Problem Solving

I'm stumped on this one and tried to figure out the explanation in the OA, but need clarification.

#59) Each week, Harry is paid x dollars per hour for the first 30 hours and 1.5x dollars for each additional hour worked that week. Each week, James is paid x dollars per hour for the first 40 hours and 2x dollars for each additional hour worked that week. Last week James worked a total of 41 hours. If Harry and James were paid the same amount last week, how many hours did Harry work last week?

(a) 35
(b) 36
(c) 37
(d) 38
(e) 39

Any thoughts on the quickest way to solve? I tried mapping it out algebraically but took too long to get the answer.

OA - d

Thanks!

James worked a total of 41 hours
So James is paid 40*x +1*2x dollars= 42x dollars

Harry and James were paid the same amount last week.
So harry is paid 42x dollars last week

in first 30 hr Harry is paid 30*x=30x dollars

and for the rest y hours he is paid 1.5xy dollars

so 30x+1.5xy=42x
or 1.5xy=12x
or y=8

so harry has worked 30+8=38 hr

Ans option D

liferocks wrote:James worked a total of 41 hours
So James is paid 40*x +1*2x dollars= 42x dollars

Harry and James were paid the same amount last week.
So harry is paid 42x dollars last week

in first 30 hr Harry is paid 30*x=30x dollars

and for the rest y hours he is paid 1.5xy dollars

so 30x+1.5xy=42x
or 1.5xy=12x
or y=8

so harry has worked 30+8=38 hr

Ans option D

Thanks for the quick reply! very concise and it's definitely much better than the OG explanation.

liferocks wrote:James worked a total of 41 hours
So James is paid 40*x +1*2x dollars= 42x dollars

Harry and James were paid the same amount last week.
So harry is paid 42x dollars last week

in first 30 hr Harry is paid 30*x=30x dollars

and for the rest y hours he is paid 1.5xy dollars

so 30x+1.5xy=42x
or 1.5xy=12x
or y=8

so harry has worked 30+8=38 hr

Ans option D

My initial equation for Harry was: 30x+1.5x(y-30)=42x

I thought that you had to subtract the initial 30 hours from the total when you multiple the additional amount of 1.5x. Any thoughts on why I shouldn't have done so and why 1.5x(y) is correct?

Thanks again!

jcbruin wrote:

liferocks wrote:James worked a total of 41 hours
So James is paid 40*x +1*2x dollars= 42x dollars

Harry and James were paid the same amount last week.
So harry is paid 42x dollars last week

in first 30 hr Harry is paid 30*x=30x dollars

and for the rest y hours he is paid 1.5xy dollars

so 30x+1.5xy=42x
or 1.5xy=12x
or y=8

so harry has worked 30+8=38 hr

Ans option D

My initial equation for Harry was: 30x+1.5x(y-30)=42x

I thought that you had to subtract the initial 30 hours from the total when you multiple the additional amount of 1.5x. Any thoughts on why I shouldn't have done so and why 1.5x(y) is correct?

Thanks again!

when we have a equation 30x+1.5x(y-30)=42x we are taking y as total hours worked by Harry.
In my case I took y as the excess to 30 hr worked by harry.That is why equation became
30x+1.5xy=42x

and in the end I have to add the value of y to 30.

if we go with 30x+1.5x(y-30)=42x
solving the equation will give us the result directly. You can use any process as per your convenience.

jcbruin wrote: My initial equation for Harry was: 30x+1.5x(y-30)=42x

I thought that you had to subtract the initial 30 hours from the total when you multiple the additional amount of 1.5x. Any thoughts on why I shouldn't have done so and why 1.5x(y) is correct?

Thanks again!

Jcbruin, both setups are correct; the discrepancy is a result of y representing different things in the two solutions. In liferocks' solution, y represents the number of overtime hours. Each of those hours earns 1.5x, so the overtime pay is 1.5xy. In contrast in your equation, y represents the total number of hours worked (regular + overtime). Thus in your setup the overtime hours only are y-30. Each of those hours earns 1.5x, so in your setup, the overtime pay is 1.5x(y-30)

This is question #59 in the OG12. Two detailed solutions (from the OG Companion) as well as a take-away lesson are attached below. People who cannot see attachments, click here.

-Patrick

jcbruin wrote:
Any thoughts on the quickest way to solve? I tried mapping it out algebraically but took too long to get the answer.

If the answer choices are numbers, and the algebra doesn't jump off the screen, you can also backsolve.

We know Harry gets paid x dollars for each of the first 30 hours worked and 1.5x for each hour thereafter.

We know James gets paid x dollars for each of the first 40 hours worked and 2x for each hour thereafter.

We also know that James worked 41 hours, and that Harry made the same amount of money as James.

To keep things simple, let x = 1. Then James made 40 + 2 = 42 dollars.

When backsolving, we should normally start with choice B or D.

Let's look at choice B. If Harry worked 36 hours, he would make 30 + 6*(1.5) = 30 + 9 = 39 dollars. Because James made 42, we are looking for a larger number, and we can eliminate choices A and B.

Now, we should check choice D--if D is too small, we know the answer is E, and if D is too large, we know the answer is C.

Looking at D: If Harry worked 38 hours, he would make 30 + 8*(1.5) = 30 + 12 = 42 dollars. Thus, the initial conditions are all satisfied, and the answer must be choice D.