Find the mean of the multiples of 7 between 15 and 55. \( \begin{array}{lllll}\text { A } 35 & \text { B } 40 & \text { C) } 45 & \text { D } 50\end{array} \)

To find the mean of the multiples of 7 between 15 and 55, we first need to identify the multiples of 7 within that range. The multiples of 7 can be found by calculating \(7n\) for integer values of \(n\). 1. **Finding the smallest multiple of 7 greater than 15:** \[ 7 \times 3 = 21 \quad (\text{since } 7 \times 2 = 14 \text{ is less than 15}) \] 2. **Finding the largest multiple of 7 less than 55:** \[ 7 \times 7 = 49 \quad (\text{since } 7 \times 8 = 56 \text{ is greater than 55}) \] 3. **Listing the multiples of 7 between 15 and 55:** The multiples of 7 in this range are: \[ 21, 28, 35, 42, 49 \] 4. **Calculating the mean:** To find the mean, we first sum these multiples: \[ 21 + 28 + 35 + 42 + 49 \] Calculating the sum step-by-step: \[ 21 + 28 = 49 \] \[ 49 + 35 = 84 \] \[ 84 + 42 = 126 \] \[ 126 + 49 = 175 \] So, the total sum is \(175\). 5. **Counting the number of terms:** There are 5 multiples of 7 in this range. 6. **Calculating the mean:** \[ \text{Mean} = \frac{\text{Sum of multiples}}{\text{Number of multiples}} = \frac{175}{5} = 35 \] Thus, the mean of the multiples of 7 between 15 and 55 is \(\boxed{35}\).