Question 243195: In a given corporation 1/4 of the hourly workers invest the 401k plan and 1/3 of the salaried workers invest in the same plan.
1/10 of the hourly workers invest the maximum allowed and 1/6 of the salaried workers invest the maximum allowed.
If there are three times as many hourly workers as salaried workers, what fraction of all the workers, who have invested in the plan, invest the maximum allowable to the 401k.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Let T = number of workers.
Let H = number of hourly workers.
Let S = number of salary workers.
T = H + S
H = 3 * S (number of hourly works are 3 times the number of salary workers).
Equation for T becomes:
T = 3S + S = 4S
This makes S = 1/4 * T
This makes H = 3/4 * T
Let P = the 401K plan.
1/4 of the hourly workers invest in it and 1/3 of the salary workers invest in it.
This means that:
(1/4 * (3/4) * T) + (1/3 * (1/4) * T) = P
Where T equals total number of workers in the firm and P equals the number of workers who invest in the 401K plan.
Simplify that equation to make it become:
(3/16 * T) + (1/12 * T) = P
Multiply 3/16 by 3/3 and multiply 1/12 by 4/4 to get:
(9/48 * T) + (4/48 * T = P
Factor out the T and combine like terms to get:
13/48 * T = P
That's the proportion of the total people in the company that invest in the 401K plan.
1/10 of the hourly works invest in the maximum of the plan and 1/6 of the salary workers invest in the maximum of the plan./
This means that:
(1/10 * 3/4 * T) + (1/6 * 1/4 * T) = M where:
M is the number of people who invest in the maximum of the 401K plan.
Simplify that equation to make it become:
(3/40 * T) + (1/24 * T) = M
Multiply 3/40 * 6/6 and multiply 1/24 by 10/10 to get:
(18/240 * T) + (10/240 * T) = M
Factor out the T and combine like terms to get:
28/240 * T = M
That's the proportion of the people in the company that invest in the maximum of the 401K plan.
We have two proportions.
They are:
13/48 * T = P
and
28/240 * T = M
Question was:
What is the proportion of the people who invested in the plan that invested in the max of the plan.
That would be:
(28/240) / (13/48)
That becomes:
(28*48)/(240*13)
Since 240/48 = 5, the equation becomes:
(28) / (5*13)
That becomes:
28/65 = the fraction of the employees who invested in the plan who also invested in the maximum allowable in the plan.
In terms of percentages it works out like this.
13/48 = .2708333 * 100% = 27.08 percent of the total employees invest in the 401K plan.
28/240 = .11666667 * 100% = 11.67 percent of the total employees invest in the maximum the 401K plan allows.
28/65 = .430769 * 100% = 43.08 percent of the employees who invest in the 401K plan invest in the maximum that the plan allows.
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