SOLUTION: The prom brought in $6300 last year. This year, they lowered the price by 3 dollars, and ended up selling 48 more tickets. They earned $5970. What was the original price per ticke

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: The prom brought in $6300 last year. This year, they lowered the price by 3 dollars, and ended up selling 48 more tickets. They earned $5970. What was the original price per ticke      Log On


   

Question 970226: The prom brought in $6300 last year.
This year, they lowered the price by 3 dollars, and ended up selling 48 more tickets. They earned $5970. What was the original price per ticket (last year), and how many tickets were sold at the lower price (this year)
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
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P=original price (last year); N=number of tickets (last year)
PN=$6300
N=$6300/P
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%28N%2B48%29%28P-%243%29=%245970 Substitute for N.
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%28%28%246300%2FP%29%2B48%29%28P-%243%29=%245970
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%246300-%2418900%2FP%2B48P-%24144=%245970
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%246156-%2418900%2FP%2B48P=%245970 Subtract $6156 from each side.
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-%2418900%2FP%2B48P=-%24186 Convert to common denominator.
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-18900%2FP%2B%2848P%5E2%29%2FP=%28-%24186P%29%2FP Multiply by P.
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-%2418900%2B48P%5E2=-%24186P Add $186P to each side.
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48P%5E2%2B%24186P-%2418900=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aP%5E2%2BbP%2Bc=0 (in our case 48P%5E2%2B186P%2B-18900+=+0) has the following solutons:

P%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=%28186%29%5E2-4%2A48%2A-18900=3663396.

Discriminant d=3663396 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-186%2B-sqrt%28+3663396+%29%29%2F2%5Ca.

P%5B1%5D+=+%28-%28186%29%2Bsqrt%28+3663396+%29%29%2F2%5C48+=+18
P%5B2%5D+=+%28-%28186%29-sqrt%28+3663396+%29%29%2F2%5C48+=+-21.875

Quadratic expression 48P%5E2%2B186P%2B-18900 can be factored:
48P%5E2%2B186P%2B-18900+=+48%28P-18%29%2A%28P--21.875%29
Again, the answer is: 18, -21.875. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+48%2Ax%5E2%2B186%2Ax%2B-18900+%29

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The price of a prom ticket last year was $18, so:
$6300/$18=350 There were 350 tickets sold last year.
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The price of the prom this year was $15 ($18-$3=$15)
$5970/$15=398 There were 398 tickets sold this year.(48 more than last year)