📚 Year 7 CAIE Maths: Case Study Practical Exercises | Year 7 CAIE 数学：案例分析实战演练
Case studies bring mathematics to life by showing how numbers and logic solve everyday problems. In these practical exercises, you will meet real-world scenarios that require addition, subtraction, multiplication, division, fractions, decimals, percentages, geometry and basic algebra. Each case study encourages you to read carefully, identify the maths hidden in words, and apply your Year 7 skills step by step.
案例分析让数学变得鲜活起来，它向我们展示了数字和逻辑如何解决日常问题。在这些实战练习中，你会遇见各种真实情境，需要用到加、减、乘、除、分数、小数、百分数、几何和基础代数。每个案例都鼓励你仔细读题，找出隐藏在文字中的数学，并一步一步地运用你七年级所学的技能。
1. Pocket Money Budget | 零花钱预算
Problem: You receive £12 pocket money each week. This week you want to buy a comic for £3.75, a pack of stickers for £1.49, and you plan to save £4. How much money will you have left for sweets?
问题：你每周得到12英镑零花钱。本周你想买一本3.75英镑的漫画书、一包1.49英镑的贴纸，并计划存下4英镑。你还剩下多少钱买糖果？
Step 1: Add up all planned spending and saving: £3.75 + £1.49 + £4.00 = £9.24.
步骤1：把计划支出的费用和储蓄加起来：3.75 + 1.49 + 4.00 = 9.24（英镑）。
Step 2: Subtract the total from your pocket money: £12.00 – £9.24 = £2.76.
步骤2：用零花钱减去总额：12.00 – 9.24 = 2.76（英镑）。
Conclusion: You can spend £2.76 on sweets. This simple multi-step addition and subtraction is something you’ll use all the time.
结论：你可以用2.76英镑买糖果。这种简单的多步加减运算你随时随地都用得上。
2. Sharing a Birthday Cake | 分享生日蛋糕
Problem: A rectangular cake is cut into 12 equal slices. At the party, 1/3 of the cake is eaten by children, and then 1/4 of the remaining cake is taken home by a friend. How many slices are left for the family?
问题：一个长方形蛋糕被切成12等份。派对上孩子们吃掉了蛋糕的1/3，之后一位朋友又把剩余蛋糕的1/4带回家。最后留给家人的还有多少块？
Step 1: Children eat 1/3 of 12 slices, which is 12 ÷ 3 = 4 slices.
步骤1：孩子们吃掉12块的1/3，即12 ÷ 3 = 4块。
Step 2: Remaining slices after children = 12 – 4 = 8 slices.
步骤2：孩子们吃完后剩下：12 – 4 = 8块。
Step 3: Friend takes 1/4 of these 8 slices: 8 ÷ 4 = 2 slices.
步骤3：朋友取走这8块的1/4：8 ÷ 4 = 2块。
Step 4: Final slices left = 8 – 2 = 6 slices.
步骤4：最终剩下的块数 = 8 – 2 = 6块。
Working with fractions of a quantity shows how important it is to keep track of what the ‘whole’ is at each stage.
处理“一个数量的几分之几”这类问题时，关键在于每一步都要清楚此时的“整体”是什么。
3. Percentage Discount on a Jacket | 夹克的折扣百分比
Problem: A sports shop has a 15% discount on all jackets. A jacket’s original price is £40. How much will you pay after the discount?
问题：一家运动用品店所有夹克打八五折（即减价15%）。一件夹克原价40英镑。打折后你需要付多少钱？
Step 1: Find 10% of £40: £40 ÷ 10 = £4. Then find 5%: half of £4 = £2. So 15% = £4 + £2 = £6.
步骤1：先求40英镑的10%：40 ÷ 10 = 4英镑。再求5%：4英镑的一半是2英镑。所以15% = 4 + 2 = 6英镑。
Step 2: Discount is £6, so new price = £40 – £6 = £34.
步骤2：折扣额是6英镑，那么实际支付价格 = 40 – 6 = 34英镑。
Alternative method: 100% – 15% = 85%, so you pay 85% of £40. 85% = 0.85 × £40 = £34.
另一种方法：100% – 15% = 85%，因此你付原价的85%。85% = 0.85 × 40 = 34英镑。
Mastering percentages helps you compare deals and stay within budget.
掌握百分数可以帮助你比较不同优惠，做到预算之内合理消费。
4. Rectangular Garden Fence | 矩形花园的围栏
Problem: A rectangular garden measures 8.5 m long and 5.2 m wide. The owner wants to build a fence around it and also lay grass over the whole area. What length of fencing is needed, and what area of grass seed must be bought?
问题：一个矩形花园长8.5米、宽5.2米。主人想沿花园四周围上栅栏，并在整个地面铺草皮。需要多长的栅栏？需要买覆盖多大面积的草籽？
Step 1: Perimeter = 2 × (length + width) = 2 × (8.5 m + 5.2 m) = 2 × 13.7 m = 27.4 m.
