📚 Year 7 OCR Maths: Case Study Practice | Year 7 OCR 数学：案例分析实战演练
Mathematics is not just about numbers on a page – it is a tool we use every day. In Year 7 OCR Maths, you will often encounter real-life scenarios where you need to apply your skills to solve problems. This case study guide walks you through ten practical challenges, each designed to build your confidence and deepen your understanding. By working through these examples step by step, you will see how topics such as arithmetic, fractions, percentages, algebra, geometry and statistics come together in everyday situations.
数学不仅仅是纸上的数字，它是我们每天使用的工具。在 Year 7 OCR 数学课程中，你会经常遇到需要运用所学技能解决实际问题的场景。这本案例分析指南将带你完成十个实践挑战，每个挑战都旨在增强你的信心并加深理解。通过逐步完成这些示例，你将看到算术、分数、百分比、代数、几何和统计等主题如何在日常生活情境中协同工作。
1. Shopping Budget | 购物预算
Emma has a £15 budget to buy snacks for a weekend trip. She picks up three packs of crisps costing £2.45 each and two bottles of fruit juice at £1.85 each. Does she have enough money, and if so, how much change does she receive?
艾玛有 15 英镑的预算，为周末旅行买零食。她拿了三包薯片，每包 2.45 英镑，以及两瓶果汁，每瓶 1.85 英镑。她的钱够吗？如果够，能找回多少零钱？
First, calculate the total cost of the crisps: 3 packs × £2.45.
首先，计算薯片的总价：3 包 × 2.45 英镑。
3 × £2.45 = £7.35
Next, work out the cost of the juice: 2 bottles × £1.85.
接着，计算果汁的费用：2 瓶 × 1.85 英镑。
2 × £1.85 = £3.70
Now add the two amounts together to find the total spending.
现在把这两个金额加起来，得出总支出。
£7.35 + £3.70 = £11.05
Since the total is less than the budget, subtract £11.05 from £15.00 to get the change.
由于总额小于预算，从 15.00 英镑中减去 11.05 英镑，得到找零。
£15.00 − £11.05 = £3.95
Emma can afford the snacks and will receive £3.95 change.
艾玛买得起这些零食，会拿到 3.95 英镑的找零。
2. Measuring Length | 测量长度
A carpenter needs to cut a wooden plank that is 2.4 m long into three pieces. The first piece must be 0.85 m and the second 0.95 m. How long is the third piece?
一位木匠需要把一块长 2.4 米的木板锯成三段。第一段必须长 0.85 米，第二段长 0.95 米。第三段有多长？
Begin by adding the lengths of the known pieces.
先把已知两段的长度加起来。
0.85 m + 0.95 m = 1.80 m
Subtract this combined length from the original plank to find the remaining piece.
从原木板总长中减去这个总长，得到剩余部分的长度。
2.4 m − 1.8 m = 0.6 m
The third piece is 0.6 m, or 60 cm. Always check by adding all three pieces: 0.85 + 0.95 + 0.6 = 2.4 m.
第三段是 0.6 米，即 60 厘米。始终通过把三段相加来检验：0.85 + 0.95 + 0.6 = 2.4 米。
3. Recipe Fractions | 食谱分数
A cookie recipe requires 3/4 cup of sugar and 2/3 cup of butter. If you want to make 1 1/2 times the recipe, how much sugar and butter do you need in total?
一份饼干食谱需要 3/4 杯糖和 2/3 杯黄油。如果想做成原食谱的 1 1/2 倍，总共需要多少糖和黄油？
First, convert the mixed number 1 1/2 to an improper fraction: 3/2.
首先，把带分数 1 1/2 化成假分数：3/2。
Multiply the sugar amount by the scaling factor.
用缩放因子乘以糖的用量。
(3/4) × (3/2) = 9/8 = 1 1/8 cups of sugar
Do the same for the butter.
对黄油做同样计算。
(2/3) × (3/2) = 6/6 = 1 cup of butter
Thus, you need 1 1/8 cups of sugar and 1 cup of butter.
因此，你需要 1 1/8 杯糖和 1 杯黄油。
4. Discount Percentages | 折扣百分比
During a sale, a £24 T-shirt is reduced by 25%. The next week, there is an extra 10% off the already reduced price. What is the final price?
在促销活动中，一件 24 英镑的 T 恤降价 25%。下一周，在已降价的基础上再打 10% 折扣。最终价格是多少？
Calculate the first discount: 25% of £24.
计算第一次折扣：24 英镑的 25%。
25% = 0.25, so 0.25 × 24 = £6
Subtract to get the reduced price: £24 − £6 = £18.
