The average of four consecutive odd numbers A, B, C and D respectively is 24. What is the product of B & D ?

Options

A. 483
B. 675
C. 621
D. 525

Answer

To solve the problem step by step, we will first identify the four consecutive odd numbers based on the given average and then calculate the product of B and D.

### Step 1: Understand the average of the numbers
The average of four consecutive odd numbers A, B, C, and D is given as 24. The average can be calculated using the formula:
\[ \text{Average} = \frac{A + B + C + D}{4} \]
Since the average is 24, we can write:
\[ \frac{A + B + C + D}{4} = 24 \]

### Step 2: Calculate the sum of the numbers
To find the sum of A, B, C, and D, we can multiply both sides of the equation by 4:
\[ A + B + C + D = 24 \times 4 = 96 \]

### Step 3: Express the numbers in terms of B
Let’s denote the four consecutive odd numbers. Since they are consecutive odd numbers, we can express them as:
- A = B - 3
- B = B
- C = B + 2
- D = B + 4

### Step 4: Substitute and simplify
Now, substituting A, C, and D in terms of B into the sum:
\[ (B - 3) + B + (B + 2) + (B + 4) = 96 \]
This simplifies to:
\[ 4B + 3 = 96 \]

### Step 5: Solve for B
Now, we can solve for B:
\[ 4B = 96 - 3 \]
\[ 4B = 93 \]
\[ B = \frac{93}{4} = 23.25 \]
Since B must be an odd integer, we realize that we need to check our expressions. The correct odd numbers should be centered around the average.

### Step 6: Identify the correct consecutive odd numbers
Since the average is 24, the middle two numbers (B and C) should be around 24. The consecutive odd numbers around 24 are:
- 21 (A)
- 23 (B)
- 25 (C)
- 27 (D)

### Step 7: Calculate the product of B and D
Now we can find B and D:
- B = 23
- D = 27

Now, we calculate the product of B and D:
\[ B \times D = 23 \times 27 \]

### Step 8: Perform the multiplication
Calculating the product:
\[ 23 \times 27 = 621 \]

### Final Answer
The product of B and D is **621**.

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