步骤1：周长 = 2 × (长 + 宽) = 2 × (8.5 + 5.2) = 2 × 13.7 = 27.4（米）。
Step 2: Area = length × width = 8.5 m × 5.2 m.
步骤2：面积 = 长 × 宽 = 8.5 × 5.2。
Step 3: Multiply: 8.5 × 5.2 = 8.5 × 5 + 8.5 × 0.2 = 42.5 + 1.7 = 44.2 m².
步骤3：计算乘法：8.5 × 5.2 = 8.5 × 5 + 8.5 × 0.2 = 42.5 + 1.7 = 44.2（平方米）。
Fencing requires 27.4 metres of material, and 44.2 square metres of grass seed. Always check units: perimeter is in metres, area in square metres.
栅栏需要27.4米长的材料，草籽需覆盖44.2平方米。要时刻注意单位：周长是米，面积是平方米。
5. Scaling a Smoothie Recipe | 按比例调配奶昔食谱
Problem: A recipe for 2 glasses of banana smoothie uses 3 bananas, 400 ml of milk and 120 ml of yoghurt. You want to make 5 glasses for your family. How much of each ingredient do you need?
问题：一款2人份香蕉奶昔食谱需要3根香蕉、400毫升牛奶和120毫升酸奶。你想给家人做5人份，每种材料各需要多少？
Step 1: Find the amount per glass: 3 bananas ÷ 2 = 1.5 bananas per glass; milk: 400 ÷ 2 = 200 ml per glass; yoghurt: 120 ÷ 2 = 60 ml per glass.
步骤1：先求一人份的量：香蕉：3 ÷ 2 = 1.5根/杯；牛奶：400 ÷ 2 = 200毫升/杯；酸奶：120 ÷ 2 = 60毫升/杯。
Step 2: Multiply by 5 for 5 glasses: bananas = 1.5 × 5 = 7.5 bananas; milk = 200 × 5 = 1000 ml (or 1 litre); yoghurt = 60 × 5 = 300 ml.
步骤2：乘以5得到5人份：香蕉 = 1.5 × 5 = 7.5根；牛奶 = 200 × 5 = 1000毫升（即1升）；酸奶 = 60 × 5 = 300毫升。
Alternatively, use the multiplier: 5 ÷ 2 = 2.5. Multiply original quantities by 2.5: bananas 3 × 2.5 = 7.5; milk 400 × 2.5 = 1000 ml; yoghurt 120 × 2.5 = 300 ml.
另一种思路：放大倍数 = 5 ÷ 2 = 2.5。将原食谱各材料乘以2.5：香蕉3 × 2.5 = 7.5；牛奶400 × 2.5 = 1000毫升；酸奶120 × 2.5 = 300毫升。
Scaling recipes is a direct application of ratio and proportion, keeping the taste exactly the same.
按比例缩放食谱是比和比例的直接应用，可以确保做出的奶昔味道一模一样。
6. Solving a Mobile Phone Equation | 解一个手机话费方程
Problem: A pay-as-you-go mobile plan charges £5 per month plus £0.03 per text message. In one month, you spent £11.24. How many text messages did you send? Let ‘t’ represent the number of texts.
问题：某预付费手机套餐每月收取固定费用5英镑，另加每条短信0.03英镑。你一个月花了11.24英镑。你发送了多少条短信？设短信数为 t。
Step 1: Write the equation: 5 + 0.03t = 11.24.
步骤1：写出方程：5 + 0.03t = 11.24。
Step 2: Subtract 5 from both sides: 0.03t = 6.24.
步骤2：方程两边同时减去5：0.03t = 6.24。
Step 3: Divide both sides by 0.03: t = 6.24 ÷ 0.03.
步骤3：两边同时除以0.03：t = 6.24 ÷ 0.03。
Step 4: To divide by a decimal, multiply both numbers by 100: 624 ÷ 3 = 208.
步骤4：除以小数的方法，可以把被除数和除数同时乘100：624 ÷ 3 = 208。
You sent 208 text messages. Forming an equation turns a word problem into a solvable puzzle.
你发了208条短信。把文字描述转变成方程，便将实际问题变成了一个可解的谜题。
7. Mean Score from a Quiz | 测验成绩的平均值
Problem: A group of six students scored these marks in a maths quiz: 12, 15, 14, 18, 11, 17. What is the mean mark? If the teacher adds a seventh student’s mark of 20, what happens to the mean?
问题：六名学生在一次数学测验中的分数分别为：12、15、14、18、11、17。平均分是多少？如果老师再加上第七名学生的成绩20分，平均分会有什么变化？
Step 1: Sum of first six marks = 12 + 15 + 14 + 18 + 11 + 17 = 87.
