15 ! : my friend sets out walking at a speed of 3 miles per hour. i set out behind her 5 minutes later at 4 miles per hour. for how many minutes will my friend have been walking when i catch up to her?

The relative speed between you two is 1 mile per hour (4-3). After five minutes your friend will have covered a distance of  3× 5/60 = 0.25 miles ( from the formula distance= speed ×time).
The distance between the two of you by the time you are starting the journey is 0.25 miles.
The time taken before you catch up with your friend will be;
 0.25 miles/ 1 mile/hr = 0.25 hours  ( distance/relative speed) equivalent to 15 minutes. Therefore, your friend will have walked for 15+5 = 20 minutes before catching up with him/her

The total distance covered is equal to the product of speed and time. The distances that should have been traveled by the two friends in this item should be equal. Let x be the total time that the first among them have walked. The time of the second person should be x - 5/60. 

The equation that would best represent their motion is,

                3(x) = 4(x - 5/60)

The value of x from the equation is equal to 0.33 hours which is equal to 20 minutes. 

20 minutes