📚 Year 8 SQA Advanced Mathematics: Case Study Masterclass | 八年级 SQA 进阶数学：案例实战特训
In Year 8 Advanced Mathematics, applying mathematical skills to real-life scenarios is essential. This case study workout challenges you with practical problems from shopping, gardening, finance, sports, travel, technology, cooking, and construction. Strengthen your problem-solving abilities by working through each example step by step.
在八年级进阶数学中，将数学技能应用于现实生活场景是至关重要的。本次案例实战特训将通过购物、园艺、理财、运动、旅行、科技、烹饪和建筑等方面的实际问题对你进行挑战。请逐步攻克每个例题，提升解题能力。
1. Best Buy Bargain | 最佳购买分析
You need to buy orange juice. Brand X: 500 ml for £1.80. Brand Y: 750 ml for £2.40. Brand Z: 1 litre (1000 ml) for £3.00. Which brand offers the best value for money? We must calculate the cost per millilitre in pence.
你需要购买橙汁。X 品牌：500 毫升售价 £1.80。Y 品牌：750 毫升售价 £2.40。Z 品牌：1 升（1000 毫升）售价 £3.00。哪个品牌性价比最高？我们必须计算每毫升的便士成本。
First, convert all prices to pence: Brand X costs 180p, Brand Y 240p, and Brand Z 300p.
首先，将所有价格转换为便士：X 品牌 180 便士，Y 品牌 240 便士，Z 品牌 300 便士。
Now divide each by its volume in millilitres.
现在将每个金额除以其毫升容积。
Brand X: 180p ÷ 500 ml = 0.36 p/ml
X 品牌：180 便士 ÷ 500 毫升 = 0.36 便士/毫升
Brand Y: 240p ÷ 750 ml = 0.32 p/ml
Y 品牌：240 便士 ÷ 750 毫升 = 0.32 便士/毫升
Brand Z: 300p ÷ 1000 ml = 0.30 p/ml
Z 品牌：300 便士 ÷ 1000 毫升 = 0.30 便士/毫升
Compare the unit prices: 0.36p, 0.32p, and 0.30p per ml. The lowest is Brand Z at 0.30p per ml, so it is the best value. Always check the unit cost when comparing packs of different sizes!
比较单价：每毫升 0.36 便士、0.32 便士和 0.30 便士。最低的是 Z 品牌的每毫升 0.30 便士，因此它性价比最高。在比较不同规格的包装时，一定记得核对单位成本！
2. Garden Makeover | 花园改造预算
A rectangular lawn measures 8 metres by 5 metres. You plan to install a fence along the perimeter and lay new turf over the whole area. Fencing costs £12 per metre and turf costs £5 per square metre. Find the total cost.
一块矩形草坪长 8 米、宽 5 米。你计划沿着周长安装围栏，并在整个区域铺设新草皮。围栏每米 £12，草皮每平方米 £5。请计算总费用。
Perimeter of a rectangle = 2 × (length + width) = 2 × (8 m + 5 m) = 2 × 13 m = 26 m.
矩形的周长 = 2 ×（长 + 宽）= 2 ×（8 m + 5 m）= 2 × 13 m = 26 m。
Fencing cost = perimeter × cost per metre = 26 m × £12/m = £312.
围栏费用 = 周长 × 每米价格 = 26 m × £12/m = £312。
Area of the lawn = length × width = 8 m × 5 m = 40 m².
草坪面积 = 长 × 宽 = 8 m × 5 m = 40 平方米。
Turf cost = area × cost per m² = 40 m² × £5/m² = £200.
草皮费用 = 面积 × 每平方米价格 = 40 m² × £5/m² = £200。
Total cost = fencing + turf = £312 + £200 = £512. Careful planning helps you stay within budget.
总费用 = 围栏 + 草皮 = £312 + £200 = £512。精心规划有助于你控制预算。
3. Savings Challenge | 储蓄挑战
You decide to save £40 every month into a bank account that pays 5% simple annual interest. After one year, how much total money will you have, including interest?
