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## How many of each kind of ticket were sold?

There were 305 tickets sold for a basketball game. The activity cardholders' tickets cost \$1.25 and the non-cardholders' tickets cost \$2.50. The total amount of money collected was \$578.75. How many of each kind of ticket were sold?

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!!!

thank you so much William, this one was a doozy for me
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