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Question

Jenny is planning a surprise party for her brother. A catering company is charging $15.95 per person for the food and there is a delivery charge of $30. She realized she could make the same food for the people at the party fir $12.75 per person but would have to spend $96 on supplies. If she invited 35 people to the party, what is the smallest number of people that would need to RSVP that they are not coming for the catering company to be the better deal?

 

Please explain and show work  

Answer 1

Let's assign a variable and write expressions for the food cost options...

 

x=number of attendees

15.95x+30....expression, cost of catering

12.75x+96....expression, cost of cooking

 

We can now write a strict inequality with the above expressions such that the cost of catering is a better deal than cooking (per the problem statement) and solve for "x", the number of attendees...

 

(cost of catering) < (cost of cooking)

15.95x+30 < 12.75x+96...substitute expressions

   3.2x+30 < 96..................subtract 12.75x, both

                                            sides

         3.2x < 66..................subtract 30, both sides

         ∴  x < 20.63.............divide both sides by 3.2

 

In the context of the problem, "x=20.63" attendees means that 20.63 attendees will produce an equal cost for catering and cooking options.  For "x<20.63" means catering in will be the better deal.

 

First integer less than 20.63 is 20 (cannot have a fraction of a person).  So maximum 20 people ARE attending.  Problem asks for "smallest number of people NOT attending.

 

(35 invitees) - (20 attendees) = (15 declines)

 

So, minimum 15 declines make catering a better deal.  I will leave you to substitute various values into the above expressions to calculate and compare costs.

 

Note:

You can also solve this problem by graphing inequalities (set above expressions equal to "y" and graph) and parameters from the problem statement.