tens digit of the numerator = t
units digit of the numerator = u
Numerator = 10t + u

>>...The two digits in the numerator are interchanged in the denominator...<<

tens digit of the denominator = u
units digit of the denominator = t
denominator = 10u + t

>>...A fraction has a value of ...<<

 = 

>>...If 11 is added to the numerator, and 22 is subtracted from the
denominator, the resulting fraction is the recipricol of the 
original fraction...<<

 = 

So we have this system of equations to solve:



We simplify the first equation:

 = 

Cross-multiply:

7(10t + u) = 4(10u + t)

  70t + 7u = 40u + 4t

       66t = 33u 

Divide both sides by 33

        2t = u 





Cross-multiply:

4(10t + u + 11) = 7(10u + t - 22)
  40t + 4u + 44 = 70u + 7t - 154
      33t + 198 = 66u 

Divide both sides by 33

          t + 6 = 2u 

So the system is now



Solve that by substitution and get

t = 2, u = 4 

numerator = 10t + u = 10(2) + 4 = 20 + 4 = 24
denominator = 10u + t = 10(4) + 2 = 40 + 2 = 42

fraction = 

Edwin