If you were stomped by a challenging question in the Merchant Taylors’ past exam paper (Q31c) on averages, then we have the perfect solution for you!
Solving mathematical problems often requires not just understanding the concepts but also applying them in creatively effective ways. In this blog post, we break down a particularly tricky question that tackles mean and median weights. We’ll explore the solution step-by-step, allowing you to grasp the thought process involved in resolving such a challenging problem.
The Problem Statement
The question reads:
“The mean weight of five children is 45 kilogrammes. The mean weight of the lightest three children is 42 kilogrammes. And the mean weight of the heaviest three children is 49 kilogrammes. What is the median weight of the children in kilogrammes?”
On the surface, this may seem like a straightforward question about averages, but it involves several layers of complexity. Let’s dive deep and find the solution while ensuring we understand each step of the way.
Initial Analysis of Merchant Taylors’ Challenging Average Question
The problem supplies us with some key information:
- The mean weight of five children is 45 kg.
- The mean weight of the lightest three children is 42 kg.
- The mean weight of the heaviest three children is 49 kg.
To effectively solve this, we must process each piece of information and utilise basic principles of mean and median.
Step-by-Step Breakdown for this Merchant Taylors’ Question on Averages
Step 1: Understanding the Mean of Five Children
The mean weight of the five children is given as 45 kg. This implies that if we sum all their weights, we should get:
[ 45 \text{ kg} \times 5 \text{ children} = 225 \text{ kg} ]
Thus, the total weight of all five children combined is 225 kg.
Step 2: Mean Weight of the Lightest Three Children
The mean weight of the lightest three children is 42 kg. This means that the combined weight of the lightest three children is:
[ 42 \text{ kg} \times 3 \text{ children} = 126 \text{ kg} ]
So, the total weight of the lightest three children is 126 kg.
Step 3: Mean Weight of the Heaviest Three Children
Similarly, for the heaviest three children, whose mean weight is given as 49 kg:
[ 49 \text{ kg} \times 3 \text{ children} = 147 \text{ kg} ]
Hence, the total weight of the heaviest three children is 147 kg.
Step 4: Summing Up the Information
At this point, we need to consolidate our information:
- The total weight of all five children: 225 kg
- The total weight of the lightest three children: 126 kg
- The total weight of the heaviest three children: 147 kg
If we add the weights of the lightest three and the heaviest three, we must remember that the middle child’s weight will be counted twice.
[ 126 \text{ kg} + 147 \text{ kg} = 273 \text{ kg} ]
Step 5: Dealing with the Overlap
Because we only have five children in total, the excess weight (273 kg – 225 kg) is due to counting the middle child twice:
[273 \text{ kg} – 225 \text{ kg} = 48 \text{ kg}]
Thus, the middle child’s weight must be 48 kg since we counted it twice.
Step 6: Identifying the Median
The median of a set of numbers is the middle number when they are arranged in ascending order. In this context, since we identified the middle child through the above calculations, the median weight is:
[\textbf{48 kg}]
Final Answer For This Merchant Taylors’ Question on Averages
Therefore, after analyzing the provided data and performing the necessary calculations, we determine that the median weight of the children is 48 kilograms.
Key Takeaways from this Merchant Taylors’ Question on Averages
- Mean is calculated by the sum of items divided by the number of items.
- Understanding how extra information (like overlap in means) affects calculations is crucial.
- With overlapping groups, excess weight or other quantities must be rationalized to solve for unknowns accurately.
By thoroughly breaking down the problem step-by-step, we’ve solved a seemingly challenging question with clarity and precision. Hopefully, this methodical approach will help you tackle similar problems in the future!
Let’s continue exploring and understanding complex math problems together. If you have any questions or other tricky problems, feel free to drop them in the comments!
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“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.”
– William Paul Thurston