Let \( d \) be the distance in miles from Ken's home to the restaurant. We can set up equations for the total cost for each app:
For the first app: \[ \text{Cost}_1 = 0.50d + 3 \]
For the second app: \[ \text{Cost}_2 = 0.75d \]
We want to find the distance \( d \) such that the costs of both apps are equal: \[ 0.50d + 3 = 0.75d \]
Next, we can solve for \( d \). First, we can subtract \( 0.50d \) from both sides: \[ 3 = 0.75d - 0.50d \] \[ 3 = 0.25d \]
Now, divide both sides by \( 0.25 \): \[ d = \frac{3}{0.25} \]
Calculating this gives: \[ d = 3 \div 0.25 = 3 \times 4 = 12 \]
Thus, the distance from Ken's home to the restaurant for both apps to cost the same is \( \boxed{12} \) miles.