Question

A theater manager predicts that $1000$ tickets to a play will be sold if each ticket costs $\$60$. The manager predicts that $20$ less tickets will be sold for every $\$1$ increase in price. For what prices can the manager predict that at least $800$ tickets will be sold?

Answer 1

Step 1 : Given data

The $1000$ tickets will be sold when each ticket costs $\$60$.

Step2: Solve for the prices at which at least $800$ tickets sold

The manager wants to sell at least $800$tickets which is maximum
$1000–800=200$ tickets less than $1000$.

Given that for $20$ less tickets will be sold for every $\$1$ increase in price.
So,$20\times 2=40$ less tickets will be sold for $\$1\times 2=\$2$ increase in price.

Proceeding in a similar way, for any natural number $n$:
$20xn$ less tickets will be sold for $\$1n=\$n$ increase in price.
Here, $200=20\times 10$, so $n=10$
So, $20\times 10$ less tickets will be sold for $\$1\times 10=\$10$ increase in price.
Hence, the manager can increase the price of each ticket up to $\$10$.
So, to sell at least $800$ tickets, the maximum price of each ticket can be $\$60+\$10=\$70$.
Hence, the value of $p$ is $p\le \$70.$