13.212. A sports club allocated $n$ rubles for the purchase of bicycles. Due to a price reduction, the cost of each bicycle decreased by $a$ rubles, which allowed the club to buy $b$ more bicycles than initially planned. How many bicycles were purchased?
## Solution.

Let $x$ be the number of bicycles bought, and $(x-b)$ be the number intended to be bought. The price of one bicycle before the price reduction: $\frac{n}{x-b}$, after the reduction: $\frac{n}{x-b}-a$.

Then $\left(\frac{n}{x-b}-a\right) x=n$, from which $x=\frac{a b+\sqrt{a^{2} b^{2}+4 n b a}}{2 a}=$

$=\frac{a b+\sqrt{a b(a b+4 n)}}{2 a}$.

Answer: $\frac{a b+\sqrt{a b(a b+4 n)}}{2 a}$.