A movie theater offers two options for membership. The first option costs $15 per ticket but an initial fee of $10. The second option costs $10 per ticket but an initial fee of $45. After how many tickets are both options the same price?
Let's assume that the number of tickets at which both options have the same price is X.
Option 1 costs $15 per ticket with an initial fee of $10. So, the total cost of X tickets would be 15X + 10.
Option 2 costs $10 per ticket with an initial fee of $45. So, the total cost of X tickets would be 10X + 45.
We can set up the equation:
15X + 10 = 10X + 45
Subtract 10X from both sides:
5X + 10 = 45
Subtract 10 from both sides:
5X = 35
Divide both sides by 5:
X = 7
So, after 7 tickets, both options would have the same price.
Two pumps are being filled by two different pumps. The first vat has 10 gallons is being filled at a rate of 12 gallons per second. The second vat has 25 gallons and is being filled at a rate of 10 gallons per second. After how many seconds will both vats have the same amount of liquid?
To find the number of tickets at which both options have the same price, we can set up the following equation:
Option 1: Total cost = Initial fee + (Cost per ticket * Number of tickets)
Option 2: Total cost = Initial fee + (Cost per ticket * Number of tickets)
For option 1:
Total cost = $10 + ($15 * Number of tickets)
For option 2:
Total cost = $45 + ($10 * Number of tickets)
Setting the two equations equal to each other:
$10 + ($15 * Number of tickets) = $45 + ($10 * Number of tickets)
Subtracting $10 and $10 * Number of tickets from both sides:
$15 * Number of tickets - $10 * Number of tickets = $45 - $10
Combining like terms:
$5 * Number of tickets = $35
Dividing both sides by $5:
Number of tickets = $35 / $5
Number of tickets = 7
Therefore, both options will have the same price after 7 tickets.
To determine after how many tickets both options are the same price, we need to set up and solve an equation.
Let's represent the number of tickets as 'x'.
For the first option, the cost per ticket is $15, and there is an initial fee of $10. So the total cost can be calculated as: 15x + 10.
For the second option, the cost per ticket is $10, and there is an initial fee of $45. So the total cost can be calculated as: 10x + 45.
Now we can set up the equation to find when the two options are equal:
15x + 10 = 10x + 45
To solve it, we need to isolate the 'x' term by moving the constants to the other side of the equation:
15x - 10x = 45 - 10
Simplifying the equation further:
5x = 35
Now, divide both sides of the equation by 5:
x = 7
So, after purchasing 7 tickets, both options will cost the same amount.