Answer :

[tex]+\left\{\begin{array}{ccc}a+b=76\\a-b=38\end{array}\right\\-----------\\.\ \ \ \ \ \ \ 2a=114\ \ \ \ \ |:2\\.\ \ \ \ \ \ \ \ \ \ a=57\\\\57+b=76\\b=76-57\\b=19\\\\\frac{a}{b}=\frac{57}{19}=\fbox{3}[/tex]
MattD
First we need to calculate how much is a and b, from the first and second equations, and then we take care of each other:
a + b = 76
a - b = 38
a / b = ?

[tex]\begin{cases}a+b=76\\a-b=38\end{cases}|+[/tex]
[tex]a+a+b-b=76+38[/tex]
[tex]2a=114|:2[/tex]

[tex]a=57[/tex]
[tex]a+b=76[/tex]
[tex]57+b=76[/tex]
[tex]b=76-57=19[/tex]

[tex]\frac{a}{b}=\frac{57}{19}=3[/tex]