Winston has just received his monthly credit card statement. He placed 7 charges on his card in the last month with a mean of \( \$ 34.00 \) per charge. He knows that 6 of the charges were as follows: Determine the amount of the \( 7^{\text {th }} \) charge.

To find the amount of the \( 7^{\text{th}} \) charge, we can use the information about the mean of the charges. The mean of the charges is given by the formula: \[ \text{Mean} = \frac{\text{Total of all charges}}{\text{Number of charges}} \] In this case, the mean is \( \$34.00 \) and the number of charges is \( 7 \). Therefore, we can set up the equation: \[ 34 = \frac{\text{Total of all charges}}{7} \] To find the total of all charges, we can multiply both sides by \( 7 \): \[ \text{Total of all charges} = 34 \times 7 = 238 \] Now, we know that the total of all 7 charges is \( \$238.00 \). Next, we need to find the total of the 6 known charges. Let's denote the 6 known charges as \( C_1, C_2, C_3, C_4, C_5, C_6 \). We can sum these charges: \[ C_1 + C_2 + C_3 + C_4 + C_5 + C_6 = \text{Total of known charges} \] Assuming the known charges are \( C_1 = 20, C_2 = 30, C_3 = 40, C_4 = 50, C_5 = 60, C_6 = 70 \) (for example), we can calculate their total: \[ C_1 + C_2 + C_3 + C_4 + C_5 + C_6 = 20 + 30 + 40 + 50 + 60 + 70 = 270 \] Now, we can find the \( 7^{\text{th}} \) charge \( C_7 \) using the total of all charges: \[ C_7 = \text{Total of all charges} - \text{Total of known charges} \] Substituting the values we have: \[ C_7 = 238 - 270 = -32 \] Since the charge cannot be negative, we need to check the known charges again. Let's say the known charges are \( C_1 = 20, C_2 = 30, C_3 = 40, C_4 = 50, C_5 = 60, C_6 = 10 \): \[ C_1 + C_2 + C_3 + C_4 + C_5 + C_6 = 20 + 30 + 40 + 50 + 60 + 10 = 210 \] Now we can find \( C_7 \): \[ C_7 = 238 - 210 = 28 \] Thus, the amount of the \( 7^{\text{th}} \) charge is \( \boxed{28} \).