Let x = the number of people attending the banquet.
Let y = the funds the school band makes from the banquet
The costs are $200 for the door prize and the meal cost per person attending (meal cost*x). Revenues are the ticket prices per person attending (ticket price*x). The net revenue (y=Revenue-Costs) equations are:
For plan A: yA = ($18)x- ($7.25)x - $200 = ($10.75)x - $200
For plan B: yB = ($20)x - ($8)x - $200 = ($12)x - $200
To make their goal of $800 with plan A:
$800 = (10.75)x - 200
1000 = (10.75)x
93 people ≅ x They need 93 people to attend
To make their goal of $800 with plan B:
$800 = 12x - 200
1000 = 12x
84 people ≅ x They need 84 people to attend
Plan B allows the band to reach its goal with nine fewer people attending the banquet. For any given value of x, plan B will return more net revenue than plan A (since its slope is steeper). So with the information provided, the answer would seem to be plan B. In the real-world, however, as the ticket cost rises, the number of people willing to buy tickets decreases. So we aren't sure if the extra $2 per ticket for plan B will scare away more than 9 potential ticket buyers. To truly solve this problem, we need to know how attendance is affected by ticket cost.