑。
Divide the total by 30 students to find the cost per student: £242.40 ÷ 30 = £8.08 exactly. If each student hands in £10, the change per student = £10.00 - £8.08 = £1.92. Checking: 30 × £1.92 = £57.60, which plus £242.40 equals £300, matching the collections.
把总费用除以30人，得出人均费用：242.40 ÷ 30 = 8.08英镑。如果每人交10英镑，则每人应找回10.00 − 8.08 = 1.92英镑。验算一下：30 × 1.92 = 57.60英镑，加上242.40英镑正好是300英镑，与收款总额一致。
Total Cost = Fixed Cost + (Number of students × Price × (1 - Discount%))
总费用 = 固定成本 + (学生人数 × 票价 × (1 - 折扣百分比))
2. Designing a Vegetable Garden | 设计一个菜园
A gardener wants to make a rectangular vegetable patch with length 6.5 m and width 4.2 m. She needs to cover the ground with turf and put up a fence around the edge. Turf costs £3.25 per m² and fencing costs £5.40 per metre. Find the total material cost. Is a budget of £80 enough? Round money answers to two decimal places where needed.
一位园艺师想建造一个长6.5米、宽4.2米的长方形菜地。她需要用草皮覆盖地面，并在四周竖起围栏。草皮每平方米售价3.25英镑，围栏每米售价5.40英镑。求材料总成本。80英镑的预算是否足够？如有必要，将金额四舍五入到两位小数。
Area for turf = length × width = 6.5 × 4.2 = 27.3 m². Turf cost = 27.3 × £3.25 = £88.725, which rounds to £88.73. Perimeter for fencing = 2 × (6.5 + 4.2) = 2 × 10.7 = 21.4 m. Fencing cost = 21.4 × £5.40 = £115.56.
草皮面积 = 长 × 宽 = 6.5 × 4.2 = 27.3 平方米。草皮费用 = 27.3 × 3.25 = 88.725英镑，四舍五入为88.73英镑。围栏周长 = 2 × (6.5 + 4.2) = 2 × 10.7 = 21.4米。围栏费用 = 21.4 × 5.40 = 115.56英镑。
Total material cost = turf cost + fencing cost = £88.73 + £115.56 = £204.29. This far exceeds the £0 budget. The gardener would need to reduce the garden size or look for cheaper materials.
材料总成本 = 草皮费用 + 围栏费用 = 88.73 + 115.56 = 204.29英镑，远远超出80英镑的预算。园艺师需要缩小菜园面积或寻找更便宜的材料。
3. Perfecting a Smoothie Recipe | 完善一份冰沙配方
A recipe for a strawberry smoothie serves 2 people and uses: 1 banana, 150 g strawberries, 200 ml yogurt and 1 tablespoon of honey. You want to make enough smoothie for 5 people. Determine how much of each ingredient is needed using proportional reasoning.
一份草莓冰沙食谱供两人享用，需要：1根香蕉、150克草莓、200毫升酸奶和1汤匙蜂蜜。你想制作供5人享用的冰沙。用比例推理确定每种配料的用量。
rries: 150 g × 2.5 = 375 g, yogurt: 200 ml × 2.5 = 500 ml, honey: 1 tbsp × 2.5 = 2.5 tbsp.
从2人份增加到5人份的放大因子为5 ÷ 2 = 2.5。将每种配料的数量乘以2.5。香蕉：1 × 2.5 = 2.5根；草莓：150 × 2.5 = 375克；酸奶：200 × 2.5 = 500毫升；蜂蜜：1 × 2.5 = 2.5汤匙。
hurt is 150:200 = 3:4 in the original, and 375:500 also simplifies to 3:4, confirming the correct scaling.
你可以用表格列出结果，也可以直接列举。始终检验比例是否一致：例如，草莓与酸奶的原比例为150:200，化简为3:4；而375:500化简后同样是3:4，证明用量调整正确。
4. Choosing a Mobile Phone Contract | 选择手机合约
Two mobile phone plans are available. Plan X charges £8 per month plus 6p per minute of calls. Plan Y charges £12 per month plus 4p per minute. After how many minutes of calls do the plans cost the same? Which plan is cheaper for 100 minutes per month? Show your working using an equation and a simple table of values.
现有两种手机套餐。套餐X月租8英镑，通话每分钟6便士。套餐Y月租12英镑，通话每分钟4便士。通话多少分钟时两种套餐费用相同？如果每月通话100分钟，哪种套餐更便宜？请用方程和简单的数值表展示你的计算过程。
Let m represent the number of minutes. Cost of Plan X: 800p + 6m pence (or £8 + £ Subtract 4m from both sides: 800 + 2m = 1200. Subtract 800: 2m = 400, so m = 200 minutes.
