The average of A , B and C is 18 and that of C, D and E is 12 and that of E and F is `6.5` and that of E and C is `3.5`. What is the average of A,B,C,D,E and F ?

Options

A. A)24
B. B)16
C. C)18
D. D)22

Answer

To solve the problem step by step, we will use the information given about the averages to find the required average of A, B, C, D, E, and F.

### Step 1: Calculate A + B + C
We know the average of A, B, and C is 18. The formula for average is:
\[ \text{Average} = \frac{\text{Sum of terms}}{\text{Number of terms}} \]
Thus, we can write:
\[ \frac{A + B + C}{3} = 18 \]
Multiplying both sides by 3 gives:
\[ A + B + C = 18 \times 3 = 54 \]

### Step 2: Calculate C + D + E
Next, we know the average of C, D, and E is 12:
\[ \frac{C + D + E}{3} = 12 \]
Multiplying both sides by 3 gives:
\[ C + D + E = 12 \times 3 = 36 \]

### Step 3: Calculate E + F
We are also given that the average of E and F is 6.5:
\[ \frac{E + F}{2} = 6.5 \]
Multiplying both sides by 2 gives:
\[ E + F = 6.5 \times 2 = 13 \]

### Step 4: Calculate E + C
The average of E and C is given as 3.5:
\[ \frac{E + C}{2} = 3.5 \]
Multiplying both sides by 2 gives:
\[ E + C = 3.5 \times 2 = 7 \]

### Step 5: Find D
Now we can use the equations we have to find D. From the equation \(C + D + E = 36\) and substituting \(E + C = 7\):
\[ D = 36 - (E + C) = 36 - 7 = 29 \]

### Step 6: Calculate A + B + C + D + E + F
Now we can find the total sum of A, B, C, D, E, and F:
\[ A + B + C + D + E + F = (A + B + C) + D + (E + F) \]
Substituting the values we found:
\[ A + B + C = 54, \quad D = 29, \quad E + F = 13 \]
Thus:
\[ A + B + C + D + E + F = 54 + 29 + 13 = 96 \]

### Step 7: Calculate the Average of A, B, C, D, E, and F
Finally, we can find the average:
\[ \text{Average} = \frac{A + B + C + D + E + F}{6} = \frac{96}{6} = 16 \]

### Conclusion
The average of A, B, C, D, E, and F is **16**.

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