8 4/8 is 8 1/2 and 2 2/8 is 2 1/4 in their simplest form. To make it easier, double each number twice: 8 1/2 becomes 17 then 34; 2 1/4 becomes 4 1/2 then 9. This step gets rid of the fractions for the time being, We must remember to halve the numbers twice when we get to the end of the calculation. So if the two numbers are a and b, a+b=34 and a-b=9. Add these two equations: 2a=43, so a=43/2=21 1/2; we get b by subtracting 21 1/2 from 34=12 1/2.
We've done the main calculation now, so we need to halve these numbers twice: 21 1/2 halved is 21 halved plus 1/2 halved, so that's 10 1/2 plus 1/4=10 3/4. Now half again: 5 3/8.; 12 1/2 halved is 6 1/4 and halved again is 3 1/8. So the answer is the two numbers are 5 3/8 and 3 1/8. Let's add them together: 8 4/8. Correct! And subtract them: 2 2/8. Correct!