An amusement park sells 25 cent, 50 cent, and $1 tickets and a teacher purchases $32.50 worth of tickets. A student remarks that there are twice as many 50 cent tickets as there are $1 tickets and that the number of 25 cent
tickets is 30 more than the number of 50
cent tickets. How many tickets of
each type are there?
Well, it seems like this amusement park is giving us quite the brain workout! Let's break it down, shall we?
Let's assume the number of $1 tickets is x. Since the teacher spent $32.50, they must have bought x tickets.
According to the student's observations, there are twice as many 50 cent tickets as $1 tickets. So, the number of 50 cent tickets is 2x.
The student also mentions that the number of 25 cent tickets is 30 more than the number of 50 cent tickets. So, we can say that the number of 25 cent tickets is 2x + 30.
Now, we need to figure out the total cost of all these tickets. The cost of each $1 ticket is $1, the cost of each 50 cent ticket is $0.50, and the cost of each 25 cent ticket is $0.25.
The total cost is given as $32.50, so we can set up an equation:
$x(1) + 2x(0.50) + (2x + 30)(0.25) = 32.50$
Simplifying this equation (I promise it won't hurt):
x + x + 30 + 0.50x + 0.25(2x + 30) = 32.50
x + x + 30 + 0.50x + 0.50x + 7.50 = 32.50
3x + 37.50 = 32.50 (Uh-oh, math emergency!)
Well, it seems like we've created a bit of a problem here. The sum on the left side of the equation is greater than the sum on the right side. So, it appears that our initial information might not be accurate, or maybe I made an error in my calculations. Either way, let's go back and double-check our numbers!
Let's solve this problem step by step:
Let's assume the number of $1 tickets as 'x'.
So, the number of 50 cent tickets would be 2x (because there are twice as many 50 cent tickets as $1 tickets).
And the number of 25 cent tickets would be (2x + 30) (because there are 30 more 25 cent tickets than 50 cent tickets).
Now, let's calculate the total cost of the tickets:
The cost of $1 tickets = x * $1 = $x
The cost of 50 cent tickets = 2x * $0.50 = $1x
The cost of 25 cent tickets = (2x + 30) * $0.25 = $0.50x + $7.50
Now, let's equate the total cost to $32.50 and solve the equation:
$x + $1x + $0.50x + $7.50 = $32.50
Combining like terms:
2.5x + $7.50 = $32.50
Subtracting $7.50 from both sides of the equation:
2.5x = $25
Dividing both sides by 2.5:
x = $10
So, there are 10 $1 tickets, 20 50 cent tickets, and (20+30) = 50 25 cent tickets.
Final Answer:
There are 10 $1 tickets, 20 50 cent tickets, and 50 25 cent tickets.
To solve this problem, we can set up a system of equations based on the given information.
Let's assume the number of $1 tickets is x.
According to the student's remark, the number of 50 cent tickets is twice the number of $1 tickets, so we can say it is 2x.
Also, the number of 25 cent tickets is 30 more than the number of 50 cent tickets, so we can represent it as 2x + 30.
Now, we can calculate the total value of the tickets. The value of each $1 ticket is $1, the value of each 50 cent ticket is $0.50, and the value of each 25 cent ticket is $0.25.
The total value of the tickets is given as $32.50. So, we can write the equation:
($1 ticket value * number of $1 tickets) + (50 cent ticket value * number of 50 cent tickets) + (25 cent ticket value * number of 25 cent tickets) = total value of tickets.
($1 * x) + ($0.50 * 2x) + ($0.25 * (2x + 30)) = $32.50
Now, we can simplify and solve this equation to find the value of x:
$x + x + 0.5(2x) + 0.25(2x + 30) = 32.50$
$x + x + x + 0.5(30) + 0.25(2x) = 32.50$
$3x + 15 + 0.5x + 0.5x = 32.50$
$4x + 15 = 32.50$
$4x = 17.50$
$x = 4.375$
Since the number of tickets must be a whole number, we need to round it to the nearest whole number. So, there are 4 $1 tickets.
Now, we can easily find the number of 50 cent tickets (2x) and the number of 25 cent tickets (2x + 30):
Number of 50 cent tickets = 2 * 4 = 8
Number of 25 cent tickets = 2 * 4 + 30 = 8 + 30 = 38
Therefore, there are 4 $1 tickets, 8 50 cent tickets, and 38 25 cent tickets.
number of $1 tickets --- x
number of 50 cent tickets ---- 2x
number of 25 cent tickets ---- 2x+30
value equation:
100x + 50(2x) + 25(2x+30) = 3250
solve for x, sub back into my definitions