📚 Year 9 CIE Maths: Case Study Practice Drills | Year 9 CIE 数学：案例分析实战演练
Welcome to this case study drill session for Year 9 CIE Mathematics. Real‑world scenarios help you master percentages, proportions, measurements, data and probability. Tackle each case step by step and build your problem‑solving confidence.
欢迎参加 Year 9 CIE 数学案例分析实战演练。通过真实场景掌握百分数、比例、测量、数据和概率。逐步解决每个案例，提升解题信心。
1. Discount Dilemma – Percentages and Multipliers | 折扣困境——百分数与乘数
You spot a jacket priced at £85. The shop advertises a 15% discount, but you must also pay 20% VAT on the reduced price. Find the final amount you pay.
你看到一件标价 £85 的夹克。商店宣传八五折，但你须在折后价上加付 20% 增值税。计算你的最终付款额。
Original price = £85. Discount multiplier = 1 − 0.15 = 0.85. Discounted price = 85 × 0.85 = £72.25.
原价 = £85。折扣乘数 = 1 − 0.15 = 0.85。折后价 = 85 × 0.85 = £72.25。
VAT multiplier = 1 + 0.20 = 1.20. Final cost = 72.25 × 1.20 = £86.70.
增值税乘数 = 1 + 0.20 = 1.20。最终价格 = 72.25 × 1.20 = £86.70。
Therefore, you pay £86.70. Notice discount and VAT do not simply cancel each other.
因此，你支付 £86.70。注意折扣和增值税并非简单抵消。
2. Recipe for a Party – Proportion and Scaling | 派对食谱——比例与缩放
A cake recipe for 4 people uses 200 g flour, 150 g sugar and 100 g butter. You need to serve 12 people. How much of each ingredient is required?
一份 4 人份的蛋糕食谱使用 200 g 面粉、150 g 糖和 100 g 黄油。你需要招待 12 人。每种原料需要多少？
Scale factor = 12 ÷ 4 = 3. Multiply every quantity by 3.
比例因子 = 12 ÷ 4 = 3。每种用量乘以 3。
Flour = 200 g × 3 = 600 g. Sugar = 150 g × 3 = 450 g. Butter = 100 g × 3 = 300 g.
面粉 = 200 g × 3 = 600 g。糖 = 150 g × 3 = 450 g。黄油 = 100 g × 3 = 300 g。
Always check if the recipe uses direct proportion; here it does, so simple scaling works perfectly.
始终检查食谱是否使用正比例；这里是的，因此简单缩放完全有效。
3. Road Trip Timing – Speed, Distance and Time | 公路旅行计时——速度、距离与时间
You drive 300 km at an average speed of 80 km/h. You also take a 20‑minute rest stop halfway. What is the total journey time?
你以 80 km/h 的平均速度行驶 300 km。中途你休息了 20 分钟。总行程时间是多少？
Driving time = distance ÷ speed = 300 ÷ 80 = 3.75 hours.
行车时间 = 距离 ÷ 速度 = 300 ÷ 80 = 3.75 小时。
minutes: 3 hours + 0.75 × 60 = 3 h 45 min. Add the rest stop of 20 min → total 4 h 5 min.
将 3.75 小时转换为小时和分钟：3 小时 + 0.75 × 60 = 3 小时 45 分钟。加上休息 20 分钟 → 总时间 4 小时 5 分钟。
If you start at 09:00, you will arrive at 13:05.
如果你 09:00 出发，将在 13:05 到达。
4. Painting a Bedroom – Area, Perimeter and Cost | 粉刷卧室——面积、周长与花费
A bedroom is 4.5 m long, 3.2 m wide and 2.8 m high. You paint all four walls, ignoring a door of area 2.0 m² and a window of 0.9 m². One litre of paint covers 10 m² and costs £12. How many litres are needed and what is the total paint cost?
一间卧室长 4.5 m、宽 3.2 m、高 2.8 m。你要粉刷所有四面墙，忽略一扇 2.0 m² 的门和一扇 0.9 m² 的窗。一升漆覆盖 10 m²，价格为 £12。需要多少升漆？总花费是多少？
= (4.5 + 3.2) × 2 = 15.4 m. Total wall area = perimeter × height = 15.4 × 2.8 = 43.12 m².
