📚 Case Study Practical Exercises in Mathematics | 数学案例分析实战演练
Welcome to a real-world mathematical case study! In this exercise, you will step into the role of a student planning a food stall for your school's cultural festival. Through a series of interconnected problems, you will apply skills in arithmetic, ratios, percentages, algebra, geometry, statistics, probability, and unit conversions. Case studies help bridge the gap between textbook theory and practical problem-solving – exactly what the CAIE Year 8 curriculum encourages.
欢迎来到真实世界的数学案例研究！在这个练习中，你将扮演一名为学校文化节策划食品摊位的学生。通过一系列相互关联的问题，你将运用算术、比例、百分比、代数、几何、统计、概率和单位换算等技能。案例研究有助于弥合课本理论与实际解题之间的鸿沟——这正是 CAIE 八年级课程所推崇的。
1. Understanding the Scenario | 理解情境
Your class has decided to run a fresh juice stall at the festival to raise money for charity. You will sell three flavours: orange, apple and grape. The stall will be set up in a rectangular area of the school playground. You have a budget of $60 to buy ingredients. Your goal is to make at least a 40% profit on the total cost. You also want to understand your classmates' preferences and make sure you have enough supplies. This case study guides you through the calculations.
你们班级决定在文化节上经营一个鲜榨果汁摊位，为慈善事业筹款。你们将出售三种口味：橙汁、苹果汁和葡萄汁。摊位将设在学校操场的一个长方形区域。你们的预算是 60 美元用于购买原料。目标是实现至少 40% 的成本利润率。你们还想了解同学们的喜好，并确保有足够的供应。本案例研究将引导你完成所有计算。
2. Budgeting and Cost Analysis | 预算与成本分析
First, you list the ingredients and their costs. You need fruit concentrates, water, cups and napkins. The prices are as follows:
首先，列出原料及其成本。你需要浓缩果汁、水、杯子和餐巾。价格如下：
- Orange concentrate: $12 for a 2-litre bottle (you will need 2 bottles)
- 橙子浓缩汁：每瓶 2 升 12 美元（你需要 2 瓶）
- Apple concentrate: $9 for a 1.5-litre bottle (need 3 bottles)
- 苹果浓缩汁：每瓶 1.5 升 9 美元（需要 3 瓶）
- Grape concentrate: $10 for a 2-litre bottle (need 1 bottle)
- 葡萄浓缩汁：每瓶 2 升 10 美元（需要 1 瓶）
- Cups (pack of 100): $4 per pack (need 2 packs)
- 杯子（100 个装）：每包 4 美元（需要 2 包）
- Napkins (pack of 50): $2 per pack (need 3 packs)
- 餐巾（50 片装）：每包 2 美元（需要 3 包）
Calculate the total cost of ingredients.
计算原料的总成本。
First, work out each item's total: orange concentrate = 2 × $12 = $24; apple = 3 × $9 = $27; grape = 1 × $10 = $10; cups = 2 × $4 = $8; napkins = 3 × $2 = $6. Then sum them up: $24 + $27 + $10 + $8 + $6 = $75. This exceeds the $60 budget! After negotiating, you manage to get a 20% discount on all supplies. 20% off the total $75 gives $75 × 0.8 = $60. Perfect! So your total cost is exactly $60.
首先，计算每项的总价：橙子浓缩汁 = 2 × 12 = 24 美元；苹果 = 3 × 9 = 27 美元；葡萄 = 1 × 10 = 10 美元；杯子 = 2 × 4 = 8 美元；餐巾 = 3 × 2 = 6 美元。然后相加：24 + 27 + 10 + 8 + 6 = 75 美元。这超出了 60 美元的预算！经过协商，你获得了所有物品 20% 的折扣。总价 75 美元打八折 = 75 × 0.8 = 60 美元。完美！因此你的总成本刚好是 60 美元。
3. Applying Ratios in Recipes | 食谱中的比例应用
t is 1 : 3, and for grape juice, it is 1 : 5. You have 2 litres of orange concentrate, 4.5 litres of apple concentrate (3 × 1.5 L), and 2 litres of grape concentrate. Calculate how many litres of each juice you can prepare, and the total amount of water needed.
每种果汁都由浓缩汁和水按特定比例混合而成。橙汁的浓缩汁与水的比例为 1 : 4。苹果汁为 1 : 3，葡萄汁为 1 : 5。你有 2 升橙汁浓缩汁、4.5 升苹果浓缩汁（3 瓶 × 1.5 升）和 2 升葡萄浓缩汁。计算你能制备多少升各种果汁，以及总共需要多少升水。
4 × 2 L = 8 L. Total orange juice = 2 + 8 = 10 L. For apple: concentrate 4.5 L, ratio 1:3 ⇒ water = 4.5 × 3 = 13.5 L, total apple juice =
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