Question

William and Debra Pierce are celebrating their 20 th anniversary by having a reception at a local reception hall. They have budgeted $ 2 comma 000 for their reception. If the reception hall charges a $ 80 cleanup fee plus $ 35 per​ person, find the greatest number of people that they may invite and still stay within their budget.

Answer 1

To find the greatest number of people William and Debra can invite while staying within their budget, we can set up the following equation based on their expenses:

1. The total budget for the reception is $2,000.
2. The reception hall charges a cleanup fee of $80.
3. The cost per person is $35.

Let \( x \) be the number of people they invite. The total cost \( C \) can be expressed as:

\[
C = 80 + 35x
\]

They want to stay within their budget, so we set up the inequality:

\[
80 + 35x \leq 2000
\]

Now, we can solve for \( x \):

1. Subtract the cleanup fee from both sides:

\[
35x \leq 2000 - 80
\]

\[
35x \leq 1920
\]

2. Divide both sides by 35 to find \( x \):

\[
x \leq \frac{1920}{35}
\]

Calculating the division:

\[
x \leq 54.857
\]

Since \( x \) must be a whole number (they cannot invite a fraction of a person), we take the largest whole number less than or equal to 54.857, which is 54.

Therefore, the greatest number of people that William and Debra may invite and still stay within their budget is:

\[
\boxed{54}
\]