Question 243195
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.