Question 72·Hard·Linear Equations in One Variable
For a school fundraiser, student tickets cost $5 each and adult tickets cost $8 each. The event collected a total of $1,184 from ticket sales. If 48 more student tickets than adult tickets were sold, which equation can be used to determine the number of adult tickets, , that were sold?
For equation-setup word problems, start by underlining what the variable represents and any key comparison phrases like "more than" or "less than." Express all other quantities in terms of the given variable (for example, "48 more student tickets than adult tickets" becomes students = if is adults). Next, write an expression for each part of the situation using "amount = price × quantity" or similar, then add the parts and set them equal to the given total. Finally, scan the answer choices and quickly eliminate any that use the wrong sign (+ vs − in the comparison) or attach the wrong coefficient (like 5 or 8) to the variable based on the prices described.
Hints
Clarify what the variable stands for
The problem says is the number of adult tickets. Think about how you would write the number of student tickets in terms of if there are 48 more student tickets than adult tickets.
Translate the relationship between student and adult tickets
"48 more student tickets than adult tickets" means you start with the number of adult tickets and then add 48 to get the number of student tickets. Ask yourself: should that be or ?
Write the money expressions
For each type of ticket, multiply the price per ticket by the number of those tickets. Then add those two expressions and set the sum equal to the total amount collected, 1184.
Desmos Guide
Enter each answer choice as an equation in Desmos
On separate lines in Desmos, type each equation exactly as written: 5a + 8(a + 48) = 1184, 8a + 5(a - 48) = 1184, 5a + 8(a - 48) = 1184, and 8a + 5(a + 48) = 1184. Desmos will treat these as equations in the variable .
Find the solution for a for each equation
Click or tap on each equation; Desmos should display the solution for (for example, in the form a = ...). Record the value of that each equation produces.
Check which solution matches the story
For each value of , add a new line in Desmos defining students = a + 48, then compute 5*students + 8*a. The correct equation is the one whose solution gives a positive number of adult tickets, a student count that is exactly 48 more than the adult count, and a total money expression 5*students + 8*a equal to 1184.
Step-by-step Explanation
Define the variable and relate the ticket counts
The problem tells you that represents the number of adult tickets.
Because there were 48 more student tickets than adult tickets, the number of student tickets is:
Write expressions for money from each type of ticket
Each adult ticket costs $8, and there are adult tickets, so the money from adult tickets is:
Each student ticket costs $5, and there are student tickets, so the money from student tickets is:
Use the total money to write the equation
The total amount collected was $1184, so the money from adults plus the money from students must equal $1184:
This is the equation that can be used to determine the number of adult tickets, , that were sold.