Question from elementary algebra I'm having trouble with

Question

If 1/3 be added to the numerator of a certain fraction the fraction will be increased by 1/21, and if 1/2 be taken from the denominator the fraction becomes 8/9: find it.

  As per my understanding the conditions that can be applied from the information given in the question is let x/y be the fraction, then 

 (X+(1/3))/y = (x/y)+(1/21)......first eqn.
 x/(y-(1/2)) = 8/9.......second eqn.

Please point out how to proceed from here or correct me if there are any mistakes. The answer is given at the end of the question to be 3/14, and this question is supposed to be of 2 equation 2 variables but the first equation by itself gives the values of both x and y so I'm confused whether what I'm doing is wrong or whether there's a flaw in the question itself

Answer 1

If u take the first eq. and then take LCM on RHS,

[x+(1/3)]/y = [x+(y/21)]/y          [the y's on the denominator cancel out]
x+(1/3) = x+(y/21)                  [the x's cancel out]
1/3 = y/21                          [multiply 21 on both sides]
y = 7                               [substitute in eq. 2]

x/[y-(1/2)] = 8/9
x/[7-(1/2)] = 8/9
x/[13/2] = 8/9                       [multiply 13/2 on both sides]
x = 52/9

The fraction is x/y = (52/9)/7 = 52/(9*7) = 52/63 Therefore, the fraction is 52/63