📚 SQA Year 9 Mathematics: Case Study Practical Exercises | SQA九年级数学：案例分析实战演练
In Year 9 SQA Mathematics, applying your skills to real-life scenarios is essential. This article presents a series of case study practical exercises that mirror the kind of multi-step problems you might encounter in exams and everyday life. Through these examples, you will learn to interpret information, choose the correct mathematical methods, and clearly present your solutions.
在SQA九年级数学中，将你的技能应用于现实生活情境至关重要。本文提供了一系列案例分析实战演练，模拟你在考试和日常生活中可能遇到的多步骤问题。通过这些例子，你将学会解读信息、选择正确的数学方法，并清晰地展示你的解答。
1. Budgeting for a School Trip | 学校旅行的预算规划
A class of 28 students is planning a trip to a science museum. The entrance fee is £6.50 per student, and the coach hire costs £210 in total. The school has already raised £150 from a bake sale. You need to find how much each student must pay to cover the remaining cost equally.
一个有28名学生的班级计划去科学博物馆。门票每位学生£6.50，租用大巴总共£210。学校已经通过烘焙义卖筹集了£150。你需要计算每位学生还需平均支付多少，才能补足剩余费用。
First, calculate the total entrance cost: 28 × £6.50 = £182. Then add the coach hire: £182 + £210 = £392. Subtract the raised funds: £392 − £150 = £242. Divide the remaining amount by the number of students: £242 ÷ 28 ≈ £8.64 (rounded to the nearest penny).
首先计算总门票费用：28 × £6.50 = £182。然后加上大巴租赁：£182 + £210 = £392。减去已筹资金：£392 − £150 = £242。将剩余金额除以学生人数：£242 ÷ 28 ≈ £8.64（四舍五入到便士）。
When budgeting, always check if rounding up leaves a small surplus for emergencies. In this case, collecting £8.65 from each student would give a total of £242.20, providing a 20p buffer.
做预算时，总是检查向上取整是否会留下一点应急备用金。这次向每位学生收取£8.65，总额为£242.20，就有了20便士的缓冲。
2. Redecorating a Bedroom | 卧室重新装修
You want to paint the walls of a rectangular bedroom measuring 4.2 m by 3.8 m, with a height of 2.5 m. The room has one door (2.1 m²) and one window (1.5 m²) that will not be painted. Each litre of paint covers 12 m² and costs £14.99. How many litres are needed and what is the total paint cost?
你要粉刷一间长4.2米、宽3.8米、高2.5米的矩形卧室墙壁。房间有一扇门（2.1平方米）和一扇窗（1.5平方米）不粉刷。每升油漆可涂12平方米，每升£14.99。需要多少升？总油漆费用是多少？
Perimeter = 2 × (4.2 + 3.8) = 16 m. Total wall area = perimeter × height = 16 × 2.5 = 40 m². Subtract door and window: 40 − 2.1 − 1.5 = 36.4 m². Litres required = 36.4 ÷ 12 ≈ 3.033, so you must buy 4 litres (since tins come in whole litres). Total cost = 4 × £14.99 = £59.96.
周长 = 2 × (4.2 + 3.8) = 16米。墙壁总面积 = 周长 × 高度 = 16 × 2.5 = 40平方米。减去门窗面积：40 − 2.1 − 1.5 = 36.4平方米。所需升数 = 36.4 ÷ 12 ≈ 3.033，因此必须购买4升（因为油漆按整升售卖）。总费用 = 4 × £14.99 = £59.96。
It is good practice to add 10% for waste; thus 4 litres is adequate but buying 5 gives extra for future touch-ups.
比较好的做法是增加10%的损耗；所以买4升够用，但买5升可以为将来补漆留有余地。
3. Planning a Journey | 规划一次旅行
A family drives from Glasgow to Inverness, a distance of 270 km. They set off at 09:15 and take a 30-minute break. If the average speed for the first 150 km is 80 km/h and they complete the remaining 120 km at an average speed of 60 km/h, what time do they arrive?
