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'Sum = Product = Quotient' printed from https://nrich.maths.org/
Answer: one pair $(a=\frac12$ and $b= -1)$
$ab=\dfrac{a}{b}\Rightarrow ab^2=a \Rightarrow b^2=\dfrac aa=1$ so $b= 1$ or $-1$.
$a+b=ab$ becomes $a+1=a$ or $a-1=-a$
imposible $\Rightarrow 2a=1\\
\Rightarrow \ a=\tfrac12$
So $a=\frac12, b=-1$ is the only pair.
(check: $\frac12\times-1=-\frac12, \quad \frac12\div-1=\frac1{-2}=-\frac12, \quad \frac12+-1=-\frac12$)