Question

Multi-Step Problem
A chauffeur can be paid in two ways.
Plan A : $500.00 plus $9.00 per hour
Plan B : $14.00 per hour
Suppose that the job takes n hours. For what values of n is Plan A better for the chauffeur than Plan B

Answer 1

Hi! Any idea where you want to start?

Answer 2

I need a starting equation for this problem, But I don't know how to set it up first.

Answer 3

Okay! I'll help you speak math!

We'll say \(n\) is the number of hours a job can take.

Then the cost of plan A, which I'll name \(C_A\) for cost of plan A, is
\(C_A=$500+\dfrac{$9\ n}{[hr]}\quad\leftarrow\)$500 plus $9/hr times the number of hours the job takes.

Do you want to take a shot at describing plan B?

Answer 4

I'll try to ^^|| sorry if math isn't my best subject

Answer 5

It's not mine either, but I got through it a while ago and have practiced it as I've gotten into more math courses! I disliked math so strongly in high school. Now I see its use! I had a few good teachers to help me like it, but some other teachers ruined it for me.

Take your time. And ask any questions you want. I think it's great that you're trying.

Answer 6

It would probably be like this? Plan B , would start out as B for the result of the plan
B = $14/{hr}

Answer 7

Well, the chauffeur is driving around and getting paid $14 each hour. So, if that chauffeur drove for, say, 5 hours, how much money would it cost?

Answer 8

so that would mean he is getting $70 for 5 hours right?

Answer 9

Yeah! How did you get that?

Answer 10

I multiplied 5 hours and the amount of money that he received and it equaled up to 70 dollars in the end

Answer 11

Right! So, if the chauffeur drives \(n\) hours, the total price for plan B will be $14/hr times \(n\), which looks like \(\dfrac{$14\ n}{[hr]}\). We'll call that \(C_B\), the cost of plan B. Does that sound good to you?

Answer 12

yea

Answer 13

Alright! So now the problem wants to know when plan A is better than plan B.

So, that would be when the cost of plan A is less than the cost of plan B. (It's better to pay less, right?)

Have you worked with inequalities before? \(\le\ \ge\ <\ >\) stuff?

Answer 14

yea, but I'm kind of getting the hang of those

Answer 15

That's good! :) Then we can use them.

Like I said, we want to talk about when the cost of plan A is less than the cost of plan B. With my variables, that is \(C_A<C_B\). Is that okay so far?

Answer 16

yea

Answer 17

Okay then. Now we look at what \(C_A\) and \(C_B\) really are.

So
\($500+\dfrac{$9\ n}{[hr]}<\dfrac{$14\ n}{[hr]}\)

Now we can solve to see for what number of hours \((n)\) the cost of plan A is really less than plan B, which the inequality says for us.

So, we should start solving for \(n\).

Answer 18

I would subtract \(\dfrac{$9\ n}{[hr]}\) from both sides to start. Sound good?

Answer 19

yes

Answer 20

\($500<\dfrac{$14\ n}{[hr]}-\dfrac{$9\ n}{[hr]}=\dfrac{$5\ n}{[hr]}\)

So

\($500<\dfrac{$5\ n}{[hr]}\)

Any questions? Double check me to make sure I didn't make a mistake!

Answer 21

no, it's good

Answer 22

Okay, thanks.

Now, \($500<\dfrac{$5\ n}{[hr]}\). To get \(n\) alone, what do you think we should do to both sides next?

Answer 23

multiply the hr to both sides?

Answer 24

Okay!
\($500\ [hr]<$5\ n\)
And now what?

Answer 25

divide 5 to both sides

Answer 26

Good plan!
\(\dfrac{$500\ [hr]}{$5}=100\ [hr]<n\)

The dollar units cancelled, and \(500\div 5=100\), so that's what I got.

So, if the cost of plan A is less than plan B, then \(n\gt100\ [\text{hours}]\).

Answer 27

So your done! Any questions?

Answer 28

nope, thanks for the help though ^^

Answer 29

You're very welcome! You did good to get through it, too :) Take care!

Answer 30

you too :3

Answer 31

Thanks! :)

Answer 32

your welcome