Actuarial and Investment companies problem.

Question

I am trying to study for the first actuarial exam, and I'm stuck with this problem. I tried to use Venn Diagrams to solve it, but I cannot reach the correct solution. The book said $135$ people have the investment plant. I need to use something else. Any help? Thanks!

A company offers a health insurance plan, a life insurance plan, and an investment insurance plan. An employee can have $0$, $1$, or $2$ plans, but cannot have both life insurance and investment plans. You are given the following information:

$450$ employees have at least one plan. $330$ employees have only one plan. $320$ employees have the health insurance plan. $45$ employees have only the life insurance plan.

There are $20$ more employees that have both health and life plans than those that have both health and investment plans.

How many people have the investment plan? Correct answer is $135$, but how?

Answer 1

Looking at it algebraically;

Let $health+investment = x$. Hence $health+life = x+20$, since there are $20$ more employees that have health and life plans than those that have health and investment plans.

Since $450$ employees have at least one plan, and $330$ have only one plan, this leaves $120$ employees that have more than one plan. Also, since the only combination of plans allowed are $health+investment$ and $health+life$, their total sum should equal $120$. Hence, $x + (x+20) = 2x+20 = 120$. Therefore, $x = 50$ (after simplification).

Since there are $320$ employees that have health plans (this is including the $health+investment$, $health+life$ and $health$ only plans), we can add them up to find number of employees that have $health$ only plans. $Health+investment = 50$, since $x = 50$. Also $health+life = 70$, since $x+20 = 50+20 = 70$. And $50+70 = 120$. Taking away $120$ from the total number of employees that have health plans $(320)$, we get $200$ employees that have only health plans.

Since $45$ employees have only life insurance plan, $200 + 45 = 245$. Thus, $245$ employees so far have only one plan. We know $330$ employees in total have only one plan, so the remaining $85$ employees have only investment plans.

The number of people who have investment plans include $investment+health$ and $investment$ only. We have found the values for both, which are: $investment+health = 50$ and $investment$ $only=85$. And $50+85 = 135$.

Therefore, the number of employees that have $investment$ plan are $135$.

Answer 2

let
$LH$ represent the number of people Having both Life and Health plans
$I$ represent the number of people Having only Investment
etc.

The answer we seek is then $x \equiv I + IH$

the information given provides a system of five equations and five unknowns

$$ \begin{eqnarray*} L + H + I + IH + LH &=& 450 \\ L + H + I &=& 330 \\ H + LH + IH &=& 320 \\ L&=& 45 \\ LH -IH &=& 20 \end{eqnarray*} $$