In the operation below, A and B represent distinct digits. What is the value of A + B

1A
xB3
_____
BA
+60
_____
6BA

(A) 4
(B) 5
(C) 9
(D) 45
(E) 54

It is difficult to follow in this strange Yahoo!Answer format.

Do you mean:

"1A" times "B3" equals "6BA"

If so, then A must be a digit where multiplying it by 3 leaves the last digit unchanged.
Therefore A is either 0 or 5.

[3*0=0, 3*1=3, 3*2=6, 3*3=9, 3*4=12, 3*5=15, 3*6=18, etc.]
Only 0 and 5 leave the last digit unchanged.
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We can check both separately:

is A=0 ?

10 times B3 = 6B0
this would force us to use B=6
10 times 63 = 660
not true.

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is A=5 ?

15 times B3 = 6B5

rewrite (I am assuming we are using base-10)

15(10B + 3) = 600 + 10B + 5
150B + 45 = 605 + 10B
140B = 560
B=4

Let's try with A=5 and B=4

15 * 43 = 645
1A * B3 = 6BA

it works!

A+B = 5+4

The BA and +60 are just the two steps of multiplying 1A by B and 3 separately, before summing the partial products to get the final result.

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Since B x 1A = 60, there are two cases: 5 x 12 (B=5 and A=2) or 4 x 15 (B=4 and A=5).

But 3 x 1A = BA, so A cannot be 2. Then A+B = 4+5 = 9.