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The profits of a business are shared among three stockholders

The profits of a business are shared among three stockholders. The first stockholder receives twice as much as the second and the second receives three times as much as the third. If the profits for last year were $ 26,400, how much did each stockholder receive?

 
Solution:
 
x = first stockholder
y = second stockholder
z = third stockholder
 
x + y + z          = 26400           [Eq. 1]
x                      = 2y                 [Eq. 2]
y                      = 3z                 [Eq. 3]
 
Rewriting Eq. 3 in terms of z will give:
y                      = 3z                 [Eq. 3]
y/3                   = z
z                       = y/3                [Eq. 4]
 
Now, we can solve the value of y by using Eq. 1 substituting the known values for x and z as follows:
x + y + z          = 26400           [Eq. 1]
            (2y) + y + (y/3) = 26400
            3y + y/3           = 26400
            3[3y + y/3]       = [26400]3     
            9y + y              = 79200
            10y                  = 79200
            y                      = 79200/10
            y                      = 7920
 
Using Eq. 2, we can derive the value of x:
x          = 2y                 [Eq. 2]
x          = 2(7920)
x          = 15840
 
Using Eq. 1, we can solve for the value z:
x + y + z                      = 26400           [Eq. 1]
15840 + 7920 + z        = 26400
23760 + z                   = 26400
z                                   = 26400 – 23760
z                                   = 2640
 
 
Therefore, there first stockholder receives 15,840 dollars, the second stockholder receives 7,920 dollars and the third stockholder receives 2,640 dollars.
 
Check:
 
x + y + z                                  = 26,400          [Eq. 1]
[The sum of the profits of the 3 stockholders is $26,400]
15,840 + 7,920 + 2,640           = 26,400
26,400                                     = 26,400
 
x                      = 2y                 [Eq. 2]
[The first stockholder’s profit is twice that of the second]
15,840             = 2(7,920)
15,840             = 15,840
 
y                      = 3z                 [Eq. 3]
[The second stockholder’s profit is three time that of the third]
7,920               = 3(2,640)
7,920               = 7,920