相减得到降价后的价格：24 − 6 = 18 英镑。
Now apply the extra 10% discount on £18.
现在在 18 英镑的基础上再打 10% 的折扣。
10% of £18 = 0.10 × 18 = £1.80
Final price: £18.00 − £1.80 = £16.20.
最终价格：18.00 − 1.80 = 16.20 英镑。
5. Number Patterns | 数字模式
Alice arranges chairs in a hall. The first row has 4 chairs, the second has 7, the third has 10, and so on. How many chairs are in the 8th row? How many chairs are there in total for the first 8 rows?
爱丽丝在大厅里摆放椅子。第一排有 4 把，第二排有 7 把，第三排有 10 把，依此类推。第 8 排有多少把椅子？前 8 排总共有多少把椅子？
Identify the pattern: each row adds 3 chairs. This is an arithmetic sequence with first term a = 4 and common difference d = 3.
找出模式：每排增加 3 把椅子。这是一个等差数列，首项 a = 4，公差 d = 3。
The nth term is given by a + (n−1)d. For n = 8, find the 8th term.
第 n 项由 a + (n−1)d 给出。当 n = 8 时，求出第 8 项。
8th row = 4 + (8−1)×3 = 4 + 21 = 25 chairs
To find the total for the first 8 rows, use the sum formula: Sn = n/2 × (first + last).
要计算前 8 排的总数，使用求和公式：Sn = n/2 × (首项 + 末项)。
S8 = 8/2 × (4+25) = 4 × 29 = 116 chairs
6. Solving Simple Equations | 解简单方程
Jake and his sister share £28 so that Jake receives £4 more than his sister. How much money does each get?
杰克和妹妹分享 28 英镑，杰克比妹妹多拿 4 英镑。每人各得多少钱？
Let the sister’s amount be x. Then Jake gets x + 4.
设妹妹的钱为 x，那么杰克得到 x + 4。
The total is x + (x + 4) = 28.
总数为 x + (x + 4) = 28。
2x + 4 = 28
Subtract 4 from both sides.
两边同时减去 4。
2x = 24
Divide by 2.
除以 2。
x = 12
The sister receives £12, and Jake receives £12 + £4 = £16.
妹妹拿到 12 英镑，杰克拿到 12 + 4 = 16 英镑。
7. Area and Perimeter | 面积与周长
Liam wants to lay new turf in his rectangular garden measuring 8.5 m by 6.2 m. A roll of turf covers an area of 2 m² and costs £7.50. How many rolls does he need, and what is the total cost?
利亚姆想在他的长方形花园里铺设新草皮，花园尺寸为 8.5 米 × 6.2 米。一卷草皮覆盖面积为 2 平方米，售价 7.50 英镑。他需要多少卷？总费用是多少？
Find the area of the garden (length × width).
计算花园的面积（长 × 宽）。
Area = 8.5 m × 6.2 m = 52.7 m²
Divide the total area by the area one roll covers, then round up to the next whole roll since you cannot buy part of a roll.
用总面积除以每卷覆盖的面积，然后向上取整，因为不可能零买一部分卷。
52.7 ÷ 2 = 26.35, so 27 rolls needed
Multiply the number of rolls by the cost per roll.
卷数乘以每卷费用。
Total cost = 27 × £7.50 = £202.50
Note: The perimeter would be useful if he were putting a fence, but here only area matters.
注意：如果他要安装围栏，周长会很有用，但这里只涉及面积。
8. Volume of a Box | 盒子的体积
A toy company packs cubes with edges of 4 cm into a cardboard box measuring 32 cm long, 20 cm wide and 16 cm high. How many cubes fit into the box without wasted space?
一家玩具公司把棱长为 4 厘米的立方体装入一个长 32 厘米、宽 20 厘米、高 16 厘米的纸箱中。不考虑浪费的空间，能装多少个立方体？
Determine how many cubes can fit along each dimension by dividing.
通过除法确定每条边能放多少个立方体。
Length: 32 ÷ 4 = 8 cubes
Width: 20 ÷ 4 = 5 cubes
Height: 16 ÷ 4 = 4 cubes
Multiply the counts together to get the total number of cubes.
将这些个数相乘，得到立方体的总数。
8 × 5 × 4 = 160 cubes
An alternative method: find the volume of the box (32×20×16 = 10,240 cm³) and the volume of one cube (4³ = 64 cm³), then 10,240 ÷ 64 = 160, giving the same result.
另一种方法：计算纸箱体积（32×20×16 = 10240 立方厘米）和一个立方体体积（4³ = 64 立方厘米），然后 10240 ÷ 64 = 160，结果相同。
9. Ratio and Proportion | 比率与比例
To make a fruit punch, orange juice and lemonade are mixed in the ratio 3:5. If you need 4.8 litres of punch altogether, how much of each ingredient is required?
