SOLUTION: Island dinner booked a party and was charging the group a fix cost of 1,680. Later the number of people in the group increased by 50 people and the average cost decreased by $10 pe

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Question 596067: Island dinner booked a party and was charging the group a fix cost of 1,680. Later the number of people in the group increased by 50 people and the average cost decreased by $10 per person. How many people were booked originally.
a. 33
b. 70
c. 120
d. 168
The answer is B. 70. Can't figure out how to get 70.
Can someone please help me out with this problem. Can't figure this one out
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let c = cost per person (before the increase of 50 people) and let n = number of people (originally -- before the increase of 50 people)


Since the total cost is 1680, and this cost is distributed evenly among 'n' people, this means that each person will pay c=1680%2Fn dollars.

Adding 50 people to the group increases the count to n+50 and it decreases the cost per person by $10. So the cost per person is now c-10


The same cost of 1680 is now distributed amongst n+50 people and the cost per person is now c-10


This means that c-10=1680%2F%28n%2B50%29


Let's use this equation and c=1680%2Fn to solve for n


c-10=1680%2F%28n%2B50%29

1680%2Fn-10=1680%2F%28n%2B50%29

n%28n%2B50%29%281680%2Fn%29-10n%28n%2B50%29=n%28n%2B50%29%281680%2F%28n%2B50%29%29

1680%28n%2B50%29-10n%28n%2B50%29=1680n

1680n%2B84000-10n%5E2-500n=1680n

1680n%2B84000-10n%5E2-500n-1680n=0

-10n%5E2-500n%2B84000=0

10n%5E2%2B500n-84000=0

Now use the quadratic formula to solve for n

n+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

n+=+%28-%28500%29%2B-sqrt%28%28500%29%5E2-4%2810%29%28-84000%29%29%29%2F%282%2810%29%29

n+=+%28-500%2B-sqrt%28250000-%28-3360000%29%29%29%2F%2820%29

n+=+%28-500%2B-sqrt%283610000%29%29%2F20

n+=+%28-500%2Bsqrt%283610000%29%29%2F20 or n+=+%28-500-sqrt%283610000%29%29%2F20

n+=+%28-500%2B1900%29%2F20 or n+=+%28-500-1900%29%2F20

n+=+1400%2F20 or n+=+-2400%2F20

n+=+70 or n+=+-120

Toss out the negative solution since you can't have a negative number of people.

Therefore, the only solution is n+=+70

So there were originally 70 people.