## Average of Two Numbers

The average of $a$ and $b$ is $x$. The average of $b$ and $x$ equals the average of $a$ and $(b+1)$. Find the value of $a-b$.
Source: NCTM Mathematics Teacher 2008

SOLUTION
Average of $a$ and $b$
$\dfrac{a+b}{2}=x\qquad\qquad\qquad\: (1)$
Average of $b$ and $x$
$\dfrac{b+x}{2}=\dfrac{a+(b+1)}{2}\qquad (2)$
Simplify Eq. $(2)$
$\dfrac{b+x}{2}=\dfrac{a+b}{2}+\dfrac{1}{2}$
Substitute the value of $\dfrac{a+b}{2}$ from Eq. $(1)$
$\dfrac{b+x}{2}=x+\dfrac{1}{2}$
Simplify
$b+x=2x+1$
$x=b-1$
Substitute the value of $x$ into Eq. $(1)$
$\dfrac{a+b}{2}=b-1$
Simplify
$a+b=2b-2$
$a-b=-2$

Answer: $-2$