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(From an actuarial exam) 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

Respuesta :

Answer:

Step-by-step explanation:

We can work with these values as a set value, and build a Venn Diagram from them.

I am going to say the set A are those that have the health insurance plan.

Set B are those that have the life insurance plan

Set C are those that have the investment plan.

We have that:

[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]

In which a is the number of employees that only have the health insurance plan, [tex]A \cap B[/tex] is the number of employees that have both the health and the life insurance plans, [tex]A \cap C[/tex] is the number of employees that have both the health insurance and the investment plans. and [tex]A \cap B \cap C[/tex] is the number of employees that have all three of those plans.

By the same logic, we have that:

[tex]B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)[/tex]

[tex]C = c + (B \cap C) + (A \cap C) + (A \cap B \cap C)[/tex]

The problem states that:

An employee cannot have both life insurance and investment plans. So:

[tex]B \cap C = 0, A \cap B \cap C = 0[/tex]

45 employees have only the life insurance plan. So:

[tex]b = 45[/tex]

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

[tex]A \cap B = A \cap C + 20[/tex]

320 employees have the health insurance plan.

[tex]A = 320[/tex]

450 employees have at least one plan

[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 450[/tex]

330 employees have only one plan

[tex]a + b + c = 330[/tex]

How many people have the investment plan?

We have to find the value of C.

Now we solve:

[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 450[/tex]

Applying what we have

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[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 450[/tex]

[tex]330 + A \cap C + 20 + A \cap C = 450[/tex]

[tex]2(A\capC) = 100[/tex]

[tex]A \cap C = 50[/tex]

[tex]A \cap B = A \cap C + 20 = 50 + 20 = 70[/tex]

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[tex]A = 320[/tex]

[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]

[tex]a + 70 + 50 = 320[/tex]

[tex]a = 200[/tex]

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[tex]b = 45[/tex]

[tex]a + b + c = 330[/tex]

[tex]245 + c = 330[/tex]

[tex]c = 85[/tex]

The number of people that have the investment plan is:

[tex]C = 85 + 50 = 135[/tex]

135 people have the investment plan