The average weight of 5 men A, B, C, D and E is 94 kg. Another man F joins the group and the average now become 96 kg. Weight of E is 6 kg less than of weight F. Now A is replaced by G then the average weight becomes 98 kg. Find the difference the weight of A and G .

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Calculate the total weight of A, B, C, D, and E
The average weight of 5 men (A, B, C, D, and E) is given as 94 kg.

To find the total weight of these 5 men, we use the formula for average:
\[ \text{Average} = \frac{\text{Total Weight}}{\text{Number of Men}} \]
So, we can rearrange this to find the total weight:
\[ \text{Total Weight} = \text{Average} \times \text{Number of Men} = 94 \, \text{kg} \times 5 = 470 \, \text{kg} \]

### Step 2: Calculate the new total weight after F joins
When man F joins the group, the average weight becomes 96 kg for 6 men (A, B, C, D, E, and F).

Using the same formula for average:
\[ \text{Total Weight with F} = 96 \, \text{kg} \times 6 = 576 \, \text{kg} \]

### Step 3: Find the weight of F
Now, we can find the weight of F by subtracting the total weight of A, B, C, D, and E from the total weight with F:
\[ \text{Weight of F} = \text{Total Weight with F} - \text{Total Weight of A, B, C, D, E} = 576 \, \text{kg} - 470 \, \text{kg} = 106 \, \text{kg} \]

### Step 4: Find the weight of E
It is given that the weight of E is 6 kg less than the weight of F:
\[ \text{Weight of E} = \text{Weight of F} - 6 \, \text{kg} = 106 \, \text{kg} - 6 \, \text{kg} = 100 \, \text{kg} \]

### Step 5: Calculate the total weight of A, B, C, and D
Now, we can find the total weight of A, B, C, and D:
\[ \text{Total Weight of A, B, C, D} = \text{Total Weight of A, B, C, D, E} - \text{Weight of E} = 470 \, \text{kg} - 100 \, \text{kg} = 370 \, \text{kg} \]

### Step 6: Replace A with G and find the new average
When A is replaced by G, the average weight becomes 98 kg for 5 men (B, C, D, E, and G):
\[ \text{Total Weight with G} = 98 \, \text{kg} \times 5 = 490 \, \text{kg} \]

### Step 7: Find the weight of G
Now we can find the weight of G:
\[ \text{Weight of G} = \text{Total Weight with G} - \text{Total Weight of B, C, D, E} = 490 \, \text{kg} - 370 \, \text{kg} = 120 \, \text{kg} \]

### Step 8: Find the weight of A
Since we know the total weight of A, B, C, D is 370 kg, we can express the weight of A as:
\[ \text{Weight of A} = 370 \, \text{kg} - (\text{Weight of B} + \text{Weight of C} + \text{Weight of D}) \]
However, we don't have the individual weights of B, C, and D, but we can find the difference between the weights of A and G.

### Step 9: Find the difference between the weights of A and G
To find the difference between the weights of A and G:
\[ \text{Difference} = \text{Weight of A} - \text{Weight of G} \]
Since we don't have the exact weight of A, we can denote it as \( x \):
\[ \text{Difference} = x - 120 \]

### Conclusion
To find the exact difference, we need the weight of A. However, we can conclude that the difference between the weights of A and G is \( x - 120 \). If we had the weight of A, we could substitute it in to find the exact difference.