Question 782884
The difference in their heights is
{{{61&1/12-59&1/9}}}
The result is {{{1&35/36}}} inches of difference in their heights (almost 2 inches).
The mother is {{{1&35/36}}} inches taller (almost 2 inches taller).
That makes sense because the approximation, calculated dropping the timy {{{1/12}}} and {{{1/9}}} fractions is
61 inches - 59 inches = 2 inches.
 
When we have to add or subtract fractions, we need a common denominator.
In this case, I would choose, {{{36}}}, because it the smallest common multiple of {{{12}}} and {{{9}}}:
{{{12*3=36}}}} and {{{9*4=36}}}
 
THE HARD WAY:
{{{61&1/12-59&1/9=61+1/12-(59+1/9)=61*12/12+1/12-(59*9+1/12)=732/12+1/12-(531/9+1/9)=733/12-532/9=733*3/(12*3)-532*4/(9*4)=2199/36-2128/36=71/36}}}
That result can be simplified;
{{{71/36=(36+35)/36=36/36+35/36=1+35/36=1&35/36}}}
 
MY WAY:
{{{61&1/12-59&1/9=61+1/12-(59+1/9)=60+(1+1/12)-59-1/9=60+13/12-59-1/9=60-59+13/12-1/9=1+13/12-1/9=1+13*3/(12*3)-1*4/(9*4)=1+39/36-4/36=1+35/36=1&35/36}}}
 
Sadly, teachers will probably favor the "HARD WAY". 
It requires more complicated calculations, and mistakes are more likely, but students can memorize and (at least temporarily) remember a recipe to calculate the result. Understanding why such calculation are done that way; how and why calculations could be done differently, and/or how to estimate the result could be valuable bonuses.