SOLUTION: Please Help!!!!Could someone help me with this problem? I am confused? Bayside Insurance offers two health plans. Under Plan A, Sam would have to pay the first $80 of his medic

Question 316883: Please Help!!!!Could someone help me with this problem? I am confused?
Bayside Insurance offers two health plans. Under Plan A, Sam would have to pay the first $80 of his medical bills, plus 35% of the rest. Under Plan B, Sam would pay the first $230, but only 20% of the rest. For what amount of medical bills will Plan B save Sam money? Assume he was over $230 in bills. Sam would save with Plan B if he had more than $_____ in bills.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Plan A:
Sam pays $80 and then 35% of the rest.
Plan B:
Sam pays first $230, then 20% of the rest.

We let T = total cost of the medical bills.
We let R = Remainder of the costs of the medical bills.

For plan A, we get TA = 80 + .35 * R
For plan B, we get TB = 230 + .20 * R

TA means total cost for plan A.
TB means total cost for plan B.

If plan B total costs are going to be less than plan A total costs, then TB must be smaller than TA, which results in the formula:

TB < TA

Since TB = 230 + .20 * R and TA = 80 + .35 * R, then we get the formula:

230 + .20 * R < 80 + .35 * R

If we subtract .20 from both sides of the equation, we get:

230 < 80 + .15 * R

If we subtract 80 from both sides of this equation, we get:

150 < .15 * R

If we divide both sides of this equation by .15, we get:

1000 < R

This equation is the same as:

R > 1000
If you wanted to prove this is correct, then do the following:

1000 < R
multiply both sides of the equation by -1 to get:
-1000 > -R
This is because multiplying both sides of an inequality by -1 reverses the inequality.
Add R to both sides of the equation to get:
R - 1000 > 0
Add 1000 to both sides of the equation to get:
R > 1000

Plan B total costs will be cheaper when R > 1000

Let's just see how that works.

We'll take R < 1000 and R = 1000 and R > 1000
Let's just work with a difference of $100.

When R = $900 ($100 cheaper than $1000), then:

TA = 80 + .35 * 900 = 395
TB = 230 + .20 * 900 = 410
Plan A is cheaper.

When R = $1000, then:

TA = 80 + .35 * 1000 = 430
TB = 230 + .20 * 1000 = 430
Plan A and B are the same.

When R = $1100 ($100 more expensive than $1000), then:
TA = 80 + .35 * 1100 = 465
TB = 230 + .20 * 1100 = 450
Plan B is cheaper.

The higher R becomes, the cheaper plan B looks in relation to plan A.

Let R = $2000

TA = 80 + .35 * 2000 = 780
TB = 230 + .20 * 2000 = 630

That cheaper cost for the remainder starts making plan B total cost cheaper than plan A total cost right after the cost of the remainder is equal to 1000.

The higher the cost of the remainder, the cheaper plan B looks in comparison to plan A.

When R = $1100, plan B cost 450 and plan A cost 465. Plan B was 96.77% of the cost of plan A.

When R = $2000, plan B cost 630 and plan A cost 780. Plan B was 80.76% of the cost of plan A.

Your answer is that Sam would save with plan B if he had more than $430 in bills.

In order for Sam to have $430 in bills, the remainder had to be $1000.

If the bills are going to be higher than $430, then the remainder has to be higher than $1000 because that's the variable part of the bills. The fixed part always stays the same.

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