The average of four consecutive odd numbers A,B, C and D respectively is 54 . What is the product of A and C ?

Options

A. 2907
B. 2805
C. 2703
D. 2915

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We need to find the product of two consecutive odd numbers A and C, given that the average of four consecutive odd numbers A, B, C, and D is 54.

### Step 2: Define the Odd Numbers
Let’s denote the first odd number as A. Since they are consecutive odd numbers, we can express them as:
- A = first odd number
- B = A + 2 (the next odd number)
- C = A + 4 (the next odd number)
- D = A + 6 (the next odd number)

### Step 3: Set Up the Average Equation
The average of these four numbers is given by the formula:
\[ \text{Average} = \frac{A + B + C + D}{4} \]
Substituting the expressions for B, C, and D:
\[ \text{Average} = \frac{A + (A + 2) + (A + 4) + (A + 6)}{4} \]
This simplifies to:
\[ \text{Average} = \frac{4A + 12}{4} \]
\[ \text{Average} = A + 3 \]

### Step 4: Set the Average Equal to 54
According to the problem, the average is 54:
\[ A + 3 = 54 \]

### Step 5: Solve for A
Now, we can solve for A:
\[ A = 54 - 3 \]
\[ A = 51 \]

### Step 6: Find C
Now that we have A, we can find C:
\[ C = A + 4 = 51 + 4 = 55 \]

### Step 7: Calculate the Product of A and C
Now we can find the product of A and C:
\[ \text{Product} = A \times C = 51 \times 55 \]

### Step 8: Perform the Multiplication
Calculating the product:
\[ 51 \times 55 = 2805 \]

### Final Answer
Thus, the product of A and C is **2805**.

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