Answer
$40$
Work Step by Step
Lets start by setting up the following data for company 1 based on the problem description.
$$
\begin{aligned}
\textbf{faxed base charge: b} &=500 \\
\textbf{cost per person: m} &=25 \\
\textbf{ size of group} &= x \\
\textbf{total cost for the tour} &= y_1
\end{aligned}
$$ The linear model for this option can be written as:
$$\begin{aligned}
y &=mx+b\\
y_1&= 25x+500.
\end{aligned}$$ We can do the same set up company 2 as follows:
$$
\begin{aligned}
\textbf{fixed base charge: b} &=700 \\
\textbf{cost per person: m} &=20 \\
\textbf{size of group} &= x \\
\textbf{total cost for the tour} &= y_2 \\
\end{aligned}
$$ The linear model for this option can be written as:
$$\begin{aligned}
y &=mx+b\\
y_2&= 20x+700.
\end{aligned}$$ Out target is to find the size of the group, $x$, of people that will lead to the same cost charged at either company. We can do this by setting the two linear functions equal and solving for $x$.
$$\begin{aligned}
y_1 &=y_2\\
25x+500&= 20x+700\\
25x-20x&= 700-500\\
5x\& = 200\\
x&= \frac{200}{5}\\
&=40.
\end{aligned}$$ Check: $$\begin{aligned}
y_1 &=25\cdot 40+500 = \$1500\\
y_2&= 20\cdot 40+700 =\$1500.
\end{aligned}$$ The size of the group should be $40$ so that the charge at either tour company be the same.
