Question

The Arnold Inn offers two plans for wedding parties. Under plan A, the inn charges $30 for each person in attendance. Under plan B, the inn charges $1300 plus $20 for each person in excess of the first 25 who attend. For what size parties will plan B cost less? I do not understand how for Plan b: 1300+20(p-25). I do not understand the part p-25

Answer 1

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,

$y_A=30x$

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,

$y_B=1300+20(p-25)$

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,

$y_A=30\cdot25=750$

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,

$y_BThis is,[tex]1300+20(p-25)<30p$

We have to solve this for p. First, apply the distributive property of multiplication over addition/subtract4ion to the 20,

$\begin{gathered} 1300+20p-20\cdot25<30p \\ 1300+20p-500<30p \end{gathered}$

Add like terms,

$\begin{gathered} (1300-500)+20p<30p \\ 800+20p<30p \end{gathered}$

Now, subtract 20p from both sides,

$\begin{gathered} 800+20p-20p<30p-20p \\ 800<10p \end{gathered}$

And divide both sides by 10,

$\begin{gathered} \frac{800}{10}<\frac{10p}{10} \\ 80

For 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}$

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,

$\begin{gathered} y_A=30\cdot81=2430 \\ y_B=1300+20(81-25)=2420 \end{gathered}$

For 81 people, plan B costs $10 less than plan A.