SOLUTION: A group of math teachers went to a restaurant. They agreed to split the bill equally. When the bill arrived, two of them discovered they had forgotten their money. The others agree

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Question 335954: A group of math teachers went to a restaurant. They agreed to split the bill equally. When the bill arrived, two of them discovered they had forgotten their money. The others agreed to make up the difference by paying an extra $1.30 each. If the total bill was $78, how many teachers were in the group?
I figured out the answer is 12 teachers but i can't figure out the equation.
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
We don't know how many teachers there are, so let's say that there are 'x' teachers (ie x = # of teachers)

Since "the total bill was $78" and "They agreed to split the bill equally", this means that 78%2Fx=c or c=78%2Fx where 'c' is the amount of money each teacher pays (before the two back out)


Since "two of them discovered they had forgotten their money" and "The others agreed to make up the difference by paying an extra $1.30 each", this means that 78%2F%28x-2%29=c%2B1.3. Basically, the bill of $78 is now divided among x-2 teachers and each teacher now pays c%2B1.3 dollars.


78%2F%28x-2%29=c%2B1.3 Start with the second equation.


78%2F%28x-2%29=78%2Fx%2B1.3 Plug in c=78%2Fx


78x=78%28x-2%29%2B1.3x%28x-2%29 Multiply EVERY term by the LCD x%28x-2%29


78x=78x-156%2B1.3x%5E2-2.6x Distribute.


780x=780x-1560%2B13x%5E2-26x Multiply every term by 10 to get every number to be a whole number.


0=780x-1560%2B13x%5E2-26x-780x Subtract 780x from both sides.


0=13x%5E2-26x-1560 Combine like terms.


Notice that the quadratic 13x%5E2-26x-1560 is in the form of Ax%5E2%2BBx%2BC where A=13, B=-26, and C=-1560


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%28-26%29+%2B-+sqrt%28+%28-26%29%5E2-4%2813%29%28-1560%29+%29%29%2F%282%2813%29%29 Plug in A=13, B=-26, and C=-1560


x+=+%2826+%2B-+sqrt%28+%28-26%29%5E2-4%2813%29%28-1560%29+%29%29%2F%282%2813%29%29 Negate -26 to get 26.


x+=+%2826+%2B-+sqrt%28+676-4%2813%29%28-1560%29+%29%29%2F%282%2813%29%29 Square -26 to get 676.


x+=+%2826+%2B-+sqrt%28+676--81120+%29%29%2F%282%2813%29%29 Multiply 4%2813%29%28-1560%29 to get -81120


x+=+%2826+%2B-+sqrt%28+676%2B81120+%29%29%2F%282%2813%29%29 Rewrite sqrt%28676--81120%29 as sqrt%28676%2B81120%29


x+=+%2826+%2B-+sqrt%28+81796+%29%29%2F%282%2813%29%29 Add 676 to 81120 to get 81796


x+=+%2826+%2B-+sqrt%28+81796+%29%29%2F%2826%29 Multiply 2 and 13 to get 26.


x+=+%2826+%2B-+286%29%2F%2826%29 Take the square root of 81796 to get 286.


x+=+%2826+%2B+286%29%2F%2826%29 or x+=+%2826+-+286%29%2F%2826%29 Break up the expression.


x+=+%28312%29%2F%2826%29 or x+=++%28-260%29%2F%2826%29 Combine like terms.


x+=+12 or x+=+-10 Simplify.


So the possible solutions are x+=+12 or x+=+-10


Ignore the negative solution as a negative amount of teachers makes no sense.


So there are originally 12 teachers.


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my website.

Jim

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
A group of math teachers went to a restaurant.
They agreed to split the bill equally.
When the bill arrived, two of them discovered they had forgotten their money. The others agreed to make up the difference by paying an extra $1.30 each. If the total bill was $78, how many teachers were in the group?
----
Let x be the expected number payers.
Average charge would be 78/x.
-----------------------------------
x-2 actually shared the bill.
Average charge would be 78/(x-2)
-----------------------------------
Equation:
New average - old average = 1.30
78/(x-2) - 78/x = 1.30
Multiply thru by x(x-2) to get:
78x -78(x-2) = 1.3x(x-2)
78*2 = 1.3x^2 - 2.6x
--------------------------
Multiply thru by 10 to get rid of the decimals:
13x^2 - 26x - 1560 = 0
---
x^2 - 2x - 120 = 0
(x-12)(x+10) = 0
Positive solution:
x = 12 (original number of payers before 2 reneged)
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Cheers,
Stan H.
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