An insurance company's revenues increased by $25 \%$ and its expenses by $15 \%$ compared to the previous year in 2006. The company's profit (revenue - expenses) increased by 40\%. What percentage of the revenues were the expenses in 2006?
Solution. Let $x$ be the total revenue of the insurance company in 2005, and $y$ be the expenses. The company's profit is given by the difference $x-y$. After a $25 \%$ increase, the revenue becomes $1.25 x$; while after a $15 \%$ increase, the expenses become $1.15 y$.

We can write the following equality:

$$
1.25 x - 1.15 y = 1.4(x - y)
$$

From this, $y = 0.6 x$, and the expenses in 2006 are $1.15 \cdot 0.6 x$. The question is: what percentage of the $1.25 x$ revenue is this.

$$
\frac{1.15 \cdot 0.6 x}{1.25 x} = 0.552
$$

Therefore, the expenses are $55.2 \%$ of the revenue.