Question 173693
I follow what you were trying to do, but I think you made this too difficult on yourself.


{{{(10t+u)/(10u+t)=4/7}}} is spot on.


But if the value of the original fraction is {{{4/7}}}, the value of the original fraction's reciprocal must be {{{7/4}}}, so for the second relationship I think you should use:


{{{(10t+u+16)/(10u+t-5)=7/4}}}


Cross-multiplying and collecting like terms until the equations are in standard {{{Ax+By=C}}} form, you should have the following:


Eq. 1: {{{66t-33u=0}}} and Eq. 2: {{{33t-66u=-99}}}


Multiply Eq. 1 by -2:  Eq. 3: {{{-132t+66u=0}}}


Add Eq. 3 to Eq. 2:  {{{-99t=-99}}} →  {{{t=1}}}


Substitute into Eq. 1:  {{{66(1)-33u=0}}} → {{{33u=66}}} → {{{u=2}}}


Therefore, the original fraction is {{{12/21}}} which has a value of {{{4/7}}} and further, {{{(12+16)/(21-5)=28/16=7/4}}} and the answer checks.