4. A humanitarian society has earned $2023 €$ from selling 150 tickets for a charity performance. More than 100 tickets were sold at full price, which has an integer value. The remaining tickets were sold at "half price". What is the revenue from tickets sold at full price?
4. A humanitarian society earned 2023 € from selling 150 tickets for a charity performance. More than 100 tickets were sold at full price, which is an integer amount. The remaining tickets were sold "at half price". What is the revenue from tickets sold at full price?

## Solution.

Let $x$ be the price of a ticket sold at full price, $x \in \mathbb{N}$.

Then $\frac{x}{2}$ is the price of a ticket sold "at half price".

Let $y$ be the number of tickets sold at full price.

Then 150 - y is the number of tickets sold "at half price".

The revenue from ticket sales is then:

$$
x \cdot y+\frac{x}{2} \cdot(150-y)=2023
$$

It follows that

$$
\begin{gathered}
x \cdot y+75 x-\frac{x}{2} \cdot y=2023 \\
2 x y+150 x-x y=4046 \\
x y+150 x=4046 \\
x \cdot(y+150)=4046
\end{gathered}
$$

The number of tickets sold at full price is greater than 100 and less than 150,

$$
100<y<150
$$

so it follows that

$$
250<y+150<300
$$

Since $x \in \mathbb{N}$, it follows that $y+150$ is a divisor of the number 4046, and since the prime factorization of 4046 is $4046=2 \cdot 7 \cdot 17 \cdot 17$, the only possibility is $y+150=289$, i.e., $y=139$.

Then $x=4046: 289=14$.

The revenue from tickets sold at full price is $14 \cdot 139=1946 €$.

Note. From the equation

$$
x \cdot(y+150)=4046
$$

it is possible to determine the solution by checking all cases

$$
x \in\{1,2,7,14,17,34,119,238,289,578,2023,4046\}
$$