How many of each kind of ticket were sold?

Question

The mount Baker theater sold 950 tickets to the musical performance of mama Mia. a premier seat cost $69.50 and a value seat cost $30.50. In all $53,740 was taken in. How many of each kind of ticket were sold?

Answer 1

Well if p, v were the number of seats sold :

p + v = 950 [ 950 seats sold ]

69.50 p + 30.50 v = 53740 [ total value ]

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You can solve these quickly on a graphing calculator or Desmos

Replace p by x and v by y

Plot the two lines x + y = 950, 69.50 x + 30.50 y = 53750

Read off x and y at the point of intersection

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If you have access to technology, might as well use it. When solving equations arising from real problems, often you can not solve easily or at all using algebra, so you have to use technology

Answer 2

Step 1: Define the variables:

Since we are being asked "how many of each kind" of ticket, let "v" be the number of Value Seats and let "p" be the number of Premier Seats.

Step 2: Write the equations:

Since there were 950 total tickets sold, we can say:

p + v = 950

Since the total taken in was $53,740, we can say:

69.50p + 30.50v = 53740

Step 3: Solve the system of equations for 1 variable:

There are a number of ways including substitution and elimination. Let's use substitution. Since p + v = 950, we can say p = 950 - v

Now, for 69.50p + 30.50v = 53740, we can plug in "950 - v" wherever we see a "p" so:

69.50(950 - v) + 30.50v = 53740 [now distribute to get:

66025 - 69.50v + 30.50v = 53740 [combine like terms to get:

66025 - 39v = 53740 [now subtract 66025 from both sides to get:

-39v = -12285 [now divide both sides by -39 to get:

v = 315

Step 4: Back substitute to find the other variable:

Now that we know v = 315, using p = 950 - v we can solve for "p":

p = 950 - 315

p = 635

Step 5: State the answer:

There were 315 Value Seats sold and 635 Premier Seats sold

Answer 3

Heya Luinda.

To make the explanation easier, I will use a similar example but with simpler numbers. From there, hopefully it will become clear to you how to work out this problem.

Let's say there were 10 total tickets sold. Premier seats cost $2 and value seats cost $1. Total money earned was $14.

So the idea for these kind of questions is to write the statements as equations. Once we have all the equations, we can solve them to find out all the information we are looking for.

The first piece of information we have is that a total of 10 tickets were sold, and there were two different types of tickets, premier and value. That means that whatever the amount of premier tickets sold plus the amount of value tickets sold must add up to 10.

As an equation:

P + V = 10

where P is the amount of premier tickets sold and V is the amount of value tickets sold. The letters used as variables do not matter. You could also use x and y if you wish.

Now let's look at the next piece of information. We have the costs for the premier and value tickets and the total money earned.

How can you calculate the total money earned? Well, it would be the money earned from premier tickets plus the money earned from value tickets.

How do you find the money earned from premier tickets?

If there were one ticket sold, the money earned from premier tickets would be $2.

If there were two tickets sold, the money earned from premier tickets would be $2 + $2 = $2 × 2 = $4.

If there were three tickets sold, the money earned from premier tickets would be $2 + $2 + $2 = $2 × 3 = $6.

etc.

So we can see that if we take the price of the premier ticket and multiply it by the amount of premier tickets sold, that gives us the money earned from premier tickets. Or $2 × P (remember we chose P to represent number of premier tickets sold).

You can follow the same logic for value tickets and you get the money earned from value tickets is $1 × V.

Going back to the total money earned, we know that it is equal to the money earned from premier tickets + the money earned from value tickets.

Let's write it as an equation. We already know the total money earned is $14 and then let's create the variables M_p = money from premier tickets, and M_v = money from value tickets.

$14 = M_p + M_v

Furthermore, we know that money earned for each ticket equals price of the ticket times number of tickets sold for each type.

Or:

M_p = $2 × P

M_v = $1 × V

Let's substitute these values for money earned for each ticket into the equation with total money earned and make the equation simpler by removing the dollar signs:

14 = M_p + M_v

14 = 2P + 1V

Now we have two equations:

P + V = 10

14 = 2P + 1V

They both have P and V in them, which is good because we ultimately want to solve for P and V (since they represent the amount of premier tickets sold and value tickets sold). Generally, the way to do is is to find a way to write an equation with only one of the variables. Then solving for one variable is much easier.

Here we can do this by isolating one variable in terms of the other. Then we can use that to substitute in the other equation. It will be clearer once you see it in action:

P + V = 10

P = 10 - V

We just subtracted V from each side. Now we have P isolated and written in terms of V. So we can substitute this "value" of P into the other equation.

14 = 2P + 1V | This is the other equation unchanged

14 = 2(10 - V) + 1V | I replace P with (10 - V), the result from before

14 = 20 - 2V + 1V | Now we just solve for V

14 = 20 - V | Add V to both sides

V + 14 = 20

V = 6

Great, now we have one of the results we are looking for. the number of value tickets sold is 6. From here, we can substitute V = 6 into one of the equations to get P.

10 = P + V

10 = P + 6

4 = P

Cool, so now we have both the number of premier tickets sold and the number of value tickets sold.

If we want to double check, we can also substitute in P and V in the second equation to make sure everything adds up correctly.

14 = 2P + 1V

14 = 2(4) + 1(6)

14 = 8 + 6

14 = 14

Wonderful it does.

So from here, you should be able to do something similar for your problem.

Hope that was helpful. Let me know if you have any further questions.

Answer 4

So this is a systems of equations problem. The first thing you want to do is set up two equations that model this scenario:

First, Identify your two variables which are premier seats and value seats.

x= number premier tickets y= number of value tickets

A total of 950 tickets were sold so we can set up our first equation as: 950= x+y

Then, a total of $53,470 was raised. Each premier seat costs 69.5 and each value seat cost 30.5

So then our second equation can be modeled as 53,740= 69.5x +30.5y

So now we have our two equations as: 950 = x+y and 53,740 = 69.5x +30.5y

The next step is to find the value of either x or y by isolating either the x or y variable in the first equation. Subtract x from both sides of the first equation to get: 950-x = y. So now that we set our y variable equal to something, we can plug that expression into the second equation.

So we have 53,470 = 69.5x + 30.5 (950-x)

Distribute the 30.5 to each term in the parentheses/ multiply each term in the parentheses by 30.5 to get:

53,470 = 69.5x + 28,975 -30.5x

Combine like terms/ add the x variables together to get : 53,470 = 28,975 +39x. Then just do simple algebra to solve for x.

Subtract 28,975 from both sides to get : 24765 = 39x

Then divide both sides by 39x to get: 635=x which means they sold 635 premier tickets.

Finally go back to the original equation which was 950 = x+y

plug in x = 635 to get: 950 = 635+y

Then subtract 635 from both sides to get: y= 315, so they sold 315 value tickets and 635 premier tickets

If you want to double check your answer, take both variables and plug them into the second equation:

53,740 = 69.5(635) + 30.5(315)

53,740 = 44132.5 + 9607.5

53,740 = 53,740