You and a friend are running on treadmills. you run 0.5 mile every 3 minutes and your friend runs 2 miles every 14 minutes. you both start and stop running at the same time and run a whole number of miles. what is the least possible number of miles you and your friend can run?

I run 1 mile every 6 minutes.
My friend runs 1 mile every 7 minutes.
Lowest Common Multiple:  LCM ( 6, 7 ) = 42 minutes.
Me : 42 : 6 = 7 miles
My friend : 42 : 7 = 6 miles. 

Start by finding how long it takes each to run 1 mile.

You: 0.5 mile every 3 minutes => 1 mile every 6 minutes

Your friend 2 miles every 14 minutes => 1 mile every 7 minutes

Now you have to calculate the minimum common multiple of 6 and 7.

For that you need to factor both numbers and take common and non common factors raised to their highest exponent:

6 = 3*2
7 = 7

Then the minimum common multiple is 2*3*7=42.

Now divide the time 42 by the humber of minutes each of you take to run 1 mile

you: 42 minutes / 6 minutes per mile = 7 miles
your friend 42 minutes / 7 minutes per mile = 6 miles.

You can run 7 miles and your friend 6 miles in the same time.

7 and 6