Sum = product = quotient
Can you find a pair of numbers such that their sum, product and quotient are all equal? Are there any other pairs?
Age
14 to 16
Challenge level
Problem
How many pairs of numbers $(a,b)$ exist such that the sum $a+b$, the product $ab$, and the quotient $\frac{a}{b}$ are all equal?
Student Solutions
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: $\frac12\times-1=-\frac12, \quad \frac12\div-1=\frac1{-2}=-\frac12, \quad \frac12+-1=-\frac12$)