Question 1164247: Some mathematics professors would like to purchase a $140 microwave oven for the department workroom. If 5 of the professors do not contribute, everyone's share will increase by $50. How many professors are in the department?
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52778)   (Show Source):
You can put this solution on YOUR website! .
Let n be the number of professors in the base scenario.
Then the contribution of each of them is  dollars.
In the hypothetical "if" scenario, there are only n-5 professors;
hence, each contributes  dollars.
The difference between these personal contributions is given as $50.
It gives you this equation
- = 50 dollars. (1)
To continue, you may multiply both sides by n*(n-5). You will get then the quadratic equation, which you should
to simplify to the standard form and solve.
But you also can solve the problem mentally and to deserve admiration of math professors.
For it, notice that the right side, 50, is a big number, comparing with 140 dollars.
It means that the divisors of the 140, n and (n-5) are comparatively small integer numbers.
Keeping it in your mind, you may quickly guess the solution to equation (1) n= 7.
Indeed,  -  =  - = 70 - 20 = 50.
ANSWER. There are 7 math professors contributing in the base scenario.
Solved.
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- Challenging word problems solved using quadratic equations
- Had they sold . . .
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https://www.algebra.com/algebra/homework/playground/test.faq.question.1163463.html
https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1163290.html
https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1162952.html
https://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.1162659.html
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Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website!
OH CRAP!!! Ikleyn did it for you before I could post this!
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Instead of doing your problem for you, I'll do one EXACTLY like yours so
that you can use it as a model for your problem:
Some mathematics professors would like to purchase a $300 microwave oven for
the department workroom. If 6 of the professors do not contribute,
everyone's share will increase by $45. How many professors are in the
department?
If all N professors shared, then everyone's share would be 300/N
If 6 professors do not contribute, the share for the N-6 professors who do
contribute will be $45 more than that or $(300/N + 45)
So the equation is:
(N-6)(300/N+45) = 300
Use FOIL to multiply that out:
N(300/N) + 45N - 1800/N + 270 = 300
300 + 45N - 1800/N + 270 = 300
45N - 1800/N + 270 = 0
Multiply through by N to clear the fraction:
45N² - 1800 + 270N = 0
45N² + 270N - 1800 = 0
Divide through by 45
N² + 6N - 40 = 0
Factor (N - 10)(N + 4) = 0
N - 10 = 0; N + 4 = 0
N = 10; N = -4
N = 10 professors. Ignore the negative answer.
Now use this as a model to do yours.
Edwin
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