SOLUTION: The members of a flying club plan to share equally the cost of a $200,000 airplane. The members want to find 5 more people to join the club so that the cost per person will decreas

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Question 207398This question is from textbook Elementary and Intermediate Algebra
: The members of a flying club plan to share equally the cost of a $200,000 airplane. The members want to find 5 more people to join the club so that the cost per person will decrease by $2000. How many members are currently in the club?
I don't know if I did it right. I did not use any formula. I took 2000/5 = 400, then I took 200,000/400 = 500. So, I got 500 current club members. Will you please check this? Thank you!! This question is from textbook Elementary and Intermediate Algebra

Found 2 solutions by jim_thompson5910, MathTherapy:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=# of people in the club (before 5 more join)


Before they get 5 people, we have the equation:


Cost Per Individual: c=200000%2Fx (since the cost is divided among "x" people)


When 5 more people join, the equation becomes


c-2000=200000%2F%28x%2B5%29 (simply replace "x" with "x+5" and subtract 2000 from "c" to form the new equation)


So we have the two equations


c=200000%2Fx Equation 1


c-2000=200000%2F%28x%2B5%29 Equation 2



c-2000=200000%2F%28x%2B5%29 Start with Equation 2


200000%2Fx-2000=200000%2F%28x%2B5%29 Plug in c=150000%2Fx (which is Equation 1)


Multiply EVERY term by the LCD x%28x%2B5%29 to clear out the fractions.


200000%28x%2B5%29-2000x%28x%2B5%29=200000x Multiply and simplify.


200000x%2B1000000-2000x%5E2-10000x=200000x Distribute


200000x%2B1000000-2000x%5E2-10000x-200000x=0 Subtract 200000x from both sides.


-2000x%5E2-10000x%2B1000000=0 Combine like terms.


Notice that the quadratic -2000x%5E2-10000x%2B1000000 is in the form of Ax%5E2%2BBx%2BC where A=-2000, B=-10000, and C=1000000


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


Plug in A=-2000, B=-10000, and C=1000000


Negate -10000 to get 10000.


x+=+%2810000+%2B-+sqrt%28+100000000-4%28-2000%29%281000000%29+%29%29%2F%282%28-2000%29%29 Square -10000 to get 100000000.


x+=+%2810000+%2B-+sqrt%28+100000000--8000000000+%29%29%2F%282%28-2000%29%29 Multiply 4%28-2000%29%281000000%29 to get -8000000000


x+=+%2810000+%2B-+sqrt%28+100000000%2B8000000000+%29%29%2F%282%28-2000%29%29 Rewrite sqrt%28100000000--8000000000%29 as sqrt%28100000000%2B8000000000%29


x+=+%2810000+%2B-+sqrt%28+8100000000+%29%29%2F%282%28-2000%29%29 Add 100000000 to 8000000000 to get 8100000000


x+=+%2810000+%2B-+sqrt%28+8100000000+%29%29%2F%28-4000%29 Multiply 2 and -2000 to get -4000.


x+=+%2810000+%2B-+90000%29%2F%28-4000%29 Take the square root of 8100000000 to get 90000.


x+=+%2810000+%2B+90000%29%2F%28-4000%29 or x+=+%2810000+-+90000%29%2F%28-4000%29 Break up the expression.


x+=+%28100000%29%2F%28-4000%29 or x+=++%28-80000%29%2F%28-4000%29 Combine like terms.


x+=+-25 or x+=+20 Simplify.


So the possible answers are x+=+-25 or x+=+20


However, a negative number of people doesn't make any sense. So we must ignore the solution x=-25



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Answer:


So the solution is x+=+20 which means that there are 20 members currently in the club (before the 5 people join).

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Let the current amount of members be M

Since the cost of the plane is $200,000, then the current cost, per
person is: 200000%2FM

Since they wish to get 5 more members, then the new amount of members then would be:
M + 5

Therefore, the new cost, then, per person would be: 200000%2F%28M+%2B+5%29

Now, since the cost per person would decrease by $2,000 when 5 new members are admitted, then we get: 200000%2FM+-+200000%2F%28M+%2B+5%29+=+2000

200,000(M + 5) - 200,000(M) = 2,000(M)(M + 5) ------> Multiplying entire equation by LCD, M(M + 5)

200000M+%2B+1000000+-+200000M+=+2000M%5E2+%2B+10000M
2000M%5E2+%2B+10000M+-+1000000+=+0
2000%28M%5E2+%2B+5M+-+500%29+=+0
M%5E2+%2B+5M+-+500+=+0

(M + 25)(M - 20) = 0

M = - 25, or 20
Ignore M = - 25, so M = 20

Therefore, the current amount of members in the club is highlight_green%2820%29, but will change to 25 (20 + 5) after 5 new ones are admitted.
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Check
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Current amount of members: 20. Cost per person = 200000%2F20 = $10,000.

New amount of members after 5 new ones are admitted: 25 (20 + 5). New cost per person then would be: 200000%2F25 = $8,000.

Reduction in price per person then would be $2,000 ($10,000 - $8,000)