### Each one of $A$ and $B$ has some money. If $A$ gives $90$ to $B$ then $B$ will have twice the money left with $A$. But, if $B$ gives $30$ to $A$ then $A$ will have thrice as much as is left with $B$. How much money does each have?

$A$ has $186$ and $B$ has $102.$

Step by Step Explanation:
1. Let us assume $A$ and $B$ have $x$ and $y$ respectively.
2. $A$ gives $90$ to $B$ and then $B$ will have twice the money left with $A$.

Money with $A$ = $$(x - 90)$ Money with $B$ =$ $(y + 90)$ \begin{aligned} \therefore \space & 2(x - 90) = (y + 90) \\ \implies & 2x - 180 = y + 90 \\ \implies & 2x - y = 270 && \ldots (1) \end{aligned}
3. $B$ gives $30$ to $A$ and then $A$ will have thrice as much as is left with $B$.

Money with $A$ = $$(x + 30)$ Money with $B$ =$ $(y - 30)$ \begin{aligned} \therefore \space & x + 30 = 3(y - 30) \\ \implies & x + 30 = 3y - 90 \\ \implies & x - 3y = -120 && \ldots (2) \end{aligned}
4. On multiplying $(1)$ by $3$ we get \begin{aligned} & 6x - 3y = 810 && \ldots (3)\end{aligned}Now, subtracting $(2)$ from $(3),$ we get \begin{aligned} & 5x = 930 \\ \implies & x = 186 \end{aligned}
5. Now, substituting $x = 186$ in $(1),$ we get\begin{aligned} & 2(186) - y = 270 \\ \implies & y = 372 - 270 = 102 \end{aligned}
6. Hence, $A$ has $186$ and $B$ has $102.$