步骤1：前六人总分 = 12 + 15 + 14 + 18 + 11 + 17 = 87。
Step 2: Mean = 87 ÷ 6 = 14.5.
步骤2：平均值 = 87 ÷ 6 = 14.5。
Step 3: After adding 20, total sum = 87 + 20 = 107.
步骤3：加入20分后，总分 = 87 + 20 = 107。
Step 4: New mean = 107 ÷ 7 = 15.2857… ≈ 15.3 (to 1 decimal place).
步骤4：新的平均值 = 107 ÷ 7 ≈ 15.3（保留一位小数）。
The mean increased from 14.5 to approximately 15.3 because the new score was higher than the original mean.
平均分从14.5上升到了约15.3，因为新加入的成绩高于原来的平均值。
8. Journey Time and Arrival | 行程时间与到达时刻
Problem: A school bus leaves at 08:15 and travels an average speed of 36 km/h for a distance of 27 km. At what time does it reach the destination?
问题：一辆校车在08:15出发，以平均36千米/时的速度行驶27千米。它什么时候到达目的地？
Step 1: Use the formula time = distance ÷ speed: t = 27 ÷ 36.
步骤1：运用公式“时间 = 路程 ÷ 速度”：t = 27 ÷ 36。
Step 2: Simplify fraction: 27/36 = 3/4 hour (since both can be divided by 9). 3/4 of an hour = 45 minutes.
步骤2：化简分数：27/36 = 3/4 小时（分子分母同除以9）。3/4 小时 = 45分钟。
Step 3: Add 45 minutes to 08:15: 08:15 + 00:45 = 09:00.
步骤3：在08:15的基础上加45分钟：08:15 + 00:45 = 09:00。
The bus arrives at 09:00. Using the speed–distance–time triangle is essential for planning journeys.
校车在09:00到达。运用速度—距离—时间三角关系对规划行程至关重要。
9. Comparing Mobile Phone Plans with Decimals | 用小数比较手机套餐
Problem: Plan A costs £8.65 per month for 1 GB data. Plan B costs £10.20 per month for 1.5 GB data. Which plan gives the lower cost per MB? 1 GB = 1000 MB.
问题：A套餐每月8.65英镑包含1 GB流量。B套餐每月10.20英镑包含1.5 GB流量。哪个套餐的每MB费用更低？（1 GB = 1000 MB）
Step 1: Convert GB to MB: Plan A provides 1000 MB, Plan B provides 1500 MB.
步骤1：将GB转换为MB：A套餐有1000 MB，B套餐有1500 MB。
Step 2: Cost per MB for Plan A: £8.65 ÷ 1000 = £0.00865 per MB (or 0.865 pence).
步骤2：A套餐每MB费用：8.65 ÷ 1000 = 0.00865英镑/MB（即0.865便士）。
Step 3: Cost per MB for Plan B: £10.20 ÷ 1500 = £0.0068 per MB (or 0.68 pence).
步骤3：B套餐每MB费用：10.20 ÷ 1500 = 0.0068英镑/MB（即0.68便士）。
Conclusion: Plan B is cheaper per MB because 0.0068 < 0.00865. Even though the total cost is higher, you get much more data for your money.
结论：B套餐每MB更便宜，因为0.0068 < 0.00865。尽管总价更高，但你花的钱换来了更多的流量。
10. Volume of a Fish Tank | 鱼缸的体积
Problem: A fish tank is a cuboid with length 60 cm, width 35 cm and height 40 cm. What is its volume in cubic centimetres? If the tank is filled to ¾ of its height, what volume of water does it hold in litres? (1 litre = 1000 cm³)
问题：一个长方体鱼缸长60厘米、宽35厘米、高40厘米。它的体积是多少立方厘米？如果水加到鱼缸高度的3/4，它能装多少升水？（1升 = 1000立方厘米）
Step 1: Volume of full tank = length × width × height = 60 × 35 × 40.
步骤1：满缸体积 = 长 × 宽 × 高 = 60 × 35 × 40。
Step 2: Calculate: 60 × 35 = 2100, then 2100 × 40 = 84000 cm³.
步骤2：计算：60 × 35 = 2100，2100 × 40 = 84000 cm³。
Step 3: Water height is ¾ of 40 cm = 30 cm. Volume of water = 60 × 35 × 30 = 60 × 35 = 2100, × 30 = 63000 cm³.
步骤3：水的高度是40厘米的3/4，即30厘米。水的体积 = 60 × 35 × 30 = 60 × 35 = 2100，2100 × 30 = 63000 cm³。
Step 4: Convert to litres: 63000 ÷ 1000 = 63 litres.
步骤4：换算成升：63000 ÷ 1000 = 63升。
Even with a lower water level, the tank holds a substantial 63 litres. Always remember to match cubic units with volume conversions.
即使水位较低，鱼缸也能装下63升水。记住体积单位换算时要对应好立方单位。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导