📚 Solving a Rectangle Perimeter & Area Problem: Cambridge Checkpoint p48_1 | 解决矩形周长与面积问题：剑桥Checkpoint p48_1
Many students working through their Cambridge Lower Secondary Checkpoint materials encounter a classic problem on page 48, question 1 of their practice books. It asks them to find the area of a rectangle given a relationship between its length and width and its perimeter. This article breaks down that exact problem, guiding you step by step from the initial reading to the final answer, while also exploring common mistakes, alternative methods and exam-style variations.
很多学生在使用剑桥初中Checkpoint练习册时，会在第48页的第1题遇到一个经典问题。题目给出了一个矩形的长与宽的关系以及它的周长，要求计算该矩形的面积。本文将详细解析这道题目，一步一步地从理解题意到得出最终答案，并探讨常见错误、替代方法和考试风格的变式题。
1. Understanding the Problem | 理解问题
The problem states: ‘A rectangle has a length that is three times its width. The perimeter of the rectangle is 64 metres. Calculate the area of the rectangle.’ Our goal is to find how many square metres the rectangle covers. We are given two key pieces of information: a multiplicative relationship between length and width, and the total distance around the shape (the perimeter).
题目表述为：“一个矩形的长是宽的三倍。矩形的周长是64米。计算该矩形的面积。”我们的目标是求出该矩形有多少平方米。题目给了两个关键信息：长与宽之间的倍数关系，以及围绕形状的总距离（周长）。
2. Defining Variables and Drawing a Diagram | 定义变量并画图
Let the width of the rectangle be w (in metres). Since the length is three times the width, we can express the length as 3w. Drawing a quick sketch helps visualise the problem: label the shorter sides as w and the longer sides as 3w.
设矩形的宽为 w（单位：米）。因为长是宽的三倍，我们可以将长表示为 3w。画一个简单的草图有助于理解题意：将较短的边标为 w，较长的边标为 3w。
3. Writing the Perimeter Equation | 写出周长方程
The perimeter (P) of a rectangle is the total length of all four sides. The standard formula is:
P = 2 × (l + w)
Substitute l = 3w and P = 64 into the formula:
64 = 2 × (3w + w)
矩形的周长 (P) 是所有四条边的总长度。标准公式为：
P = 2 × (l + w)
将 l = 3w 和 P = 64 代入公式：
64 = 2 × (3w + w)
4. Solving the Equation Step by Step | 逐步解方程
Simplify inside the brackets first: 3w + w = 4w. The equation becomes 64 = 2 × 4w, which is 64 = 8w. To find w, divide both sides by 8:
w = 64 ÷ 8 = 8
So the width is 8 metres. The length, which is three times the width, is 3 × 8 = 24 metres.
首先化简括号内：3w + w = 4w。方程变为 64 = 2 × 4w，即 64 = 8w。为了求出 w，两边同时除以 8：
w = 64 ÷ 8 = 8
因此宽为 8 米。长是宽的三倍，所以长为 3 × 8 = 24 米。
5. Calculating the Area | 计算面积
Now that we know both dimensions, we can find the area (A) using the rectangle area formula:
A = l × w
Substitute l = 24 and w = 8: A = 24 × 8 = 192. Don’t forget the units: the area is in square metres.
Area = 192 m²
现在我们知道了两个尺寸，可以使用矩形面积公式计算面积 (A)：
A = l × w
代入 l = 24 和 w = 8：A = 24 × 8 = 192。不要忘记单位：面积以平方米为单位。
面积 = 192 m²
6. Verification | 验证答案
It’s always good practice to check your answer. Plug the found length and width back into the perimeter formula: P = 2 × (24 + 8) = 2 × 32 = 64 m, which matches the given perimeter. This confirms our values are correct, and therefore the area is also correct.
验证答案总是个好习惯。把求出的长和宽代回周长公式：P = 2 × (24 + 8) = 2 × 32 = 64 米，与所给周长一致。这证实我们的数值正确，因此面积也正确无误。
7. Common Pitfalls and How to Avoid Them | 常见错误与避免方法
Many students lose marks on such questions because of small slips. Here are the most frequent mistakes and how to steer clear of them:
许多学生在这类题目上丢分是因为一些小失误。以下是最常见的错误以及如何避免：
- Forgetting to multiply by 2 in the perimeter formula, writing P = l + w instead of 2(l + w).
- 忘记在周长公式中乘以 2，错误地写成 P = l + w 而不是 2(l + w)。
- Confusing area and perimeter formulas and using A = 2(l + w).
- 混淆面积和周长公式，使用了 A = 2(l + w)。
- Misinterpreting ‘three times’ as ‘three more than’, leading to l = w + 3.
- 将“三倍”误解为“多三”，导致 l = w + 3。
- Stopping after finding the width and forgetting to compute the area.
- 求出宽度后就停止，忘记计算面积。
- Omitting or squaring the units incorrectly, e.g., writing m instead of m² for area.
