Respuesta :
ANSWER
81 people
EXPLANATION
Let p be the number of people that attend the party.
Under plan A, the inn charges $30 for each person, so the value y of a party for p people is,
[tex]y_A=30x[/tex]Then, under plan B, the cost is $1300 for a maximum of 25 people - this means that if 1 to 25 people attend the party, the cost is the same, $1300. For each person in excess of the first 25 - this means for 26, 27, 28, etc, the inn charges $20 each. The cost for plan B is,
[tex]y_B=1300+20(p-25)[/tex]The last part, (p - 25), is the part of the equation that separates the first 25 attendees. This equation works for 25 people or more, but it is okay to solve this problem. Note that for p = 25, the cost for plan A is,
[tex]y_A=30\cdot25=750[/tex]Which is less than the cost of plan B ($1300).
We have to find for what number of people attending the party, the cost of plan B is less than the cost of plan A,
[tex]y_BThis is,[tex]1300+20(p-25)<30p[/tex]We have to solve this for p. First, apply the distributive property of multiplication over addition/subtract4ion to the 20,
[tex]\begin{gathered} 1300+20p-20\cdot25<30p \\ 1300+20p-500<30p \end{gathered}[/tex]Add like terms,
[tex]\begin{gathered} (1300-500)+20p<30p \\ 800+20p<30p \end{gathered}[/tex]Now, subtract 20p from both sides,
[tex]\begin{gathered} 800+20p-20p<30p-20p \\ 800<10p \end{gathered}[/tex]And divide both sides by 10,
[tex]\begin{gathered} \frac{800}{10}<\frac{10p}{10} \\ 80For 80 people, the costs of the plans are,
[tex]\begin{gathered} y_A=30\cdot80=2400 \\ y_B=1300+20(80-25)=1300+20\cdot55=1300+1100=2400 \end{gathered}[/tex]Both have the same cost. The solution to the inequation was the number of people, p, is more than 80. This means that for 81 people the cost of plan B should be less than the cost of plan A,
[tex]\begin{gathered} y_A=30\cdot81=2430 \\ y_B=1300+20(81-25)=2420 \end{gathered}[/tex]For 81 people, plan B costs $10 less than plan A.
