CEMC Galois, 2022

The ratio of Alice’s contribution to Bello’s contribution was
$3:8$.

This means that for every $\$3+\$8=\$11$ contributed to starting the
business, Bello contributed $8, and so Bello contributed $\dfrac{8}{11}$ of the cost of starting the
new business.

If the cost of starting the new business was $9240, then Bello’s
contribution to this cost was $\dfrac{8}{11}\times\$9240=\$6720$.
The ratio of Alice’s share of the first year profits to Bello’s
share was $3:8$.

This means that for every $\$3+\$8=\$11$ of profit in the first year,
Alice’s share was $3.

Let the first year profit be $\$P$.

Since Alice’s share of the first year profit was $1881, then $\dfrac{3}{11}\times P=1881$ or $P=\dfrac{1881\times11}{3}$, and so $P=\dfrac{20\,691}{3}=6897$.

The total profit of the business for the first year was $6897.
The ratio of Alice’s share of the second year profits to Bello’s
share was $3:(8+x)$.

This means that for every $\$3+\$(8+x)=\$(11+x)$ of profit in the
second year, Bello’s share was $\$(8+x)$.

Since Bello’s share of the second year profit was $5440, and the profit
that year was $6400, then $\dfrac{(8+x)}{(11+x)}\times 6400=5440$.

Solving this equation, we get $$\begin{align*} \dfrac{(8+x)}{(11+x)}\times 6400&=5440\\ 6400(8+x) &= 5440(11+x)\\ 20(8+x) &= 17(11+x) & \text{(dividing both sides by 320)}\\ 160+20x&=187+17x\\ 3x&=27\end{align*}$$ and so $x=9$.