Question 102·Hard·Linear Equations in One Variable
A local theatre sold tickets for a play at two price levels: balcony tickets for $18 each and orchestra tickets for $32 each. The theatre sold a total of 240 tickets. The amount of money collected from orchestra tickets was $1,008 more than the amount collected from balcony tickets.
Which equation can be used to find the number of orchestra tickets, , that were sold?
For equation-translation word problems, first choose a clear variable and immediately express all other quantities in terms of that variable, using the total or fixed amounts given. Next, write algebraic expressions for each relevant amount (like revenues, lengths, or ages), and then carefully translate relational phrases such as 'more than', 'less than', or 'is' into operations and an equals sign. Finally, build the equation by substituting your expressions into that relationship and choose the answer option whose structure (addition vs. subtraction, correct totals, correct coefficients) exactly matches your translation.
Hints
Relate the two ticket counts
If is the number of orchestra tickets and there were 240 tickets total, how can you write the number of balcony tickets in terms of ?
Write money expressions
How do you write the amount of money from orchestra tickets using 32 dollars per ticket and tickets? How do you write the amount of money from balcony tickets using 18 dollars per ticket and the number of balcony tickets from the previous hint?
Focus on the phrase about 'more than'
The problem says the orchestra amount was 1008 dollars more than the balcony amount. Does that suggest adding the two amounts together, or taking one minus the other?
Set up the final equation structure
Once you have expressions for orchestra revenue and balcony revenue, think about which one should go first in a subtraction so that the result equals 1008.
Desmos Guide
Define revenue expressions in Desmos
In Desmos, use in place of .
Type the following on separate lines:
f(x) = 32x(orchestra revenue)g(x) = 18(240 - x)(balcony revenue)
These show how much money comes from each type of ticket for any ticket count .
Represent the difference between revenues
Now type a third expression:
h(x) = f(x) - g(x)
This gives the difference (orchestra revenue minus balcony revenue) for each value of . Look at the formula Desmos displays for ; this is the left-hand side of the equation that matches the word statement.
Match with the answer choices
The problem says the orchestra revenue was 1008 dollars more than the balcony revenue, so you want the equation where the expression for f(x) - g(x) is set equal to 1008. Compare the algebraic form of h(x) with the left side of each answer choice and pick the one that sets that expression equal to 1008.
Step-by-step Explanation
Define the variable and the two ticket amounts
We are told that is the number of orchestra tickets sold.
Since there were 240 tickets in total, the number of balcony tickets must be (because orchestra tickets + balcony tickets = total tickets).
Write expressions for the money from each type of ticket
Each orchestra ticket costs 32 dollars, so the amount of money from orchestra tickets is
- orchestra revenue: .
Each balcony ticket costs 18 dollars, and there are of them, so the amount of money from balcony tickets is
- balcony revenue: .
Translate the phrase about one amount being more than the other
The problem says the amount collected from orchestra tickets was 1008 dollars more than the amount collected from balcony tickets.
That means:
- orchestra revenue − balcony revenue = 1008.
In symbols, using the expressions from the previous step, this becomes:
- .
Substitute the expressions and match to the choices
Now substitute the revenue expressions into the relationship from Step 3:
- orchestra revenue is
- balcony revenue is
So the equation becomes
This matches answer choice D.