College Algebra, Chapter 1, 1.5, Section 1.5, Problem 76

Suppose that a group of friends decide to buy a vacation home for $\$ 120,000$, sharing the cost equally. If they can find one more person to join them, each person's contribution will drop by $\$ 6000$. How many people are in the group?

If we let $n$ be the number of the people, then the cost per head is $\displaystyle \frac{120,000}{n}$. If the number is increased by 1 then the cost per head will be $\displaystyle \frac{120,000}{n + 1}$. So,



$
\begin{equation}
\begin{aligned}

\frac{120,000}{n + 1} =& \frac{120,000}{n} - 6000
&& \text{Model}
\\
\\
\frac{120,000}{n} - \frac{120,000}{n + 1} =& 6000
&& \text{Add } 6000 \text{ and subtract } \frac{120,000}{n + 1}
\\
\\
\frac{20}{n} - \frac{20}{n + 1} =& 1
&& \text{Divide both sides by } 6000
\\
\\
20(n + 1) - 20n =& n(n + 1)
&& \text{Multiply the LCD } n(n + 1)
\\
\\
20n + 20 - 20n =& n^2 + n
&& \text{Simplify}
\\
\\
n^2 + n - 20 =& 0
&& \text{Cancel out like terms and subtract } 20
\\
\\
(n + 5)(n - 4) =& 0
&& \text{Factor out}
\\
\\
n + 5 =& 0 \text{ and } n - 4 = 0
&& \text{Zero Product Property}
\\
\\
n =& -5 \text{ and } n = 4
&& \text{Solve for } n
\\
\\
n =& 4
&& \text{Choose } n > 0, \text{ the group consists of 4 members}

\end{aligned}
\end{equation}
$