周长 = (4.5 + 3.2) × 2 = 15.4 m。墙体总面积 = 周长 × 高 = 15.4 × 2.8 = 43.12 m²。
Paintable area = 43.12 − (2.0 + 0.9) = 40.22 m².
可粉刷面积 = 43.12 − (2.0 + 0.9) = 40.22 m²。
required = 40.22 ÷ 10 = 4.022 → you need 5 litres (must buy whole tins). Total cost = 5 × £12 = £60.
所需升数 = 40.22 ÷ 10 = 4.022 → 你需要 5 升（必须整桶购买）。总花费 = 5 × £12 = £60。
5. Filling a Fish Tank – Volume and Capacity | 注满鱼缸——体积与容积
cm by 40 cm. Water is filled to a depth of 35 cm. How many litres of water are in the tank? If you later fill it to the top, what extra volume is added? (1 litre = 1000 cm³)
一个长方体鱼缸长 80 cm、宽 30 cm、高 40 cm。注水深度为 35 cm。鱼缸里有多少升水？若之后注满至顶部，额外加入多少体积？（1 升 = 1000 cm³）
Volume of water now = 80 × 30 × 35 = 84 000 cm³. In litres = 84 000 ÷ 1000 = 84 litres.
当前水体积 = 80 × 30 × 35 = 84 000 cm³。换算升 = 84 000 ÷ 1000 = 84 升。
tank capacity = 80 × 30 × 40 = 96 000 cm³ = 96 litres. Extra volume = 96 − 84 = 12 litres.
鱼缸总容积 = 80 × 30 × 40 = 96 000 cm³ = 96 升。额外体积 = 96 − 84 = 12 升。
Always check unit conversions: 1 ml = 1 cm³, 1 litre = 1000 ml.
始终检查单位换算：1 ml = 1 cm³，1 升 = 1000 ml。
6. Choosing a Phone Contract – Linear Equations and Graphs | 选择手机合约——线性方程与图像
Contract A costs £15 per month plus £0.05 per minute. Contract B costs £10 per month plus £0.08 per minute. After how many minutes do the two contracts cost the same? Which is cheaper for 200 minutes?
合约 A 月费 £15 加每分钟 £0.05。合约 B 月费 £10 加每分钟 £0.08。多少分钟后两合约费用相同？若使用 200 分钟，哪个更便宜？
of minutes. Cost A = 15 + 0.05m; Cost B = 10 + 0.08m. Set equal: 15 + 0.05m = 10 + 0.08m.
设 m 为分钟数。费用 A = 15 + 0.05m；费用 B = 10 + 0.08m。令两者相等：15 + 0.05m = 10 + 0.08m。
m both sides: 15 = 10 + 0.03m → 5 = 0.03m → m = 5 ÷ 0.03 = 166²/₃ minutes (about 167 min).
两边减去 0.05m：15 = 10 + 0.03m → 5 = 0.03m → m = 5 ÷ 0.03 = 166²/₃ 分钟（约 167 min）。
For m = 200: A = 15 + 0.05×200 = £25; B = 10 + 0.08×200 = £26. So A is cheaper by £1.
当 m = 200：A = 15 + 0.05×200 = £25；B = 10 + 0.08×200 = £26。因此 A 便宜 £1。
7. Sports Day Results – Averages and Charts | 运动会成绩——平均数与图表
The long jump results (in metres) for 10 students are: 3.2, 3.5, 3.2, 4.0, 3.8, 3.2, 3.6, 3.9, 3.5, 3.1. Calculate the mean, median and mode. Then draw a simple frequency table.
10 名学生的跳远成绩（米）为：3.2, 3.5, 3.2, 4.0, 3.8, 3.2, 3.6, 3.9, 3.5, 3.1。计算平均数、中位数和众数。然后绘制一个简单的频数表。
3.8, 3.9, 4.0. Mean = (3.1+3.2+3.2+3.2+3.5+3.5+3.6+3.8+3.9+4.0) ÷ 10 = 35.0 ÷ 10 = 3.50 m.
按顺序排列：3.1, 3.2, 3.2, 3.2, 3.5, 3.5, 3.6, 3.8, 3.9, 4.0。平均数 = (3.1+3.2+3.2+3.2+3.5+3.5+3.6+3.8+3.9+4.0) ÷ 10 = 35.0 ÷ 10 = 3.50 m。
average of 5th and 6th values = (3.5 + 3.5) ÷ 2 = 3.50 m. Mode = 3.2 m (appears 3 times).
