Question

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?

Answer 1

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 * 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((c-2000))_(color(blue)("new cost per sailor")) * underbrace((x + 3))_(color(orange)("new group of sailors")) = 72000`

Expand the parantheses for this second equation to get

`cx + 3c - 2000x - 6000 = 72000`

Use the first equation to replace the total cost

`color(red)(cancel(color(black)(cx))) + 3c - 2000x - 6000 = color(red)(cancel(color(black)(cx)))`

Use the first equation again to replace `c` with

`c = 72000/x`

This will get you

`3 * 72000/x - 2000x - 6000 = 0`

Provided that `x!=0`, you can multiply the second and third terms by `1 = x/x` to get rid of the denominator

`216000/x -2000x * x/x - 6000 * x/x = 0`

`216000 - 2000x^2 - 6000x = 0`

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

`x^2 + 3x - 108 = 0`

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

`x_(1,2) = (-3 +- sqrt(3^2 - 4 * 1 * (-108)))/(2 * 1)`

`x_(1,2) = (-3 +- sqrt(441))/2`

`x_(1,2) = (-3 +- 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 = (-3 + 21)/2 = 18/2 = color(green)(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.