一个家庭从格拉斯哥开车到因弗内斯，全程270公里。他们09:15出发，途中休息30分钟。如果前150公里平均速度为80公里/小时，剩余的120公里以平均60公里/小时行驶，他们什么时候到达？
Time for first part = 150 ÷ 80 = 1.875 hours = 1 hour 52.5 minutes. Time for second part = 120 ÷ 60 = 2 hours. Total driving time = 3 hours 52.5 minutes. Add break: 30 minutes, so total journey time = 4 hours 22.5 minutes. Start at 09:15, arrival time: 09:15 + 4 h 22.5 min = 13:37 (rounded to nearest minute).
第一部分时间 = 150 ÷ 80 = 1.875小时 = 1小时52.5分钟。第二部分时间 = 120 ÷ 60 = 2小时。总驾驶时间 = 3小时52.5分钟。加上休息30分钟，总行程时间 = 4小时22.5分钟。在09:15出发，到达时间：09:15 + 4小时22.5分钟 = 13:37（四舍五入到分钟）。
Always include the break in your time calculations and convert decimals to minutes: 0.875 hours × 60 = 52.5 minutes. This helps avoid common timing errors.
始终将休息时间纳入计算，并将小数小时转换为分钟：0.875小时 × 60 = 52.5分钟。这有助于避免常见的时间计算错误。
4. Analysing Sports Data | 分析体育数据
A basketball player’s points over 10 games are: 18, 22, 15, 30, 28, 10, 25, 23, 21, 19. Calculate the mean, median, and range of this data. Which measure best represents the player’s typical performance if one game was unusually low due to injury?
一名篮球运动员在10场比赛中的得分：18, 22, 15, 30, 28, 10, 25, 23, 21, 19。计算这组数据的平均数、中位数和全距。如果有一场比赛因伤得分异常低，哪个度量最能代表球员的典型表现？
Mean = (18+22+15+30+28+10+25+23+21+19) ÷ 10 = 211 ÷ 10 = 21.1. To find median, order: 10, 15, 18, 19, 21, 22, 23, 25, 28, 30. Median = (21+22) ÷ 2 = 21.5. Range = 30 − 10 = 20.
平均数 = (18+22+15+30+28+10+25+23+21+19) ÷ 10 = 211 ÷ 10 = 21.1。求中位数，排序：10, 15, 18, 19, 21, 22, 23, 25, 28, 30。中位数 = (21+22) ÷ 2 = 21.5。全距 = 30 − 10 = 20。
The score of 10 is an outlier caused by injury. The mean is pulled down to 21.1, while the median (21.5) is less affected. Here the median gives a better picture of the player’s typical scoring level.
10分是因受伤造成的异常值。平均数被拉低到21.1，而中位数（21.5）受影响较小。在此中位数更能反映球员的典型得分水平。
5. Designing a Community Garden | 设计社区花园
A rectangular plot of 10 m by 8 m is to be turned into a garden. A circular flower bed of radius 1.5 m sits centrally. The rest will be lawn. Find the area of lawn and the length of fencing needed to enclose the entire plot (excluding gates). Use π ≈ 3.14.
一块10米乘8米的长方形地块将改造成花园。中央有一个半径1.5米的圆形花坛。其余部分为草坪。求草坪面积和围住整个地块所需栅栏长度（不含大门）。取π ≈ 3.14。
Area of plot = 10 × 8 = 80 m². Area of flower bed = π × 1.5² = 3.14 × 2.25 = 7.065 m². Lawn area = 80 − 7.065 = 72.935 m². Perimeter of plot = 2 × (10 + 8) = 36 m. So 36 metres of fencing is needed, but subtract gate width if specified.
地块面积 = 10 × 8 = 80平方米。花坛面积 = π × 1.5² = 3.14 × 2.25 = 7.065平方米。草坪面积 = 80 − 7.065 = 72.935平方米。地块周长 = 2 × (10 + 8) = 36米。因此需要36米栅栏，但如果指定门宽则需减去。
When planning materials, always use the exact perimeter and then round up. You might also need extra area calculations for turf or seed quantities.
规划材料时，始终使用精确周长并向上取整。你可能还需要额外的面积计算来确定草皮或种子的数量。
6. Comparing Mobile Phone Contracts | 比较手机合约
Contract A: £15 per month plus 5p per minute of calls. Contract B: £22 per month with 200 free minutes, then 8p per minute. If you estimate using 350 minutes per month, which contract is cheaper over a year?