制作一种水果宾治，需要将橙汁和柠檬水按 3:5 的比例混合。如果总共需要 4.8 升宾治，每种配料需要多少？
The total number of parts is 3 + 5 = 8 parts.
总份数是 3 + 5 = 8 份。
One part equals the total volume divided by total parts.
一份等于总体积除以总份数。
1 part = 4.8 litres ÷ 8 = 0.6 litres
Orange juice is 3 parts: 3 × 0.6 = 1.8 litres.
橙汁占 3 份：3 × 0.6 = 1.8 升。
Lemonade is 5 parts: 5 × 0.6 = 3.0 litres.
柠檬水占 5 份：5 × 0.6 = 3.0 升。
Check: 1.8 + 3.0 = 4.8 litres, and the ratio 1.8:3.0 simplifies to 3:5.
检验：1.8 + 3.0 = 4.8 升，且 1.8:3.0 化简后为 3:5。
10. Interpreting Data | 数据解读
A class of 30 students voted for their favourite fruit. The results were: apple 9, banana 6, orange 8, and grapes 7. Create a frequency table and find the mean number of votes each fruit might be expected to receive if the votes were shared equally, then comment on the distribution.
一个 30 人的班级投票选出最喜爱的水果。结果如下：苹果 9 票，香蕉 6 票，橙子 8 票，葡萄 7 票。建立一个频数表，并计算如果票数均等分配每种水果期望获得的票数平均值，然后对分布情况做出评论。
Frequency table:
频数表：
| Apple | 9 |
| Banana | 6 |
| Orange | 8 |
| Grapes | 7 |
The mean number of votes per fruit (if shared equally) is total votes divided by the number of fruit types.
每种水果的平均票数（如果均等分配）是总票数除以水果种类数。
Mean = (9+6+8+7) ÷ 4 = 30 ÷ 4 = 7.5 votes
Compare each fruit to this mean: apple has 1.5 more, banana 1.5 fewer, orange 0.5 more, grapes 0.5 fewer. The range is 9−6 = 3, showing a relatively even spread, with apple being the mode.
将每种水果与均值比较：苹果多 1.5 票，香蕉少 1.5 票，橙子多 0.5 票，葡萄少 0.5 票。极差为 9−6 = 3，显示分布相对均匀，苹果是众数。
11. Time and Timetables | 时间与时刻表
A train departs from station at 08:45 and arrives at 11:23. How long is the journey? If the return journey takes 15 minutes longer, at what time must you leave the destination to arrive back by 19:00?
一列火车在 08:45 从车站出发，11:23 到达。行程有多长时间？如果返程需多花 15 分钟，为了在 19:00 前回到出发站，必须在几点离开目的地？
Find the duration by counting from 08:45 to 11:23.
计算从 08:45 到 11:23 经过的时间。
From 08:45 to 11:45 is 3 hours, but 11:23 is 22 minutes earlier, so 3h − 22min = 2h 38min.
从 08:45 到 11:45 是 3 小时，但 11:23 早了 22 分钟，所以是 3 小时减 22 分钟 = 2 小时 38 分钟。
Alternatively: 11:23 − 08:45 = 2 hours 38 minutes.
另一种算法：11:23 − 08:45 = 2 小时 38 分钟。
Return journey takes 2h 38m + 15m = 2h 53m.
返程需要 2 小时 38 分 + 15 分 = 2 小时 53 分。
To arrive by 19:00, subtract the duration from 19:00.
要在 19:00 前到达，从 19:00 减去行程时间。
19:00 − 2h 53m = 16:07
You must leave at 16:07.
必须在 16:07 离开。
12. Angles and Shapes | 角度与图形
A window is designed as a regular pentagon. What is the size of each interior angle? If three of these pentagons are placed side by side, meeting at a point, will they fit exactly without a gap?
一扇窗户被设计成正五边形。每个内角是多少度？如果将三个这样的五边形并排放置，并在一点相接，它们能完全密合而不留空隙吗？
The sum of interior angles of a pentagon is (5−2) × 180° = 540°.
五边形的内角和为 (5−2) × 180° = 540°。
Each interior angle = 540° ÷ 5 = 108°
Three such angles meeting at a point would sum to 3 × 108° = 324°, which is less than 360°, so they would leave a gap of 36°. Therefore, they do not fit exactly.
三个这样的角在一点相接的总和为 3 × 108° = 324°，小于 360°，因此会留下 36° 的空隙。所以它们不能完全密合。
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