中位数：第 5 和第 6 个数值的平均 = (3.5 + 3.5) ÷ 2 = 3.50 m。众数 = 3.2 m（出现 3 次）。
Frequency table: 3.1 (1), 3.2 (3), 3.5 (2), 3.6 (1), 3.8 (1), 3.9 (1), 4.0 (1). A bar chart can easily be drawn from this.
频数表：3.1 (1), 3.2 (3), 3.5 (2), 3.6 (1), 3.8 (1), 3.9 (1), 4.0 (1)。据此可轻松绘出柱状图。
8. Lucky Wheel Spin – Probability and Expectation | 幸运转盘——概率与期望
A fair spinner has five equal sectors: two win £5, two win £2, one wins £10. It costs £4 to play. What is the expected profit (or loss) per spin?
一个均匀转盘分为 5 个等份：2 份赢得 £5，2 份赢得 £2，1 份赢得 £10。每次玩收费 £4。每次旋转的期望利润（或亏损）是多少？
Probability of £5 = 2/5, of £2 = 2/5, of £10 = 1/5. Expected win = (2/5)×5 + (2/5)×2 + (1/5)×10 = 2 + 0.8 + 2 = £4.80.
赢得 £5 的概率 = 2/5，£2 = 2/5，£10 = 1/5。期望赢额 = (2/5)×5 + (2/5)×2 + (1/5)×10 = 2 + 0.8 + 2 = £4.80。
Expected profit = £4.80 − £4.00 = £0.80. On average you gain 80p per spin, so the game is favourable.
期望利润 = £4.80 − £4.00 = £0.80。平均每次旋转赚 80p，因此游戏有利可图。
Over 50 spins you would expect to win about 50 × 0.80 = £40 profit.
若旋转 50 次，预期可赢约 50 × 0.80 = £40 利润。
9. Reading a Map – Scale Drawings | 读地图——比例尺绘图
A map uses a scale of 1 : 25 000. Two villages are 8 cm apart on the map. What is the actual distance in kilometres? A lake area of 2 cm² on the map represents what real area in km²?
一张地图使用比例尺 1 : 25 000。两个村庄在地图上相距 8 cm。实际距离是多少千米？地图上 2 cm² 的湖泊代表实际多大面积（km²）？
Real distance = 8 cm × 25 000 = 200 000 cm. Convert to km: 200 000 cm = 2000 m = 2 km.
实际距离 = 8 cm × 25 000 = 200 000 cm。换算为 km：200 000 cm = 2000 m = 2 km。
→ 62 500 m²; to km²: ÷1 000 000 → 0.0625 km². So 2 cm² represents 2 × 0.0625 = 0.125 km².
对于面积，需平方比例因子：图上 1 cm² = 实际 (25 000)² cm² = 625 000 000 cm²。换算 m²：÷10 000 → 62 500 m²；换算 km²：÷1 000 000 → 0.0625 km²。因此 2 cm² 代表 2 × 0.0625 = 0.125 km²。
Always remember: linear scale is 1 : n; area scale is 1 : n².
始终记住：长度比例尺为 1 : n；面积比例尺为 1 : n²。
10. Saving for a Bike – Simple Interest and Growth | 为自行车储蓄——单利与增长
You deposit £200 in a savings account. Bank X offers 4% per year simple interest. Bank Y offers 4% per year compound interest. How much is in each account after 3 years?
你在储蓄账户存入 £200。银行 X 提供年单利 4%。银行 Y 提供年复利 4%。3 年后每个账户有多少钱？
Simple interest: Interest per year = 200 × 0.04 = £8. After 3 years: total interest = 3 × £8 = £24. Amount = £200 + £24 = £224.
单利：每年利息 = 200 × 0.04 = £8。3 年后：总利息 = 3 × £8 = £24。总额 = £200 + £24 = £224。
Compound interest: Year 1: £200 × 1.04 = £208. Year 2: £208 × 1.04 = £216.32. Year 3: £216.32 × 1.04 = £224.97 (to nearest penny).
复利：第 1 年：£200 × 1.04 = £208。第 2 年：£208 × 1.04 = £216.32。第 3 年：£216.32 × 1.04 = £224.97（精确到便士）。
Compound interest gives £224.97, which is slightly more than simple interest (£224.00) due to earning interest on interest.
复利得到 £224.97，略高于单利的 £224.00，因为利息也产生利息。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图