📚 Case Study in Year 7 Cambridge Maths: Practical Problem Solving | 剑桥7年级数学案例分析实战演练
This article presents a series of real-life case studies covering key topics in the Year 7 Cambridge Mathematics curriculum. Each case is designed to show how mathematical concepts are applied in practical situations, helping students build confidence in problem-solving. Follow the steps carefully and practise similar questions to strengthen your skills.
本文通过一系列贴近生活的案例，覆盖剑桥7年级数学课程的核心内容。每个案例都展示了如何在实际情境中运用数学概念，旨在帮助学生建立解题信心。请仔细阅读每个步骤并多加练习，以巩固你的数学技能。
1. Case 1: Calculating Percentages in a Sale | 案例1：打折中的百分比计算
A shop offers a 15% discount on a jacket originally priced at £60. What is the sale price?
一家商店对原价60英镑的夹克提供15%的折扣。折后价是多少？
Step 1: Find 15% of £60. 15% = 15/100 = 0.15. Multiply 0.15 × 60 = 9. The discount is £9.
步骤1：求出60英镑的15%。15% = 15/100 = 0.15。计算 0.15 × 60 = 9。折扣为9英镑。
Step 2: Subtract the discount from the original price. 60 − 9 = 51. The sale price is £51.
步骤2：从原价中减去折扣。60 − 9 = 51。折后价是51英镑。
You can also calculate directly: 85% of £60 = 0.85 × 60 = 51.
你也可以直接计算：60英镑的85% = 0.85 × 60 = 51。
2. Case 2: Solving a Simple Linear Equation | 案例2：解简单的一次方程
Three friends share the cost of a gift equally. If each pays £7.50 and the total cost is £x, write an equation and solve for x.
三个朋友平摊一份礼物的费用。如果每人支付7.50英镑，总费用为x英镑，请写出方程并求解x。
Step 1: Total cost divided by 3 equals 7.50. The equation is x ÷ 3 = 7.50.
步骤1：总费用除以3等于7.50。方程为 x ÷ 3 = 7.50。
Step 2: Multiply both sides by 3: x = 7.50 × 3. So x = 22.50. The total cost is £22.50.
步骤2：两边同乘以3：x = 7.50 × 3。因此 x = 22.50。总费用为22.50英镑。
Check: 22.50 ÷ 3 = 7.50, which is correct.
检验：22.50 ÷ 3 = 7.50，正确。
3. Case 3: Converting Units of Length | 案例3：长度单位的换算
A piece of ribbon is 2.4 metres long. How many centimetres is this? How many millimetres?
一条丝带长2.4米。这是多少厘米？多少毫米？
Step 1: 1 m = 100 cm, so 2.4 m = 2.4 × 100 = 240 cm.
步骤1：1米 = 100厘米，所以2.4米 = 2.4 × 100 = 240厘米。
: 1 m = 1000 mm, so 2.4 m = 2.4 × 1000 = 2400 mm. Or since 1 cm = 10 mm, 240 cm = 2400 mm.
步骤2：1米 = 1000毫米，所以2.4米 = 2.4 × 1000 = 2400毫米。或者因为1厘米 = 10毫米，240厘米 = 2400毫米。
4. Case 4: Area and Perimeter of a Rectangle | 案例4：矩形的面积与周长
A rectangular garden has length 8.5 m and width 3.2 m. Find its perimeter and area.
一个矩形花园长8.5米，宽3.2米。求它的周长和面积。
Perimeter = 2 × (length + width) = 2 × (8.5 + 3.2) = 2 × 11.7 = 23.4 m.
周长 = 2 × (长 + 宽) = 2 × (8.5 + 3.2) = 2 × 11.7 = 23.4米。
ength × width = 8.5 × 3.2. 8.5 × 3 = 25.5, 8.5 × 0.2 = 1.7, so total 27.2. Area = 27.2 m².
面积 = 长 × 宽 = 8.5 × 3.2。8.5 × 3 = 25.5，8.5 × 0.2 = 1.7，合计27.2。面积 = 27.2平方米。
5. Case 5: Mean, Median and Mode | 案例5：平均数、中位数和众数
The marks of 7 students in a test were: 14, 11, 18, 14, 9, 16, 14. Find the mean, median, and mode.
7名学生的测验成绩为：14, 11, 18, 14, 9, 16, 14。求平均数、中位数和众数。
ean: Sum = 14+11+18+14+9+16+14 = 96. Number of values = 7. Mean = 96 ÷ 7 ≈ 13.71 (2 d.p.).
平均数：总和 = 14+11+18+14+9+16+14 = 96。数据个数 = 7。平均数 = 96 ÷ 7 ≈ 13.71（精确到两位小数）。
n: Arrange in order: 9, 11, 14, 14, 14, 16, 18. The middle value (4th) is 14. Median = 14.
中位数：按顺序排列：9, 11, 14, 14, 14, 16, 18。中间值（第4个）是14。中位数 = 14。
Mode: 14 appears most often (3 times). Mode = 14.
众数：14出现次数最多（3次）。众数 = 14。
6. Case 6: Adding and Subtracting Negative Numbers | 案例6：负数的加减
The temperature at midnight was -5°C. By noon it rose by 12°C. In the evening it fell by 7°C. What was the evening temperature?
