SOLUTION: Some students planned for a get-together. The budget for food was $500. Five of the students failed to come because of the distance and therefore the cost of food for each member i

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Question 200500: Some students planned for a get-together. The budget for food was $500. Five of the students failed to come because of the distance and therefore the cost of food for each member increased by $5. How many students attended the get-together?

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let c=cost per individual (the cost before the students failed to arrive) and n=number of students


So the total cost is divided among n people to get

c=500%2Fn


Since 5 failed to attend, this means that n-5 students showed up. Now the new cost is

c%2B5=500%2F%28n-5%29


c%2B5=500%2F%28n-5%29 Start with the given equation.


500%2Fn%2B5=500%2F%28n-5%29 Plug in c=500%2Fn


500%28n-5%29%2B5n%28n-5%29=500n Multiply every term by the LCD n%28n-5%29. This cancels out every denominator.


500n-2500%2B5n%5E2-25n=500n Distribute


500n-2500%2B5n%5E2-25n-500n=0 Subtract 500n from both sides.


5n%5E2-25n-2500=0 Subtract 500n from both sides.


Let's use the quadratic formula to solve for n


n+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


n+=+%28-%28-25%29+%2B-+sqrt%28+%28-25%29%5E2-4%285%29%28-2500%29+%29%29%2F%282%285%29%29 Plug in a=5, b=-25, and c=-2500


n+=+%2825+%2B-+sqrt%28+%28-25%29%5E2-4%285%29%28-2500%29+%29%29%2F%282%285%29%29 Negate -25 to get 25.


n+=+%2825+%2B-+sqrt%28+625-4%285%29%28-2500%29+%29%29%2F%282%285%29%29 Square -25 to get 625.


n+=+%2825+%2B-+sqrt%28+625--50000+%29%29%2F%282%285%29%29 Multiply 4%285%29%28-2500%29 to get -50000


n+=+%2825+%2B-+sqrt%28+625%2B50000+%29%29%2F%282%285%29%29 Rewrite sqrt%28625--50000%29 as sqrt%28625%2B50000%29


n+=+%2825+%2B-+sqrt%28+50625+%29%29%2F%282%285%29%29 Add 625 to 50000 to get 50625


n+=+%2825+%2B-+sqrt%28+50625+%29%29%2F%2810%29 Multiply 2 and 5 to get 10.


n+=+%2825+%2B-+225%29%2F%2810%29 Take the square root of 50625 to get 225.


n+=+%2825+%2B+225%29%2F%2810%29 or n+=+%2825+-+225%29%2F%2810%29 Break up the expression.


n+=+%28250%29%2F%2810%29 or n+=++%28-200%29%2F%2810%29 Combine like terms.


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


Since a negative amount of people doesn't make any sense, this means that the only answer is n+=+25


So 25 people were going to attend but only 20 actually attended.



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