你决定每月向一个年利率为 5% 的银行账户存入 £40，按单利计算。一年后，包括利息在内，你总共会有多少钱？
First, calculate the total deposits after 12 months: £40 × 12 = £480.
首先，计算 12 个月后的总存款：£40 × 12 = £480。
Simple interest formula: Interest = Principal × Rate × Time. Here Principal = £480, Rate = 5% = 0.05, Time = 1 year.
单利公式：利息 = 本金 × 利率 × 时间。此处本金 = £480，利率 = 5% = 0.05，时间 = 1 年。
Interest = £480 × 0.05 × 1 = £24
利息 = £480 × 0.05 × 1 = £24
Total amount after one year = deposits + interest = £480 + £24 = £504. Regular saving combined with interest makes your money grow.
一年后的总金额 = 存款 + 利息 = £480 + £24 = £504。定期储蓄加上利息让你的钱不断增长。
4. Basketball Stats | 篮球数据统计
A basketball player scores the following points in eight games: 12, 15, 8, 20, 15, 10, 18, 15. Calculate the mean, median, mode, and range of the scores.
一名篮球运动员在八场比赛中的得分如下：12, 15, 8, 20, 15, 10, 18, 15。请计算这些得分的平均数、中位数、众数和极差。
. Sum = 12+15+8+20+15+10+18+15 = 113. Number of games = 8. Mean = 113 ÷ 8 = 14.125 points.
平均数：将所有得分相加后除以比赛场数。总和 = 12+15+8+20+15+10+18+15 = 113。比赛场数 = 8。平均数 = 113 ÷ 8 = 14.125 分。
number of values, median = average of the 4th and 5th values (15 and 15), so median = 15.
求中位数时，将得分按顺序排列：8, 10, 12, 15, 15, 15, 18, 20。数据个数为偶数，中位数 = 第 4 和第 5 个数值（15 和 15）的平均数，因此中位数 = 15。
Mode is the most frequent score, which is 15 (appears three times).
众数即出现最频繁的得分，为 15（出现三次）。
score = 20 - 8 = 12 points. These statistics help coaches analyse performance consistency.
极差 = 最高分 - 最低分 = 20 - 8 = 12 分。这些统计数据有助于教练分析表现的一致性。
5. Road Trip Planner | 自驾旅行规划
You plan to drive from Edinburgh to London, a distance of 330 miles. You estimate an average speed of 55 miles per hour. How long will the journey take without breaks? If you leave at 08:30 and take a 45-minute rest stop, when will you arrive?
你计划从爱丁堡驾车前往伦敦，全程 330 英里。你估计平均速度为每小时 55 英里。如果不休息，旅程将耗时多久？如果你在 08:30 出发并休息 45 分钟，将在何时到达？
Use the speed formula: Time = Distance ÷ Speed. Time = 330 miles ÷ 55 mph = 6 hours.
使用速度公式：时间 = 距离 ÷ 速度。时间 = 330 英里 ÷ 55 mph = 6 小时。
Without breaks, arrival time would be 08:30 + 6 hours = 14:30.
不休息的话，到达时间为 08:30 + 6 小时 = 14:30。
in. Adding this to 08:30 gives 15:15. Always include rest breaks when planning long trips.
加上 45 分钟休息，总行程时间 = 6 小时 + 45 分钟 = 6 小时 45 分钟。将时长加至 08:30 得到 15:15。规划长途旅行时，一定要把休息时间考虑在内。
6. Phone Plan Puzzle | 手机套餐谜题
Two mobile phone plans are available. Plan A: £10 monthly fee plus 5p per minute of calls. Plan B: £15 monthly fee plus 2p per minute. How many minutes of calls per month make the total cost the same for both plans?
现有两种手机套餐。套餐 A：月租 £10，通话每分钟 5 便士。套餐 B：月租 £15，通话每分钟 2 便士。每月通话多少分钟时，两个套餐的总费用相同？
Let m be the number of minutes. Write both costs in pence for easier comparison.
