2530.52 – Defined Operation 4; Do Composition

Here are two made-up operations, $g$ and $r$ . If $10 g = 16$ , $100 g = 196$ , and $4 g = 4$ ; and also if $12 r = 5$ , $300 r = 101$ , $30 r = 11$ , then what is $n$ if $n g r = 9$ ?

(Notation: $n g r$ means we do $g$ to $n$ and then $r$ to that. So $n g r$ means $(ng)r$ .)

Solution

$\begin{align*}g:&& 10 &\rightarrow 16 \\&&100 &\rightarrow 196 \\&&4 &\rightarrow 4 \\&&x &\rightarrow 2x - 4\end{align*}$

$\begin{align*}r:&& 12 &\rightarrow 5 \\&&300 &\rightarrow 101 \\&&30 &\rightarrow 11 \\&&x &\rightarrow \dfrac{x}{3} + 1\end{align*}$

$\begin{align*}g r:&& n \rightarrow \left(\dfrac{2n - 4}{3}\right) + 1 &= 9 \\[1em]&&\dfrac{2n - 4}{3} + 1 &= 9 \\[1em]&&2n - 4 &= 24 \\[1em]&& n &= 14\end{align*}$

Check: $14 g r = \dfrac{2 \times 14 - 4}{3} + 1 = \dfrac{24}{3} + 1 = 9$