
Word Problem
Tom Smith hires three types of laborers (I, II, III) and pays them $8, $6, and $5 per hour, respectively. If the total amount paid is $2,800 for a total of 500 hours of work, find the possible number of hours put in by the three categories of workers if category III workers must put in the maximum number of hours?

Re: Word Problem
Let us say we have x, y and z hours put be the three types of laborers, I, II, III respectively
Thus we get x + y + z = 500  ( 1 )
where 'z' has to be maximum
Also we are given the expenditure.
8x + 6y + 5z = 2800  ( 2 )
From (1) we get x = 500  y  z
Plugging it in equation (2), we get
8( 500  y  z ) + 6y + 5z = 2800
4000  8y  8z + 6y + 5z = 2800
2y + 3z = 1200
Since z has to be largest and total number of hours is 500 we try z = 300
That will give
y = 150
Plugging in these values in (1) we get x = 50.
We can verify that this solution which satisfies the given conditions