午夜气温为 -5°C。到中午上升了12°C。傍晚又下降了7°C。傍晚的气温是多少？
Step 1: Midnight temperature = -5. Rise of 12 means -5 + 12 = 7°C at noon.
步骤1：午夜气温 = -5。上升12表示 -5 + 12 = 7°C，即中午气温。
Step 2: A fall of 7 means 7 − 7 = 0°C in the evening.
步骤2：下降7表示 7 − 7 = 0°C，即傍晚气温。
Evening temperature: 0°C.
傍晚气温：0°C。
7. Case 7: Ratio and Proportion in Recipes | 案例7：食谱中的比和比例
A pancake recipe requires flour and milk in the ratio 3 : 2. If 240 g of flour is used, how much milk is needed?
一份煎饼食谱要求面粉和牛奶的比为3 : 2。如果使用了240克面粉，需要多少牛奶？
rts of flour there are 2 parts of milk. Total parts = 3+2=5, but we use direct proportion.
比3 : 2意味着每3份面粉对应2份牛奶。总份数 = 3+2=5，但这里用正比例关系。
3 : 2. So 3 parts flour = 240 g, 1 part = 240 ÷ 3 = 80 g. Milk (2 parts) = 2 × 80 = 160 g.
面粉 : 牛奶 = 3 : 2。所以3份面粉 = 240克，1份 = 240 ÷ 3 = 80克。牛奶（2份）= 2 × 80 = 160克。
Alternatively, set up a proportion: 3/2 = 240/x, solve for x: 3x = 480, x = 160 g.
或者列比例式：3/2 = 240/x，解得 3x = 480，x = 160克。
8. Case 8: Fraction Operations | 案例8：分数的运算
A pizza is cut into 8 equal slices. Sam eats 3/8 of the pizza and Jane eats 1/4. What fraction is left?
一个披萨被切成8等份。Sam吃了3/8，Jane吃了1/4。还剩下几分之几？
Step 1: Convert 1/4 to eighths: 1/4 = 2/8.
步骤1：将1/4转换为八分之几：1/4 = 2/8。
Step 2: Total eaten = 3/8 + 2/8 = 5/8.
步骤2：总共吃了 = 3/8 + 2/8 = 5/8。
Step 3: Fraction left = 1 − 5/8 = 8/8 − 5/8 = 3/8.
步骤3：剩余部分 = 1 − 5/8 = 8/8 − 5/8 = 3/8。
9. Case 9: Interpreting a Bar Chart | 案例9：解读条形图
A bar chart shows the number of books read by four students: Ali (6), Bella (10), Carlos (8), Dana (12). Use the data to answer questions.
一张条形图显示了四名学生阅读的书籍数量：Ali (6), Bella (10), Carlos (8), Dana (12)。请根据数据回答问题。
How many more books did Dana read than Ali? 12 − 6 = 6 books.
Dana比Ali多读了多少本书？12 − 6 = 6本。
What is the total number of books read by all students? 6 + 10 + 8 + 12 = 36 books.
所有学生一共读了多少本书？6 + 10 + 8 + 12 = 36本。
What is the mean number of books read? 36 ÷ 4 = 9 books.
平均每人读了多少本书？36 ÷ 4 = 9本。
(You could include a simple table here, but no need in this case.)
（此处可包含一个简单表格，但本案例不必要。）
10. Case 10: Area of a Triangle | 案例10：三角形的面积
A triangular flag has a base of 60 cm and a perpendicular height of 24 cm. What is its area?
一面三角形旗帜的底边长60厘米，垂直高度为24厘米。它的面积是多少？
Area of triangle = ½ × base × height = ½ × 60 × 24.
三角形面积 = ½ × 底 × 高 = ½ × 60 × 24。
Calculate: 60 × 24 = 1440, then half of 1440 is 720. So area = 720 cm².
计算：60 × 24 = 1440，1440的一半是720。所以面积 = 720平方厘米。
Remember to use the perpendicular height, not the slant height.
记住要用垂直高度，而不是斜高。
11. Case 11: Multiplying Decimals in Real Life | 案例11：生活中小数乘法
A fruit seller sells apples at £1.45 per kilogram. How much do 3.5 kg of apples cost?
一个水果商贩以每千克1.45英镑出售苹果。3.5千克苹果要多少钱？
1.45 and one in 3.5, total three decimal places. So 1.45 × 3.5 = 5.075. Round to currency: £5.08 (since the third decimal is 5, round up).
乘法：1.45 × 3.5。145 × 35 = 5075。1.45有两位小数，3.5有一位小数，共三位小数。所以 1.45 × 3.5 = 5.075。按货币四舍五入：£5.08（因为第三位小数是5，进位）。
She pays £5.08.
她需要支付5.08英镑。
12. Case 12: Simplifying Algebraic Expressions | 案例12：简化代数表达式
Simplify the expression: 4a + 3b − 2a + 5b
简化表达式：4a + 3b − 2a + 5b
Collect like terms: for 'a': 4a − 2a = 2a. For 'b': 3b + 5b = 8b. So simplified expression is 2a + 8b.
合并同类项：含a的：4a − 2a = 2a。含b的：3b + 5b = 8b。所以化简结果为 2a + 8b。
Always combine terms that have exactly the same variable part.
始终把变量部分完全相同的项合并。
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