Milltown has 1600 phones of which 3/4 are dial. If 1/3 of these are replaced by touch tone phones and 300 additional touch tone phones are installed. What fractions of phone are now manually operated?

Answer 1

Let's break down the problem step by step.

1. Calculate total number of dial phones: There are 1600 phones in total, and \( \frac{3}{4} \) of them are dial phones. \[ \text{Number of dial phones} = \frac{3}{4} \times 1600 = 1200 \]

2. Determine how many dial phones are replaced: One-third of the dial phones are replaced by touch tone phones. \[ \text{Number of dial phones replaced} = \frac{1}{3} \times 1200 = 400 \]

3. Calculate remaining dial phones: After replacing 400 dial phones with touch tone phones, the remaining number of dial phones is: \[ \text{Remaining dial phones} = 1200 - 400 = 800 \]

4. Calculate the number of touch tone phones after replacement: Initially, there were 0 touch tone phones. After replacing 400 dial phones with touch tone phones, the total number of touch tone phones is: \[ \text{Touch tone phones after replacement} = 0 + 400 = 400 \] Then, we add the 300 additional touch tone phones: \[ \text{Total touch tone phones} = 400 + 300 = 700 \]

5. Calculate the total number of phones: The total number of phones remains 1600.

6. Determine the fraction of manually operated phones: The manually operated phones are the remaining dial phones. We have: \[ \text{Number of manually operated phones} = 800 \] Thus, the fraction of phones that are manually operated is: \[ \text{Fraction of manually operated phones} = \frac{800}{1600} = \frac{1}{2} \]

Therefore, the fraction of phones that are now manually operated is \( \frac{1}{2} \).