Let's represent Nemo's age with an n.
Barney is 8 years older so his age is nemo's plus 8. That is, Barney's age is n + 8
In seven years they will each be the age they are now plus 7. So, Nemo will be n + 7 and Barney will be n + 8 + 7. If the sum of their ages will be 46 we know that when we add these expressions we should get 46, so let's set the sum to 46 and solve for n.
n + 7 + n + 8 + 7 = 46
2n + 22 = 46
2n = 46 - 22
2n = 24
n = 12
That means that Nemo's age is 12.
Since Barney is 8 years older his age is 12 + 8 = 20.
To check our work we see that in 7 years Nemo will be 12 + 7 = 19 and Barney will be 20 + 7 = 27. Together that is 19 +27 = 46 as expected.