SOLUTION: Some of Aaron's friends are planning to buy him a gift worth 270, dividing the cost equally among themselves. Six more of his friends decided to share in the expenses and so each o

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Question 747745: Some of Aaron's friends are planning to buy him a gift worth 270, dividing the cost equally among themselves. Six more of his friends decided to share in the expenses and so each one's share is decreased by 12. How many friends were originally part of the plan?
Found 2 solutions by lynnlo, ikleyn:
Answer by lynnlo(4176) About Me  (Show Source):
Answer by ikleyn(53339) About Me  (Show Source):
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Some of Aaron's friends are planning to buy him a gift worth 270, dividing the cost equally among themselves.
Six more of his friends decided to share in the expenses and so each one's share is decreased by 12.
How many friends were originally part of the plan?
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Let 'n' be the number of friends originally.

Then each of n friends contributes  270%2Fn.


In the extended version, the number of friends is (n+6),

and every contributes  270%2F%28n%2B6%29.


Write equation as you read the problem

    270%2Fn - 270%2F%28n%2B6%29 = 12.


For simplicity, divide both sides by 3

    90%2Fn - 90%2F%28n%2B6%29 = 4.


Multiply both sides by n*(n+6)

    90(n+6) - 90n = 4n*(n+6),

    90n + 540 - 90n = 4n*(n+6),

          540       = 4n*(n+6).


Divide both sides by 4

          135 = n*(n+6),

          n^2 + 6n - 135 = 0.


Factor

          (n-9)*(n+15) = 0.


Thus the roots of the quadratic equation are 9 and - 15,
and we select positive n= 9 as the solution to the problem.


ANSWER.  The original number of friends was 9.

Solved.