设通话分钟数为 m。为便于比较，将费用均以便士表示。
Plan A cost (pence) = 1000p + 5m. Plan B cost = 1500p + 2m.
套餐 A 费用（便士）= 1000p + 5m。套餐 B 费用 = 1500p + 2m。
= 1500 + 2m. Solve for m: 5m - 2m = 1500 - 1000 → 3m = 500 → m = 500 ÷ 3 ≈ 166.67 minutes.
令两者相等：1000 + 5m = 1500 + 2m。解出 m：5m - 2m = 1500 - 1000 → 3m = 500 → m = 500 ÷ 3 ≈ 166.67 分钟。
So at about 167 minutes of calls, the two plans cost the same. Below this, Plan A is cheaper; above it, Plan B becomes better value. Using algebra makes the decision clear.
因此，大约通话 167 分钟时，两种套餐费用相等。低于此分钟数，套餐 A 更便宜；高于它，套餐 B 更划算。借助代数能让决策变得清晰明了。
7. Cake Fractions | 蛋糕中的分数与比例
A cake recipe requires 2/3 cup of sugar and 1 1/2 cups of flour. You only want to bake three-quarters of the original recipe. How much sugar and flour are needed?
一份蛋糕食谱需要 2/3 杯糖和 1 1/2 杯面粉。你只想制作原食谱四分之三的量。需要多少糖和面粉？
Multiply each ingredient amount by the scale factor 3/4.
将每种原料的用量乘以缩放因子 3/4。
Sugar: (3/4) × (2/3) = (3×2) / (4×3) = 6/12 = 1/2 cup.
糖：(3/4) × (2/3) = (3×2) / (4×3) = 6/12 = 1/2 杯。
Flour: convert 1 1/2 to an improper fraction: 3/2. Then (3/4) × (3/2) = 9/8 = 1 1/8 cups.
面粉：将 1 1/2 转化为假分数 3/2。然后 (3/4) × (3/2) = 9/8 = 1 1/8 杯。
You will need ½ cup of sugar and 1 1/8 cups of flour. Mastering fractions is essential for adjusting recipes and sharing quantities accurately.
你将需要 ½ 杯糖和 1 1/8 杯面粉。掌握分数对于准确调整食谱和分配份量至关重要。
8. Ramp Safety Check | 坡道安全检查
You are designing a wheelchair ramp with a vertical rise of 0.6 metres and a horizontal length of 2.4 metres. Safety regulations state the slope (rise ÷ run) must be no steeper than 1:12. Is this ramp safe? Also calculate the actual length of the ramp slope using Pythagoras' theorem.
你正在设计一个轮椅坡道，垂直升高 0.6 米，水平长度 2.4 米。安全规范要求坡度（升高 ÷ 水平长度）不得陡于 1:12。这个坡道安全吗？同时使用勾股定理计算坡道的实际长度。
2.4 m = 0.25. Express as a ratio: 0.25 = 1:4. The standard is 1:12 (approximately 0.083).
坡度 = 升高 / 水平长度 = 0.6 m / 2.4 m = 0.25。用比值表示为 1:4。标准是 1:12（约 0.083）。
Since 1:4 is steeper than 1:12, the ramp is NOT safe for wheelchair users. A longer horizontal run is needed.
由于 1:4 比 1:12 更陡，该坡道对轮椅使用者来说不安全。需要更长的水平长度。
Ramp slope length = √rise² + run² = √0.6² + 2.4² = √0.36 + 5.76 = √6.12 ≈ 2.47 metres.
坡道斜面长度 = √升高² + 水平长度² = √0.6² + 2.4² = √0.36 + 5.76 = √6.12 ≈ 2.47 米。
This calculation shows how geometry ensures inclusive design and public safety.
这种计算体现了如何利用几何学来确保包容性设计和公共安全。
Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com
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