Average Weight Problem: New Boy's Weight

This question involves calculating the weight of a new individual when the average weight of a group changes after a replacement. We need to find the weight of the new boy given the change in the average weight of 5 boys.

Average Weight Calculation Basics

The average (or mean) is calculated by dividing the sum of all values by the number of values. In this case:

$$ \text{Average Weight} = \frac{\text{Total Weight of Boys}}{\text{Number of Boys}} $$

When one boy is replaced, the total weight changes, which can affect the average weight.

New Boy's Weight Calculation Steps

1. Initial Values and Goal

   • Number of boys = 5
   • Weight of the boy being replaced ($W_{old}$) = 39 kg
   • The average weight increased by 9 kg.
   • Goal: Find the weight of the new boy ($W_{new}$).

2. Calculating the Total Weight Change

   An increase in the average weight means the total weight of the group has also increased. The total increase in weight is the increase in average weight multiplied by the number of boys.

   Increase in Average Weight = 9 kg

   Number of Boys = 5

   Total Increase in Weight = (Increase in Average Weight) $ \times $ (Number of Boys)

   $$ \text{Total Increase} = 9 \text{ kg} \times 5 = 45 \text{ kg} $$

   This total increase of 45 kg represents the difference between the weight of the new boy and the weight of the boy who was replaced.

3. Calculating the New Boy's Weight

   The change in total weight is the difference between the incoming boy's weight ($W_{new}$) and the outgoing boy's weight ($W_{old}$).

   Total Increase in Weight = $W_{new} - W_{old}$

   We know the Total Increase is 45 kg and $W_{old}$ is 39 kg.

   $$ 45 \text{ kg} = W_{new} - 39 \text{ kg} $$

   To find the weight of the new boy ($W_{new}$), we rearrange the equation:

   $$ W_{new} = 45 \text{ kg} + 39 \text{ kg} $$

   $$ W_{new} = 84 \text{ kg} $$

Final Answer Verification

The calculation shows that the new boy's weight must be 84 kg to cause the average weight of the 5 boys to increase by 9 kg after replacing the 39 kg boy.