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How many of each kind of ticket were sold?

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6 Answers

Let x = number of activity cardholder's tickets sold
Let y = number of non-activity cardholder's tickets sold
 
We know that x + y = 305     Eq. (1)
 
We also know that ($1.25)*x + ($2.50)*y = $578.75     Eq. (2)
 
Solve Eq. 1 for y and substitute into Eq. 2:
 
y = 305 - x and ($1.25)x + ($2.50)(305 - x) = $578.75
 
($1.25)x + $762.5 - ($2.50)x =  $578.75
 
(-$1.25)x = $578.75 - $762.5
 
x = 147 tickets
 
Therefore, y = 158 tickets
 
Check
 
Does ($1.25)*(147) + ($2.50)(158) = $578.75?  Yes!!!
 
 

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You can solve this by using two equations, each with two unknowns, or variables.
 
The variables are the number of cardholders tickets (C), and the number of non-cardholders tickets (N).
 
There were 305 tickets sold, so the total of cardholders plus noncardholders is 305:
 
N + C = 305
 
Each cardholders ticket costs 1.25, so the amount from sales of cardholders tickets is 1.25 * C.
Each non-cardholders ticket costs 2.50, so the amount from sales of non-cardholders tickets is 2.50 * N.
The total sales is 578.75:
 
1.25C + 2.5N = 578.75
 
You now have two equations in two unknowns. Now, change the first equation to express one of the variables in term of the other (by subtracting C from both sides of the equation:
 
N + C = 305
N = 305 - C
 
Now, use substitution to eliminate N from the second equation:
 
1.25C + 2.5N = 578.75
 
substitute for N:
1.25C + 2.5(305 - C) = 578.75
 
distributive property:
1.25C + (2.5 * 305 - 2.5 * C) = 578.75
 
perform multiplication inside the parentheses
1.25C + (762.5 - 2.5C) = 578.75
 
eliminate the parentheses
1.25C - 2.5C + 762.5 = 578.75
 
combine variable terms
-1.25C + 762.5 = 578.75
 
subtract 762.5 from both sides of the equation
-1.25C = -183.75
 
divide both sides by -1.25
C = 147
 
Now that you know how many cardholder tickets were sold, use the original equation to determine the non-cardholder tickets:
 
N + C = 305
 
substitute for C:
N + 147= 305
 
subtract 147 from both sides of the equation:
N = 305 - 147
N = 158
 
C = 147, N = 158
 
There were 147 cardholder tickets sold, and 158 non-cardholder tickets sold. Double check by plugging those values into the second equation:
 
1.25C + 2.5N = 578.75
1.25 * 147 + 2.5 * 158 = 578.75
183.75 + 395 = 578.75
578.75 = 578.75
Kevin,
 
If we let x=number of activity cardholders and y= the number on non-activity cardholders the we were told that:
 
x+y=305    total number of tickets sold
 
and:
 
1.25x+2.5y=578.75    total revenue from ticket sales. This gives 2 equations with 2 unknowns x and y.
 
If you now multiply both sides of the first equation by 1.25 and subtract it from the second equation we have
 
1.25y=197.50, or y=158 and since x+y=305 x must be 147
 
The sales from activity cardholders =1.25*147 = 183.75
The sales from non act.cardholders=2.5*158 = 395 which totals to $578.75
 
Regards
Jim
Hi Kevin;
x=quantity of cardholders' tickets
$1.25x+$2.50(305-x)=$578.75
$1.25x+($762.50-$2.50x)=$578.75
$762.50-$1.25x=$578.75
Let's subtract $762.50 from both sides as we proceed to isolate x...
-$762.50+$762.50-$1.25x=$578.75-$762.50
-$1.25x=-183.75
Let's divide both sides by -$1.25 as we proceed to isolate x...
(-$1.25x)/-$1.25=(-$183.75)/(-$1.25)
 
x=147 cardholders' tickets
305-147=158 non-cardholders' tickets
 
Let's check our work...
[147($1.25)]+[158($2.50)]=$578.75
$183.75+$395.00=$578.75
$578.75=$578.75
 
Please note that I always indicate all information of cents as well as dollars for each monetary figure.  Consistency is very important, and all initial figures of $1.25, $2.50 and $578.75 include such information.  Therefore, all supplemental information must include such too, even if it is 00 cents.
 
 
 
 
Let C be the number of cardholders.
 
1.25C + 2.5(305 - C) = 578.75
 
1.25C + 762.5 - 2.5C = 578.75
 
-1.25C = 578.75 - 762.5
 
-1.25C = -183.75
 
C = 147
 
Cardholders 147, Non Cardholders 158
 
 
 
 
 
 
x=cardholder ticket
y=non cardholder ticket
Do a system of equations:
 
x+y=305
1.25x+2.50y=578.75
 
Now multiply the top equation to get rid of one of the variables in order to solve the other:
 
-1.25x-1.25y=-381.25
1.25x+2.50y=578.75
 
This leads to: 
1.25y=197.5 -> y=1197.5/1.25=158
 
Plug the y-value back into x+y=305 ->x+158=305 -> x=147
 
So there were 147 cardholder and 158 non-cardholder tickets sold