Question 1164247
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<pre>

Let n be the number of professors in the base scenario.

Then the contribution of each of them is  {{{140/n}}}  dollars.


In the hypothetical  "if"  scenario, there are only n-5 professors;

hence, each contributes  {{{140/(n-5)}}}  dollars.


The difference between these personal contributions is given as $50.  
It gives you this equation


    {{{140/(n-5)}}} - {{{140/n}}} = 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,  {{{140/(7-5)}}} - {{{140/7}}} = {{{140/2}}} - {{{140/7}}} = 70 - 20 = 50.


<U>ANSWER</U>.  There are 7 math professors contributing in the base scenario.
</pre>

Solved.


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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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