设m代表通话分钟数。套餐X的费用：800便士 + 6m便士（或8英镑 + 0.06m英镑）。套餐Y的费用：1200便士 + 4m便士。令二者相等：800 + 6m = 1200 + 4m。两边减去4m：800 + 2m = 1200。再减去800：2m = 400，得到m = 200分钟。
6m - 4m = 1200 - 800 → 2m = 400 → m = 200
6m − 4m = 1200 − 800 → 2m = 400 → m = 200
For 100 minutes, Plan X costs 800 + 6(100) = 1400p (£14.00) and Plan Y costs 1200 + 4(100) = 1600p (£16.00). So Plan X is cheaper below 200 minutes, and Plan Y is cheaper above 200 minutes.
通话100分钟时，套餐X费用为800 + 6×100 = 1400便士（14.00英镑），套餐Y为1200 + 4×100 = 1600便士（16.00英镑）。因此，当通话时间少于200分钟时套餐X更便宜，超过200分钟时套餐Y更便宜。
5. Analysing Class Test Scores | 分析班级测验成绩
The marks out of 20 for a Year 8 maths test are: 12, 15, 14, 18, 11, 17, 15, 16, 19, 13, 15, 14, 16, 10, 18, 15, 12, 14, 17, 15. Calculate the mean, median, mode and range of these scores. Which measure best describes the typical performance?
一次Year 8数学测验的分数（满分20分）如下：12, 15, 14, 18, 11, 17, 15, 16, 19, 13, 15, 14, 16, 10, 18, 15, 12, 14, 17, 15。请计算这些分数的平均数、中位数、众数和极差。哪个统计量最能描述典型成绩？
ode is 15 (it appears 5 times). To find the median, order the scores: 10,11,12,12,13,14,14,14,15,15,15,15,15,16,16,17,17,18,18,19. The median is the average of the 10th and 11th values: both are 15, so median = 15. The range = 19 - 10 = 9.
首先，求所有分数之和：12+15+14+18+11+17+15+16+19+13+15+14+16+10+18+15+12+14+17+15 = 300。共有20个分数，所以平均数 = 300 ÷ 20 = 15。众数是15（出现了5次）。求中位数时，将分数排序：10,11,12,12,13,14,14,14,15,15,15,15,15,16,16,17,17,18,18,19。中位数是第10和第11个数的平均值，两者都是15，所以中位数 = 15。极差 = 19 − 10 = 9。
The mean, median and mode are all 15, indicating a symmetrical distribution. The range of 9 shows moderate spread. In this case, any of the averages accurately reflects typical performance, but the median is robust if there were outliers.
平均数、中位数和众数都是15，表明分数呈对称分布。极差为9，说明数据的离散程度中等。在这个案例中，任何一个平均数都能准确反映典型成绩，但如果存在异常值，中位数会更加稳健。
6. Interpreting a Map Scale | 解读地图比例尺
An Ordnance Survey map uses a scale of 1 : 25 000. The distance between two villages on the map measures 8.4 cm. What is the actual straight-line distance in kilometres? Also, if a footpath is 5 km long, how many centimetres would it cover on this map?
某地形测量地图使用1:25 000的比例尺。地图上两个村庄之间的直线距离是8.4厘米。实际直线距离是多少千米？另外，如果一条步行道实际长5千米，它在这张地图上应画多长（厘米）？
vert to metres by dividing by 100: 2100 m. Then to kilometres by dividing by 1000: 2.1 km.
根据比例尺，地图上1厘米代表实际25 000厘米。因此实际距离 = 8.4 × 25 000 = 210 000厘米。除以100转化为米：2100米，再除以1000转化为千米：2.1千米。
ale factor = 500 000 ÷ 25 000 = 20 cm. Always ensure units are consistent before dividing.
对于5千米长的步行道，先把千米化为厘米：5 km = 5000 m = 500 000 cm。地图上的距离 = 实际距离 ÷ 比例因子 = 500 000 ÷ 25 000 = 20厘米。切记在相除之前统一单位。
7. Saving for a Games Console | 攒钱买游戏机
A new games console costs £350. At the moment you have £80 saved. You plan to save £25 each month. Write an equation to find how many whole months you must save to afford it. Also, if the price increases by 5% next year, how much would the console cost then?
一台新游戏机售价350英镑。目前你已存了80英镑，并计划每月存25英镑。写出方程，求出你需要存多少个完整的月份才够买游戏机。此外，如果明年游戏机价格上涨5%，届时价格会是多少？
y 25: m = 10.8. Since you can only save whole months, you need 11 months to reach at least £350. After 10 months you would have 80 + 250 = £330, still £20 short.
设m为月份数。m个月后的存款为：80 + 25m = 350。两边减去80：25m = 270。除以25：m = 10.8。由于只能按整月储蓄，你需要11个月才能攒够至少350英镑。10个月后你只有80 + 250 = 330英镑，还差20英镑。
A 5% price increase means the new price = £350 × 1.05 = £367.50. That would require an extra £17.50, taking about one more month of saving.
价格上涨5%意味着新价格 = 350 × 1.05 = 367.50英镑。这将需要额外存款17.50英镑，大约再多存一个月。
New Price = Original Price × (1 + Percentage increase/100)
新价格 = 原价 × (1 + 上涨百分比/100)
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