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The average weight of a certain number of persons in a group was 75.5 kg. Later on, 4 persons weighing 72.6 kg. 74 kg, 73.4 kg and 70 kg joined the group. As a result, the average weight of all persons in the group reduced by 500g. The number of persons in the group, initially, was

A

20

B

16

C

18

D

24

Text Solution

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The correct Answer is:
To solve the problem step by step, we need to find the initial number of persons in the group based on the information provided. ### Step 1: Define Variables Let \( N \) be the initial number of persons in the group. The average weight of these persons is given as 75.5 kg. ### Step 2: Calculate Total Initial Weight The total weight of the initial group can be calculated using the formula for average: \[ \text{Total Weight} = \text{Average Weight} \times \text{Number of Persons} \] Thus, the total weight \( W \) of the initial group is: \[ W = 75.5 \times N \] ### Step 3: Calculate the Total Weight of New Persons Four new persons join the group with weights: - 72.6 kg - 74 kg - 73.4 kg - 70 kg Calculating the total weight of these new persons: \[ \text{Total Weight of New Persons} = 72.6 + 74 + 73.4 + 70 = 290 kg \] ### Step 4: Calculate the New Total Weight The new total weight of the group after the new persons join is: \[ \text{New Total Weight} = W + 290 = 75.5N + 290 \] ### Step 5: Calculate the New Average Weight After the four new persons join, the total number of persons becomes \( N + 4 \). The new average weight is given to be 500 grams less than the initial average weight. Since 500 grams is equal to 0.5 kg, the new average weight is: \[ \text{New Average Weight} = 75.5 - 0.5 = 75 kg \] ### Step 6: Set Up the Equation for New Average Using the new average weight, we can set up the equation: \[ \text{New Average Weight} = \frac{\text{New Total Weight}}{\text{Total Number of Persons}} \] Substituting the values we have: \[ 75 = \frac{75.5N + 290}{N + 4} \] ### Step 7: Cross-Multiply and Simplify Cross-multiplying gives: \[ 75(N + 4) = 75.5N + 290 \] Expanding the left side: \[ 75N + 300 = 75.5N + 290 \] ### Step 8: Rearranging the Equation Rearranging the equation to isolate \( N \): \[ 300 - 290 = 75.5N - 75N \] \[ 10 = 0.5N \] ### Step 9: Solve for \( N \) Dividing both sides by 0.5: \[ N = \frac{10}{0.5} = 20 \] ### Conclusion The initial number of persons in the group was \( N = 20 \).
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