Answer to Question #5826 in Complex Analysis for neelakshay

Question

there are100 people and 100 plates.among 100 people they are divided into three categories rich which take 4plates each,service class 2 plate each and poor .4 poor eat in 1 plate.calculate no of persons of each categories?

Answer 1

Let's denominate R, S and P the number of members in each class. So,

R + S + P = 100, (total number of people) (1)
4R + 2S + P/4 = 100. (total number of plates) (2)

Also, R,S,P are integers and 0<=R<=100/4=25, 0<=S<=100 0<=P<=100& (*).

(1) ==> R = 100 - S - P
(2): 4(100 - S - P) + 2S + P/4 = 100

400 - 4S - 4P + 2S + P/4 = 100
2S + 3.75P = 300 ==> S = 150 - 1.875P, or S = 150 - 15P/8.

As we see, P must be a multiple of 8.
These pairs satisfy obtained relation and restrictions (*):

P = 8*4, S = 150-15*8*4/8 = 90;
P = 8*5, S = 75;
P = 8*6, S = 60;
P = 8*7, S = 45;
P = 8*8, S = 30;
P = 8*9, S = 15;
P = 8*10, S = 0.

As R = 100 - S - P, we'll get

P = 32, S = 90 ==> R = -22<0;
P = 40, S = 75 ==> R = -15<0;
P = 48, S = 60 ==> R = -8<0;
P = 56, S = 45 ==> R = -1<=0;
P = 64, S = 30 ==> R = 6;
P = 72, S = 15 ==> R = 13;
P = 80, S = 0& ==> R = 20.

So, there are three possible opportunities:
6 rich, 30 service class and 64 poor people;
13 rich, 15 service class and 72 poor people;
20 rich, no service class and 80 poor people.