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GMATGuruNY
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anilnandyala
Health insurance Plan A requires the insured to pay $1000 or 50% of total cost, whichever is lower. Plan B requires the insured to pay the initial $300, but then pays 80% of the cost over $300. Which of the following is a cost level for which both insurance plans pay out the same amount?

A. $600
B. $1000
C. $3800
D. $5300
E. $6200

Case 1: What Insured pay is
1000 (as per plan A) = 300+(20/100)*(C-300)
Solving this equation, C = 3800

Case 2: What Insured pay is
(50/100)*C (as per plan A) = 300+(20/100)*(C-300)
Solving this equation, C = 800

Answer: Option C

Hi, I don't really under stand the plan B equation that you formulate. 300+0.2(C-300) << what I understand is 0.2 here is what the insurance company is paying, but the for the insured would pay 80% (0.8)?

So, should't it be 300+(80/100)*(C-300) if the case is "What Insured pay is"

Thanks.
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wasario
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Leaving this answer for myself, since I had a hard time understanding what the question was saying:

The question is asking at which cost does the INSURANCE COMPANY pay the same amount; in this case, if we have a total T, and the insured pays a price I, then the amount that the company has to pay is \(T - I\).

Plan A says that the person insured pays either 1000 or 50% of the total cost, whichever is cheaper. This means that if the total price P is >1000, the COMPANY must pay \(P - 1000\) since \(1000 > .5P\) for the INSURED all values of P bigger than 1000, otherwise if \(P <= 1000\), the company must pay \(P - .5P = .5P\).

Plan B says that the person insured only pays 300 dollars, but then the insurance company must pay 80% of the REST of the money. This means that for a total price P, the company must pay \(.8(P - 300)\).

We want to find the value at which the 2 values are the same. You can either do this algebraically, or just plug in the answer choices and see which one gives the correct answer.

Doing it algebraically, let's say that we're assuming that \(P > 1000\). Then we have the following system of equations and can solve as follows:

\(\\
P - 1000 = .8(P - 300)\\
\\
P - 1000 = .8P - 240\\
\\
.2P = 760\\
\\
P = 3800\\
\)

Answer choice C.

Let's say that for the case of \(P <= 1000\). Then we have this different system of equations which can be solved as such:

\(\\
.5P = .8(P - 300)\\
\\
.5P = .8P - 240\\
\\
.3P = 240\\
\\
P = 800\\
\)

However, because 800 isn't an answer choice, this case isn't applicable to this problem.

Alternatively, we could also just plug in the answer choices and see which one works. I won't write them all out, but for C, which is the correct answer, we get:

Plan A
\(P = 3800 - 1000 = 2800\)

Plan B
\(P = .8(3800 - 300) = 2800\)
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