# A group of sailors plans to share equally the cost and use of a $72,000 boat. If they can get 3 more sailors to join their group, the cost per person will be reduced by$2000. How many original sailors are there?

Sep 3, 2015

The group originally had 9 members.

#### Explanation:

Let's say that the original group consists of $x$ sailors.

For this group, each sailor has to pay a cost of $c$ to get the total amount covered, which means that you can write

$c \cdot x = 72000$

Now, if the group increases by 3 sailors, the cost per sailor will decrease by 2,000$. The key here is to realize that the cost of the boat does not change, it will still be 72,000$.

This means that you can write

${\underbrace{\left(c - 2000\right)}}_{\textcolor{b l u e}{\text{new cost per sailor")) * underbrace((x + 3))_(color(orange)("new group of sailors}}} = 72000$

Expand the parantheses for this second equation to get

$c x + 3 c - 2000 x - 6000 = 72000$

Use the first equation to replace the total cost

$\textcolor{red}{\cancel{\textcolor{b l a c k}{c x}}} + 3 c - 2000 x - 6000 = \textcolor{red}{\cancel{\textcolor{b l a c k}{c x}}}$

Use the first equation again to replace $c$ with

$c = \frac{72000}{x}$

This will get you

$3 \cdot \frac{72000}{x} - 2000 x - 6000 = 0$

Provided that $x \ne 0$, you can multiply the second and third terms by $1 = \frac{x}{x}$ to get rid of the denominator

$\frac{216000}{x} - 2000 x \cdot \frac{x}{x} - 6000 \cdot \frac{x}{x} = 0$

$216000 - 2000 {x}^{2} - 6000 x = 0$

Divide all the terms by $2000$ and rearrange in quadratic form

${x}^{2} + 3 x - 108 = 0$

Use the quadratic formula to find the two solutions to this equation

${x}_{1 , 2} = \frac{- 3 \pm \sqrt{{3}^{2} - 4 \cdot 1 \cdot \left(- 108\right)}}{2 \cdot 1}$

${x}_{1 , 2} = \frac{- 3 \pm \sqrt{441}}{2}$

${x}_{1 , 2} = \frac{- 3 \pm 21}{2}$

Since you can't have a negative number of sailors in the group, the only valid solution to this equation will be

$x = \frac{- 3 + 21}{2} = \frac{18}{2} = \textcolor{g r e e n}{9}$

The group originally contained 9 sailors, each having to contribute 8000$ to get the boat. The new group would have 12 sailors and each would only have to pay 6000$, which is indeed 2000\$ less per sailor for the same total cost.