Two numbers are in the ratio 5 : 6.

Let the two numbers be x and y.

Then, by first Condition, ratio of these two numbers = 5:6

$x: y=5: 6$

$\Rightarrow$ $\frac{x}{y}=\frac{5}{6} \Rightarrow y=\frac{6 x}{5}$ $\ldots$ (i)

and by second condition, then, 8 is subtracted from each of the numbers, then ratio becomes $4: 5$.

$\frac{x-8}{y-8}=\frac{4}{5}$

$\Rightarrow \quad 5 x-40=4 y-32$

$\Rightarrow \quad 5 x-4 y=8$ ...(ii)

Now, put the value of $y$ in Eq. (ii), we get

$5 x-4\left(\frac{6 x}{5}\right)=8$

$\Rightarrow \quad 25 x-24 x=40$

$\Rightarrow \quad x=40$

Put the value of $x$ in Eq. (i), we get 

$y=\frac{6}{5} \times 40$

$=6 \times 8=48$

Hence, the required numbers are 40 and 48.