in a given training week a swimmer completes 1/4 mile on day one 2/5 mile on day two and 3/7 mile on day three,, how many total miles does the athlete swim in the given week

We'll have to add up all the fractions. 
First, find the LCM of the denominators.
To do this, we'll use the listing method. 

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, all the way to 140.
5: 5, 10, 15, 20, 25, 30, 35, all the way to 140.
7: 7, 14, 21, 28, 35, 42, 49, all the way to 140.

So, 140 is the Least Common Multiple. 

Multiply the numerator of 1 as many times you multiplied 4 to get to 140, which is 35.
1 x 35 = 35.

Multiply the numerator of 2 as many times you multiplied 5 to get to 140, which is 28.
2 x 28 = 56

Multiply the numerator of 3 as many times you multiplied 7 to get to 140, which is 20.
3 x 20 = 60.

So: 35/140 + 56/140 + 60/140 
=35/140 + 116/140 
=151/140
=1 11/140

So, the swimmer swam 1 and 11/140th of a mile.