Question 1059·Easy·Math

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A school club sold a total of 80 tickets to its play. Student tickets cost $4 each and adult tickets cost $6 each. The club collected $400 in ticket sales. How many adult tickets were sold?

A known total split between two ticket types can become a one-variable equation: if $x$ tickets are one type, the rest are the total minus $x$ . Multiply each count by its own price, set the sum equal to the money collected, and use a Desmos regression to solve. Don't use the total ticket count as the student count too; that counts adult tickets twice.

Hints

Hint 1
Let $x$ count adult tickets. The two ticket counts must add to $80$ , so the student count is what's left after taking away $x$ . How would you write that count?
Hint 2
Each revenue term is a price multiplied by the number of tickets sold at that price. Keep the $6 price with adults and the $4 price with students, then add the two amounts to get the total collected.

Step-by-step

Approach 1: Write one equation and solve in Desmos

Step 1
Write the student count
Let $x$ be the number of adult tickets. All $80$ tickets are either adult or student tickets, so the student count is the total minus the adult count:
$\text{student tickets}=80-x.$
Step 2
Write the money equation
Adult sales bring in $6x$ dollars, and student sales bring in $4(80-x)$ dollars. Their sum is the $400 collected: $6x+4(80-x)=400$ . Don't put $80$ in place of $80-x$ : the $80$ already includes the adult tickets.
Step 3
Solve for the adult count
Type $6x_1+4(80-x_1)\sim400$ in Desmos. Use $x_1$ instead of $x$ so Desmos solves for a number rather than graphing with $x$ ; $\sim$ tells it to match the two sides. Under PARAMETERS, it shows $x_1=40$ . Since $x_1$ counts adult tickets, the club sold $40$ adult tickets. Choice B.

Approach 2: Count the extra money from adult tickets

Step 1
Find an all-student starting total
Imagine all $80$ tickets cost the student price of $4. The sales total would be:
$80\times4=320.$
Step 2
Find the money above that starting total
Subtract that starting total from the actual sales of $400: $400-320=80$ . Adult tickets account for this extra $80.
Step 3
Find the extra money per adult ticket
Each adult ticket replaces one $4 student ticket with a $6 ticket, so find the price difference: $6-4=2$ . Each adult ticket adds $2 to the starting total, not $4.
Step 4
Count the adult tickets
Divide the extra $80 by the $2 added per adult ticket: $80\div2=40$ . The club sold $40$ adult tickets. Choice B.

Question 1059·Easy·Math

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