The average of 5 consecutive odd numbers is 27. What is the product of the first and the last number?

Options

A. 621
B. 667
C. 713
D. 725

Answer

To solve the problem, we need to find the product of the first and last numbers of 5 consecutive odd numbers whose average is 27. Here’s a step-by-step solution:

### Step 1: Understand the average of consecutive odd numbers
The average of a set of numbers is the sum of the numbers divided by the count of the numbers. In this case, we have 5 consecutive odd numbers.

### Step 2: Calculate the total sum of the numbers
Since the average of the 5 consecutive odd numbers is 27, we can find the total sum of these numbers:
\[ \text{Total Sum} = \text{Average} \times \text{Number of Terms} = 27 \times 5 = 135 \]

### Step 3: Identify the middle number
For 5 consecutive odd numbers, the middle number is also the average. Therefore, the middle number is 27.

### Step 4: Determine the 5 consecutive odd numbers
The 5 consecutive odd numbers can be expressed as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \) (which is 27)
- Fourth number: \( x + 6 \)
- Fifth number: \( x + 8 \)

From the third number, we have:
\[ x + 4 = 27 \implies x = 27 - 4 = 23 \]

Thus, the 5 consecutive odd numbers are:
- First number: 23
- Second number: 25
- Third number: 27
- Fourth number: 29
- Fifth number: 31

### Step 5: Calculate the product of the first and last numbers
Now we need to find the product of the first number (23) and the last number (31):
\[ \text{Product} = 23 \times 31 \]

### Step 6: Perform the multiplication
Calculating the product:
\[ 23 \times 31 = 23 \times (30 + 1) = 23 \times 30 + 23 \times 1 = 690 + 23 = 713 \]

### Final Answer
The product of the first and last number is \( 713 \).

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