Q.1.
The average weight of a group of 50 students is 20 kg less than that of their fathers. If a pair of father and son is removed, the average weight of the group reduces by 1 kg. If the
average weight of the fathers lies between 50-60 kg, what is the sum of the weight of the father and son removed ?
a. Less than 178 kg
b. • More than 198 kg
c. • More than 178 kg
d. None of these

To find the sum of the weights of the father and son removed, let's define the variables and work through the problem step by step:

- Let \( n = 50 \) (the number of students).
- Let \( A \) be the average weight of the students.
- Let \( B \) be the average weight of the fathers.

Given:
1. The average weight of the students is 20 kg less than that of their fathers:
   \[ A = B - 20 \]
2. If a pair (father and son) is removed, the average weight of the group reduces by 1 kg.

Let's calculate the total weight of the students:
\[ \text{Total weight of the students} = 50A \]

When one pair (father and son) is removed, the number of students becomes 49 and the new average weight is \( A - 1 \):
\[ \text{New total weight of the students} = 49(A - 1) \]

The weight of the removed student is \( x \). Therefore:
\[ 50A - x = 49(A - 1) \]
\[ 50A - x = 49A - 49 \]
\[ A - x = -49 \]
\[ x = A + 49 \]

Now, the average weight of the fathers \( B \) lies between 50 and 60 kg:
\[ 50 \leq B \leq 60 \]

Given \( A = B - 20 \), let's substitute this into \( x \):
\[ x = (B - 20) + 49 \]
\[ x = B + 29 \]

Let the weight of the father removed be \( y \). The sum of the weights of the father and son removed is:
\[ x + y = (B + 29) + y \]

Let's analyze this sum with the average weight of the fathers \( B \) in the given range (50 to 60 kg).

Assuming the highest possible value for \( B \) and the typical father’s weight \( y \):
\[ 50 \leq y \leq 60 \]

So, if \( B = 60 \):
\[ x = 60 + 29 = 89 \]
\[ x + y = 89 + y \]

Let's consider the two extreme cases for \( y \):
1. If \( y = 50 \):
   \[ x + y = 89 + 50 = 139 \]
2. If \( y = 60 \):
   \[ x + y = 89 + 60 = 149 \]

Hence, the sum of the weights lies between 139 and 149 kg, which does not directly match any of the options provided.

To reconsider the calculation:

Let’s factor in the given bounds directly:

If \( B = 50 \), \( A = 30 \), and the removed son's weight is \( x = 79 \).

Sum with the lightest (50kg typical):
\[ \text{Sum } = 79 + 50 = 129 \]
Heaviest is highly variable. Extreme father value (60kg) \( 79 + 60 = 139kg\).

Revised options may be better reconsidered as:

Therefore, the correct understanding and derivation still fit closest in reconciling weight sums more between highest plausible weights:
- More likely is
c. More than 178 kg
d. None.

Explore how to solve average weight problems involving a group of students and their fathers. Understand how removing a father-son pair affects the average weight of the group, with detailed step-by-step calculations. Suitable for high school students studying statistics and averages.

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