Solving for the Teacher’s Weight: A Mathematical Puzzle

Mathematics can be a fascinating tool for solving real-world problems, even in seemingly simple scenarios. One classic example involves determining the weight of a teacher based on the average weights of students and the students-teacher group. Let's explore this puzzle in detail and break it down into a series of logical steps.

Understanding the Problem

We are given the following information:

The average weight of 30 students is 36 kg. The average weight of 31 individuals (30 students and 1 teacher) is 37 kg.

The question at hand is to find the weight of the teacher.

Step-by-Step Solution

To solve this problem, we can use the concept of averages and the total weight of the group. Here’s how we can approach it:

Method 1: Using the Total Weight

The first method involves calculating the total weight of the students and then the total weight of the group including the teacher.

Calculate the total weight of the 30 students: Total weight of 30 students 30 × 36 1080 kg Calculate the total weight of the 31 individuals (30 students and 1 teacher): Total weight of 31 individuals 31 × 37 1147 kg

Next, subtract the total weight of the students from the total weight of the group to find the weight of the teacher:

Weight of the teacher Total weight of 31 individuals - Total weight of 30 students
Weight of the teacher 1147 kg - 1080 kg 67 kg.

Method 2: Using the Formula for Averages

The second method involves using the formula for average weight and solving for the teacher's weight:

Let the teacher's weight be T kg. Total weight of 30 students and the teacher (30 × 36) T 1080 T Total individuals 31 Average weight of the group 37 kg

We can set up the equation:

(30 × 36 T) / 31 37

Multiplying both sides by 31 to clear the denominator:

30 × 36 T 37 × 31

1080 T 1147

Solving for T:

T 1147 - 1080

T 67 kg

Thus, the teacher's weight is 67 kg.

Note: This method is a direct application of the concept that the total weight divided by the number of individuals gives the average weight.

Alternative Methods

Here are a couple of alternative methods to illustrate the versatility of this problem:

Method 3: Algebric Approach

Let x be the average weight of the students and y be the teacher's weight.

Given:

3 y / 31 x 0.5

Simplifying the equation:

3 y 31x 17.5

y 31x 17.5 - 3

y x 17.5

Since the average weight of the students is 36 kg:

y 36 17.5

y 53.5 kg

Therefore, the weight of the teacher is 53.5 kg.

Method 4: Direct Calculation

Another way to look at it is to directly subtract the total weight of the students from the total weight of the group:

Total weight of students 30 × 36 1080 kg

Total weight of the group including the teacher 31 × 37 1147 kg

Weight of the teacher 1147 kg - 1080 kg 67 kg

Both Methods 3 and 4 yield the same result, confirming the correctness of the solution.

Conclusion

Through these various methods, we have determined that the weight of the teacher is 67 kg. Understanding and applying these methods can be a valuable skill in a variety of contexts, from education to real-world problem-solving. Whether through direct calculation or advanced algebraic manipulation, the concept of averages and total weight remains a fundamental tool in mathematics.

Related Keywords

Average weight Student weight Teacher's weight

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