The fractional part x-a will necessarily be less than 1, and so if we want to maximise x, we should use our highest number as a, so a=7. To maximise the fractional part (x-a) we need to minimise b+1/(c+1/d), and so we should use our smallest possible value for b, so b=1. Then c should be the larger of what is left, c=5, and so d=4.
Then working backwards to evaluate the fraction:
x=7+1/(1+1/(5+1/4))
x=7+1/(1+4/21)
x=7+21/25
x=7.84
