Unknown fraction

Since $A < B$ and the difference between $A$ and $B$ is $2$ , we can write $B = A + 2$ . This turns the original fraction into $\frac{A}{A+2}$ . We can write the information in the question as $\frac{A}{A+2} + \frac{1}{6} = \frac{(A) + 2}{(A+2) + 3}$ $\frac{A}{A+2} + \frac{1}{6} = \frac{A + 2}{A+5}$ Multiplying through by $6(A+2)(A+5)$ , $6A^2 + 30A + A^2 + 7A + 10 = 6A^2 + 24A + 24$ $A^2 + 13A - 14 = 0$ Hence, $A = -14$ or $A = 1$ . The only sensible solution is then $\boxed{\frac{1}{3}}$