The School Festival Ticket Booth
A class runs a ticket booth at the school festival. Adult tickets cost 300 yen and student tickets cost 150 yen. On Saturday they sold 120 tickets in total and collected 27,000 yen. On Sunday the class wants to collect exactly 30,000 yen. They plan to sell 140 tickets in total, but they are unsure how many of each type to aim for. The class president says, 'If we just sell more adult tickets than Saturday, we will reach 30,000 yen.' A member notes that the number of tickets sold must be a whole number and that neither type can be negative.
1. Find how many adult and student tickets were sold on Saturday. Show the equations you set up.
Read the worked answer
Let adult tickets be $x$ and student tickets be $y$.
$$x + y = 120$$ $$300x + 150y = 27000$$
Divide the second equation by 150: $2x + y = 180$.
Subtract the first equation: $(2x + y) - (x + y) = 180 - 120$, so $x = 60$.
Then $y = 120 - 60 = 60$.
Answer: 60 adult and 60 student tickets.
Check: $300(60) + 150(60) = 18000 + 9000 = 27000$. Correct.
2. For Sunday, solve for the number of adult tickets $x$ and student tickets $y$ if the total is 140 tickets and the total money is 30,000 yen. Is a whole-number solution possible? Explain.
Read the worked answer
Equations:
$$x + y = 140$$ $$300x + 150y = 30000$$
Divide the second by 150: $2x + y = 200$.
Subtract: $x = 200 - 140 = 60$, so $y = 80$.
Check: $300(60) + 150(80) = 18000 + 12000 = 30000$. Correct.
A whole-number solution exists: 60 adult and 80 student tickets.
Note this is a valid solution, but it is not the only possible pair if the total ticket count were allowed to change.
3. The president's claim is that selling more adult tickets than Saturday guarantees reaching 30,000 yen. Using the constraints (140 total tickets, non-negative counts), test whether this claim is true. Give a counterexample or justify it.
Read the worked answer
On Saturday, 60 adult tickets were sold. Suppose Sunday sells 61 adult tickets and the rest student tickets: $y = 140 - 61 = 79$.
Revenue: $300(61) + 150(79) = 18300 + 11850 = 30150$ yen.
This reaches 30,000 yen, so in this case the claim holds. But consider selling 61 adult tickets with only 130 total tickets: $y = 69$, revenue $= 18300 + 10350 = 28650$ yen, which is below 30,000.
The claim depends on the total ticket count, not just on having more adult tickets than Saturday. With the stated plan of 140 tickets, more adult tickets than 60 does push revenue above 30,000, but the president's reasoning is incomplete because it ignores the total-ticket constraint.
Judgement: the claim is not generally true; it only works together with the 140-ticket condition.