[latex]frac{1}{10} = frac{10^{12}}{frac{ab}{a+b}+10^{12}} iff frac{ab}{a+b}+10^{12} = 10^{13} iff frac{ab}{a+b} = 10^{13}-10^{12}iff \\iff ab = (10^{13}-10^{12})a+(10^{13}-10^{12})b iff \\iff ab-(10^{13}-10^{12})a =aleft(b-10^{13}+10^{12} ight)= (10^{13}-10^{12})b iff\\iff a= frac{(10^{13}-10^{12})b}{b-10^{13}+10^{12}}[/latex]
[latex] dfrac{1}{10} = dfrac{10^{12} }{ dfrac{ab}{a+b}+ 10^{12} } \ dfrac{ab}{a+b}+10^{12}=10cdot10^{12}\ dfrac{ab}{a+b}=10^{13}-10^{12}\ ab=(10^{13}-10^{12})(a+b)\ ab=(10^{13}-10^{12})a+(10^{13}-10^{12})b\ ab-(10^{13}-10^{12})a=(10^{13}-10^{12})b\ a(b-(10^{13}-10^{12}))=(10^{13}-10^{12})b\ a=dfrac{(10^{13}-10^{12})b}{b-(10^{13}-10^{12})}\ oxed{a=dfrac{(10^{13}-10^{12})b}{b-10^{13}+10^{12}}} [/latex]
