Our television was priced at \( \$ 564.00 \). However, our trade-in was valued at \( \frac{1}{3} \) of the price of our new television. How much did we actually pay for the new set?

To find out how much you actually paid for the new television after the trade-in, we first need to determine the value of the trade-in. The price of the new television is \( \$564.00 \). The trade-in value is given as \( \frac{1}{3} \) of the price of the new television. We can calculate the trade-in value as follows: \[ \text{Trade-in value} = \frac{1}{3} \times 564.00 \] Calculating this: \[ \text{Trade-in value} = \frac{564.00}{3} = 188.00 \] Now, we subtract the trade-in value from the price of the new television to find out how much you actually paid: \[ \text{Amount paid} = \text{Price of new television} - \text{Trade-in value} \] Substituting the values we have: \[ \text{Amount paid} = 564.00 - 188.00 = 376.00 \] Thus, the amount you actually paid for the new television is \[ \boxed{376.00} \]