The sum of five consecutive odd numbers is 585. What is the sum of second largest number and twice the smallest number?

Options

A. 335
B. 332
C. 328
D. None of these

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Define the Consecutive Odd Numbers
Let the first odd number be \( x \). The five consecutive odd numbers can be represented as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \)
- Fourth number: \( x + 6 \)
- Fifth number: \( x + 8 \)

### Step 2: Set Up the Equation
The sum of these five consecutive odd numbers is given to be 585. Therefore, we can write the equation:
\[ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 585 \]

### Step 3: Simplify the Equation
Combine like terms:
\[ 5x + (2 + 4 + 6 + 8) = 585 \]
Calculating the sum of the constants:
\[ 2 + 4 + 6 + 8 = 20 \]
So the equation becomes:
\[ 5x + 20 = 585 \]

### Step 4: Solve for \( x \)
Subtract 20 from both sides:
\[ 5x = 585 - 20 \]
\[ 5x = 565 \]
Now, divide by 5:
\[ x = \frac{565}{5} = 113 \]

### Step 5: Determine the Five Odd Numbers
Now that we have \( x \), we can find the five consecutive odd numbers:
- First number: \( 113 \)
- Second number: \( 115 \)
- Third number: \( 117 \)
- Fourth number: \( 119 \)
- Fifth number: \( 121 \)

### Step 6: Identify the Second Largest and Smallest Numbers
The second largest number is \( 119 \) and the smallest number is \( 113 \).

### Step 7: Calculate Twice the Smallest Number
Twice the smallest number is:
\[ 2 \times 113 = 226 \]

### Step 8: Find the Final Sum
Now, we need to find the sum of the second largest number and twice the smallest number:
\[ 119 + 226 = 345 \]

### Final Answer
The sum of the second largest number and twice the smallest number is \( 345 \).

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