Let’s call the sum of the set of five numbers S and the sixth number N. We know the mean of the set of five numbers is 3k, so we can write the equation S/5 = 3k or S = 15k. When we add the sixth number the mean increases by k, so we can write the equations (S + N)/6 = 4k or S + N = 24k. Using S = 15k, we find that N = 9k. The ratio of the sixth number to the sum of the set of five numbers is therefore N/S = 9k/15k = 3/5.