Question

In a casino on a Saturday, 90% of the visitors lost $100 each, 9% of the visitors lost $1,000 each, and the rest won $10,000 each. If the profit of the casino is $80,000, how many people visited the casino?

Answer 1

x = total amount of visitors

so we have 90% that lost $100, 9% that lost $1000, now 90+9 = 99%, what's left is just 1% then, so 1% won $10,000.

Profit is Revenue - Cost.

so the profit for the casino, is not who wins, because from the wins the casino needs to pay that to the winner, so they're losing that much, so for the casino, profits is when a player loses, so their profit will come from the 99% that lost money then.

keeping in mind that

$\bf \begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em} \begin{array}{llll} \stackrel{\textit{90\% of x}}{\left( \cfrac{90}{100} \right)x}\implies 0.9x\qquad \stackrel{\textit{9\% of x}}{\left( \cfrac{9}{100} \right)x}\implies 0.09x \\\\\\ \stackrel{\textit{1\% of x}}{\left( \cfrac{1}{100} \right)x}\implies 0.01x \end{array}$

since we know that 90% lost $100, that means 0.9x(100) is the total amount lost by all of them, since we also know that 9% lost $1000, then 0.09x(1000) is the total amount lost by them, and the wins are of course just 0.01x(100000).

$\bf \stackrel{\textit{90\% at \ 100}}{0.9x(100)}~~+~~\stackrel{\textit{9\% at \ 1000}}{0.09x(1000)}~~-~~\stackrel{\textit{1\% at \ 10000}}{0.01x(10000)}~~=~~\stackrel{\textit{casino's profit}}{80000} \\\\\\ 90x+90x-100x=80000\implies 80x=80000\\\\\\ x=\cfrac{80000}{80}\implies x=1000$