The easiest way to find an answer closest to 1 is to find the LCM of all the denominators first, for which I used prime factorisation:
6 = 2*3
25= 5*5 (5^2)
5 = 5
20 = 2*2*5 (2^2*5)
15 = 3*5
8 = 2*2*2 (2^3)
To find the LCM we need to multiply together the highest number of 2s, 3s and 5s present in each set, overall. From above, we know that the highest number of 2s is 3 (2*2*2), the highest number of 3s is 1 (2*3 and 3*5) and the highest number of 5s is 2 (5*5).
Therefore we multiply 2*2*2*3*5*5 = 600, which is the LCM.
Then we can put all the fractions over the LCM:
100/600
24/600
360/600
90/600
160/600
75/600
To find the fractions that add to give an answer closest to 1, I first added all the above fractions together, giving 809/600 (remember 1 = 600/600).
I needed 209/600 less to make 1.
The fractions that add to make the closest to 209/600 were: 100/600, 24/600 and 75/600 (1/6, 1/25 and 5/8), summing to 199/600.
Therefore, I needed to add other fractions to give me the answer closest to 1:
3/5+3/20+4/15
360/600+90/600+160/600 = 610/600 or 1 1/60, the closest answer to 1.