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A school band is planning an honoree and fundraiser banquet To encourage attendance a $200 cash door prize will be given The caterer has offered them two choices for the meal Plan A The caterer will charge 7.25 per meal and the band will charge $18 for a ticket to the banquet, and plan B the caterer will charge $8 per meal and the choir will charge $20 per ticket. The band plans of making $800 with the banquet. Determine which meal plan the band should choose. explain the reason for your decision

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Philip P. | Effective and Affordable Math TutorEffective and Affordable Math Tutor
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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.