AlgebraVery hardInterpreting the slope of a constraint line
Algebra practice question
A cinema sells adult tickets for $12 and child tickets for $8. The equation 12a + 8c = 480 gives the combinations of a adult tickets and c child tickets that bring in exactly $480. If c is graphed against a in the ac-plane, what does the slope of the graph represent?
- A. For each additional adult ticket sold, 1.5 fewer child tickets are needed to reach $480.Correct
- B. For each additional adult ticket sold, 1.5 more child tickets are needed to reach $480.
- C. Each adult ticket brings in $1.50 more than each child ticket.
- D. The total number of tickets sold is 1.5 times the number of adult tickets.
Answer: A. For each additional adult ticket sold, 1.5 fewer child tickets are needed to reach $480.
Solve for c: 8c = 480 − 12a, so c = 60 − 1.5a. The slope is −1.5: every extra adult ticket replaces 1.5 child tickets' worth of revenue, so 1.5 fewer child tickets are needed.
Why the other answers are wrong
- B. For each additional adult ticket sold, 1.5 more child tickets are needed to reach $480.
- The slope is negative: more adult tickets means FEWER child tickets for the same total.
- C. Each adult ticket brings in $1.50 more than each child ticket.
- An adult ticket brings in $4 more than a child ticket (12 − 8), not $1.50. The 1.5 is a ratio of tickets, 12/8.
- D. The total number of tickets sold is 1.5 times the number of adult tickets.
- The slope relates the CHANGE in c to the change in a; it says nothing about the total number of tickets.
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