- 单位遗漏或错误平方，比如面积写成 m 而不是 m²。
8. Alternative Approach Using Ratios | 使用比例的替代方法
You can also solve this problem by thinking in terms of ratios. The length-to-width ratio is 3:1. The semi-perimeter (half the perimeter) is 64 ÷ 2 = 32 m. This semi-perimeter is made up of one length and one width, which together correspond to 3 + 1 = 4 equal parts. Therefore, one part = 32 ÷ 4 = 8 m. Width = 1 part = 8 m, Length = 3 parts = 24 m. This method is very quick once you are comfortable with ratios.
你也可以通过比例思维来解这道题。长与宽的比例为 3:1。半周长（周长的一半）为 64 ÷ 2 = 32 米。这个半周长由一个长和一个宽组成，它们一共对应 3 + 1 = 4 等份。因此，一份 = 32 ÷ 4 = 8 米。宽 = 1 份 = 8 米，长 = 3 份 = 24 米。一旦你对比例感到熟练，这种方法非常快捷。
9. Exam-Style Variations | 考试风格变式
Examiners often change the wording slightly. Consider these variants and practise adapting the approach:
考官常常稍微变换措辞。思考以下变式并练习调整解题方法：
Variation A: ‘The length of a rectangle is 5 times its width. The perimeter is 120 cm. Find the area.’ Solution: w = unknown, l = 5w. P = 2(5w + w) = 12w = 120, so w = 10 cm, l = 50 cm, A = 500 cm².
变式A：“一个矩形的长是宽的5倍。周长是120厘米。求面积。”解答：设宽为未知数，l = 5w。P = 2(5w + w) = 12w = 120，所以 w = 10 厘米，l = 50 厘米，A = 500 cm²。
Variation B: ‘A rectangle’s length is 8 m more than its width. The perimeter is 56 m. Find the area.’ Here the relationship is additive: l = w + 8. Then 56 = 2(w + 8 + w) = 2(2w + 8) = 4w + 16, so 4w = 40, w = 10 m, l = 18 m, A = 180 m².
变式B：“一个矩形的长比宽多8米。周长是56米。求面积。”这里的关系是加法：l = w + 8。然后 56 = 2(w + 8 + w) = 2(2w + 8) = 4w + 16，所以 4w = 40，w = 10 米，l = 18 米，A = 180 m²。
Such modifications test whether you truly understand how to translate words into algebraic expressions.
这类改动考查你是否真正理解如何将文字转化为代数表达式。
10. Key Formulas and Summary Table | 关键公式与总结表格
Keep these essential relationships at your fingertips. The table below summarises the core formulas and common unit conversions that appear in KS3 Cambridge exams.
将这些基本关系牢记于心。下表总结了核心公式以及 KS3 剑桥考试中常见的单位换算。
| Concept / 概念 | Formula / 公式 |
|---|---|
| Perimeter of rectangle / 矩形周长 | P = 2(l + w) |
| Area of rectangle / 矩形面积 | A = l × w |
| Length from width and ratio / 已知宽和比例求长 | l = k × w (where k is the multiplying factor) |
| Semi-perimeter / 半周长 | P/2 = l + w |
| 1 m = 100 cm | 100 cm = 1 m; 1 m² = 10,000 cm² |
11. Practice Questions | 练习题
Test yourself with these two problems. Try to solve them without looking at the hints first.
用下面两道题来测试自己。先尝试不借助提示解答。
Question 1: The length of a rectangle is four times its width. If the perimeter is 80 m, what is the area?
问题1：一个矩形的长是宽的四倍。如果周长是 80 米，面积是多少？
Question 2: A rectangle has a perimeter of 48 cm. Its length is twice its width. Find its area in cm².
问题2：一个矩形的周长是 48 厘米。它的长是宽的两倍。求其面积（以 cm² 为单位）。
Try the ratio method for Question 2: semi-perimeter = 24 cm, parts = 3, one part = 8 cm. Check your answers: Q1 area = 256 m², Q2 area = 128 cm².
问题 2 可尝试比例法：半周长 = 24 厘米，总份数 = 3，每份 = 8 厘米。核对答案：问题1 面积 = 256 m²，问题2 面积 = 128 cm²。
12. Conclusion | 总结
This problem from the Cambridge Checkpoint practice set teaches you to translate a word problem into an algebraic equation, solve for one dimension, and then compute the required quantity. Always identify the relationship, write the correct perimeter equation, solve carefully, and verify with the original statement. With regular practice, questions like p48_1 will become straightforward and help you secure high marks in the KS3 Checkpoint maths exam.
这道来自剑桥Checkpoint练习题集的题目教会你如何将文字题转化为代数方程，解出一个尺寸，然后计算所需的量。始终要明确边长关系，写出正确的周长方程，仔细求解，并用原始条件验证。通过经常练习，像 p48_1 这样的问题将变得简单明了，并帮助你在 KS3 Checkpoint 数学考试中取得高分。
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