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?

1 Answer
Sep 3, 2015

Answer:

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.