合约A：每月£15，通话每分钟5便士。合约B：每月£22，包含200分钟免费通话，超出后每分钟8便士。如果你估计每月使用350分钟，哪种合约在一年内更便宜？
Contract A monthly cost = 15 + 0.05 × 350 = 15 + 17.50 = £32.50. Contract B: 350 − 200 = 150 extra minutes. Cost = 22 + 0.08 × 150 = 22 + 12 = £34.00. Over a year, A = 32.50 × 12 = £390; B = 34.00 × 12 = £408. Contract A saves £18 per year.
合约A月费用 = 15 + 0.05 × 350 = 15 + 17.50 = £32.50。合约B：350 − 200 = 150额外分钟。费用 = 22 + 0.08 × 150 = 22 + 12 = £34.00。一年内，A = 32.50 × 12 = £390；B = 34.00 × 12 = £408。合约A每年节省£18。
You could also set up an equation: cost for x minutes where x > 200 for B: 22 + 0.08(x − 200). This is a simple linear comparison. Always check your usage estimate carefully.
你还可以建立方程：当x > 200时，B的成本 = 22 + 0.08(x − 200)。这是简单的线性比较。务必仔细检查你的使用估算。
7. Scaling Up a Recipe | 按比例放大食谱
A recipe for 6 people uses 300 g of pasta, 2 eggs, 150 ml of milk, and 80 g of cheese. You need to serve 16 people. Find the new ingredient quantities, keeping proportions the same. Then convert 150 ml into litres for bulk buying.
一份6人食用的食谱需要300克意大利面、2个鸡蛋、150毫升牛奶和80克奶酪。你需要供应16人。求出保持比例不变下的新食材用量。然后将150毫升转换为升，便于批量采购。
Scale factor = 16 ÷ 6 = &frac83; ≈ 2.667. Pasta: 300 × &frac83; = 800 g. Eggs: 2 × &frac83; = 16/3 = 5⅓, so you need 6 eggs (round up). Milk: 150 ml × &frac83; = 400 ml. Cheese: 80 × &frac83; = 640/3 = 213.33 g, so about 213 g. 150 ml = 0.15 litres.
比例因子 = 16 ÷ 6 = &frac83; ≈ 2.667。意大利面：300 × &frac83; = 800克。鸡蛋：2 × &frac83; = 16/3 = 5⅓，所以需要6个鸡蛋（向上取整）。牛奶：150毫升 × &frac83; = 400毫升。奶酪：80 × &frac83; = 640/3 = 213.33克，约213克。150毫升 = 0.15升。
When scaling recipes, eggs are discrete units so rounding up is practical. For liquids, you can measure exactly 400 ml using a measuring jug.
当按比例缩放食谱时，鸡蛋是离散单位，因此向上取整更为实际。对于液体，你可以用量杯准确量取400毫升。
8. Interpreting a Business Graph | 解读商业图表
A small cafe’s weekly sales (in £) over 6 weeks: Week 1: 400, Week 2: 460, Week 3: 430, Week 4: 520, Week 5: 490, Week 6: 550. Plot a line graph and find the week with the biggest increase in sales. What might explain the spike in Week 4?
一家小咖啡馆6周的周销售额（£）：第1周：400，第2周：460，第3周：430，第4周：520，第5周：490，第6周：550。绘制折线图，找出销售额增长最大的一周。什么可能解释第4周的飙升？
Increases: W2−W1 = 60, W3−W2 = −30 (decrease), W4−W3 = 90, W5−W4 = −30, W6−W5 = 60. The biggest rise is from Week 3 to Week 4 (+£90). Possible reasons: a local event, holiday weekend, or new marketing campaign.
增长量：第2周−第1周 = 60，第3周−第2周 = −30（下降），第4周−第3周 = 90，第5周−第4周 = −30，第6周−第5周 = 60。最大增长出现在第3周到第4周（+£90）。可能原因：本地活动、假期周末或新的营销活动。
A line graph would show an overall upward trend with a dip in Week 3 and Week 5. You could also calculate the mean weekly sales: (400+460+430+520+490+550) ÷ 6 = 2850 ÷ 6 = £475. Most weeks are around this average.
折线图会显示整体上升趋势，第3周和第5周出现下降。你还可以计算周平均销售额：(400+460+430+520+490+550) ÷ 6 = 2850 ÷ 6 = £475。多数周围绕这个平均值波动。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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