So we need 2x+ 4y+ z/4= 100 with the understanding that x, y, and z are integers.

Multiplying through by 4, 8x+ 16y+ z= 100. I would also write that as 8x+ 16z= 100- z. Since the left side is obviously divisible by 8, so is 100- z. Write 100- z= 8a and divide the equation by 8: x+ 2y= a.

That is a Diophantine equation (maybe not all

Further, it is easy to see that x= -a+ 2k, y= a- k is also a solution for any integer k: x+ 2y= (-a+ 2k)+ (2a-2k)= a.

But a was equal to 100- z so we have x= z- 100- k and y= 100- z+ k. Those are again Diophantine equations: z- x= 100+ k and y+ z= 100- k. Solve those for x and z. Put them back into the original equation to determine k.

they included Engineers, Scientists and Workers.

He had 100 plates for the dinner.

Now Engineers saw it and thought why don't we play our game.

They said we will need two plates per person for the dinner.

Seeing that Scientists too couldn't help,

they said we will need 4 plates per person for the dinner.

Now Workers thought, they will fail the dinner.

So they said we need only 1 plate per four person.

At last the dinner was successful all 100 plates were used by the 100 persons.

So can you tell me the respective number of Engineers, Scientists and Workers?

................................................

2 solutions:

E = 15, S = 13, W = 72

E = 30, S = 6, W = 64

To have a unique solution, you need another information,

like "2 more E's than S's".

I'm puzzled. I can see:

So we need 2x+ 4y+ z/4= 100 with the understanding that x, y, and z are integers.

Multiplying through by 4, 8x+ 16y+ z= 100. I would also write that as 8x+ 16z= 100- z. Since the left side is obviously divisible by 8, so is 100- z. Write 100- z= 8a and divide the equation by 8: x+ 2y= a.

That is a Diophantine equation (maybe not allthat"straight forward"). In particular, taking x= -1, y= 1 we have x+ 2y= -1+ 2= 1. Multiplying by a, if x= -a, y= 2a, x+ 2y= -a+ 2a= a.

Further, it is easy to see that x= -a+ 2k, y= a- k is also a solution for any integer k: x+ 2y= (-a+ 2k)+ (2a-2k)= a.

But a was equal to 100- z so we have x= z- 100- k and y= 100- z+ k. Those are again Diophantine equations: z- x= 100+ k and y+ z= 100- k. Solve those for x and z. Put them back into the original equation to determine k.

2x+ 4y+ z/4= 100

but multiplying thru by 4 I get:

8x+ 16y+ z= 400

Is there something I've missed?