Question 67920
The total cost of the party is =  $ 75


Let x be the number of people that have attended the party. 


Let the price divided by all the people who have attended the party is =  y 


Hence, the equation will be: 


{{{(75/x) = y}}}  --------------->(1) 


Now it is given that 5 more people attended the party. So it is x + 5 


And the amount to be shared is reduced by 50 cents.


So it is, y - 0.50 


Hence, the equation will be: 


{{{(75/(x+5)) = y - 0.50}}}   ----------------->(2) 


Substituting EQN (1) in EQN (2), we get: 



{{{(75/(x+ 5)) = (75/x) - 0.50}}} 


{{{ 0.50 = (75/x)- (75/(x+ 5))}}} 



{{{ 0.50 = (75(x + 5) - 75x)/(x(x+5))}}} 


{{{0.50 = (75x + 375 - 75x)/(x(x + 5))}}} 


{{{0.50 = 375/x(x + 5)}}}


{{{0.50x(x + 5) = 375}}} 


{{{(1/2)x(x + 5) = 375}}}


x(x + 5) = 375 * 2 


{{{x^2}}} + 5x = 750 


{{{x^2}}} + 5x - 750 = 0 


This is a quadratic equation, hence by the Method Of Factorization we solve for the value of x. 


We observe that the factors of 750 are 25 * 30 


So, the equation becomes: 


{{{x^2}}} + 30x - 25x - 750 = 0 


x(x + 30) - 25(x + 30) = 0 


(x - 25)(x + 30) = 0 


This implies x = 25 or x = -30 


Since 30 is neagtive we cannot take the value. Hence, we take the value 25.


Thus, the total number of people that attended tha party is = 25.


Therefore, y = 25 


So when the number of people of increased by 5 then the total number of people who attended the party is y = 30 


Hence, the cost per people whould be $ 3 when the number of peopl is 25. 


So when there are 30 people, the share of money to be given to the party is $2.50


I hope the steps are clear. 


If you have any more queries you can get back to me on this ID. tutyfruty9@yahoo.com


Happy Calculating!!!!