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 170726This question is from textbook intermediate algebra
: 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?
Not sure how to do these word problems and will appreciate any help, so far I have a B in class :) This question is from textbook intermediate algebra

Found 2 solutions by jim_thompson5910, Mathtut:
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.

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
alright so we would normally find out an average cost per person by taking the cost and dividing by number of people....lets call the original # of people attending x so avg cost would be 500/x but now x is decreasing by 5 and the average cost is increasing by 5 so
:
500%2Fx=%28500%2F%28x-5%29-5%29 mulitply all terms by x(x-5)
:
500%28x-5%29=500x-5x%5E2%2B25x
:
500x-2500=500x-5x^2+25x}}}
:
5x%5E2-25x-2500=0divide by 5
:
x%5E2-5x-500=0
:
system%28x=25%2Cx=-20%29
:
throw out the negative and we have the original amount of people of 25
:
therefore the number of people attending the party ishighlight%2825-5=20%29
:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-5x%2B-500+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-5%29%5E2-4%2A1%2A-500=2025.

Discriminant d=2025 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+2025+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+2025+%29%29%2F2%5C1+=+25
x%5B2%5D+=+%28-%28-5%29-sqrt%28+2025+%29%29%2F2%5C1+=+-20

Quadratic expression 1x%5E2%2B-5x%2B-500 can be factored:
1x%5E2%2B-5x%2B-500+=+1%28x-25%29%2A%28x--20%29
Again, the answer is: 25, -20. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-5%2Ax%2B-500+%29