Problem Solving

Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?

A) 4
B) 6
C) 8
D) 10
E) 12

OA: E

fambrini wrote:Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?

A) 4
B) 6
C) 8
D) 10
E) 12

Let w = 60 widgets, implying that (5/4)w = (5/4)(60) = 75 widgets.
Since X and Y together can produce (5/4)w widgets in 3 days, the combined rate for X and Y = w/t = 75/3 = 25 widgets per day.

We can PLUG IN THE ANSWERS, which represent X's time to produce 2w widgets.
When the correct answer choice is plugged in, the combined rate for X and Y will be 25 widgets per day.

D: 10
Here, X can produce 2w widgets in 10 days.
Thus, the time for X to produce w widgets = 5 days.

Since X takes 5 days to produce w=60 widgets, X's rate = w/t = 60/5 = 12 widgets per day.
Since X takes 2 days longer than Y to produce w widgets, Y's time to produce w widgets = 3 days.
Since Y takes 3 days to produce w=60 widgets, Y's rate = w/t = 60/3 = 20 widgets per day.
Combined rate for X and Y = 12+20 = 32 widgets per day.

Here, X and Y are working TOO FAST.
Implication:
X must take LONGER to produce 2w widgets, with the result that X and Y will work more SLOWLY.

The correct answer is E.

fambrini wrote:Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y. At these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets?

A) 4
B) 6
C) 8
D) 10
E) 12

OA: E

Another alternative solution:

Let machine Y takes z days to complete W widgets.

Then machine X takes z+2 days to complete W widgets.

Combined work for both produce 5w/4 in 3 days

Translating the above into rate-time-work equation

[w/z + w/(z+2)]* 3 = 5w/4

1/z + 1/(z+2) = 5/12

We need to find z that create 12 in denominator

z = 4

If it takes 6 days to produce w, then it will take 12 days to produce 2w.

Answer: E

Good Evening-

I am looking for expert help as I am not sure what I am doing wrong. I tried two ways, Algebra & Smart #, and still can't solve the problem. The main difference I see between the setup between the answers in the forum and mine is that I labeled the T (see chart) as D+2 for X and D for Y VS D for X an D-2 for Y. I realize that the second would have been wiser as the answer is asking for X, but i believe both would be valid. Can someone help me understand my error? Thanks much!

schelljo wrote:Good Evening-

I am looking for expert help as I am not sure what I am doing wrong. I tried two ways, Algebra & Smart #, and still can't solve the problem. The main difference I see between the setup between the answers in the forum and mine is that I labeled the T (see chart) as D+2 for X and D for Y VS D for X an D-2 for Y. I realize that the second would have been wiser as the answer is asking for X, but i believe both would be valid. Can someone help me understand my error? Thanks much!

You did nothing wrong! Here's where you abandoned ship: 5d^2 -14d - 24 = 0.

All that's left to do is factor: (5d + 6)(d - 4) = 0; d = -6/5 or d = 4. (Obviously d can't be negative, so d = 4.)

If d = 4, d + 2 = 6. If Machine X can do w widgets in 6 days, it can do 2w widgets in 12 days. You were two steps away.

Takeaway: trust yourself!

Hi Experts,

Can you please explain by 1/k approach?

Thanks,

Rashi

rsarashi wrote:Hi Experts,

Can you please explain by 1/k approach?

Thanks,

Rashi

I'm not quite sure what you're asking here, Rashi. Do you mean you'd like to see the setup in the event that k represents that number of days it takes X to complete 1 job, and so X's rate would be 1 job/ k days, or 1/k ?

I'm not quite sure what you're asking here, Rashi. Do you mean you'd like to see the setup in the event that k represents that number of days it takes X to complete 1 job, and so X's rate would be 1 job/ k days, or 1/k ?

Hi David,

Thanks for your reply.

Sorry for the confusion. I mean if A did a job in x days, then in A day it will be 1/x days. This approach i was talking about.

Please explain me in this approach.

Thanks,

Rashi

We can use the "add up the rates to get the combined rate" method to build an equation. (rsarashi, I think this is what you are referring to -- in your example, 1/k is a rate; something can do 1 job in k hours, or 1/k jobs per hour)

X's Rate + Y's rate = Combined rate

Since Rate = (Work) / (Time), we get


X's Rate + Y's rate = Combined rate --->
w/(x) + w/(y) = (5w/4)/3 [x represents X's time to make w widgets, y represents Y's]
w/(x) + w/(x-2) = (5w/4)/3 [If X takes two more days, then Y takes two fewer]
1/(x) + 1/(x-2) = 5/12

Algebra looks unpleasant (though it would certainly be possible), so let's switch to working with the answer choices. Somehow we want a 12 in the denominator, so I'll start with an answer choice that seems likely to lead to that.

Let's pretend B is the answer. The question is asking how long it would take x to make 2w widgets. If X takes 6 days to make 2w, it would take 3 days to make w (that's what our x represents in our equation above, so lets plug in:

1/(3) + 1/(3-2) = 5/12 --> No good. the left is much bigger than the right. To make the left smaller, let's pick a number that will lead to bigger denominators on the left.

Let's pretend E is the answer (it will lead to smaller values on the left and seems likely to lead to a 12 in the denominator). The question is asking how long it would take X to make 2w widgets. If X takes 12 days to make 2w, it would take 6 days to make w (that's what our x represents in our equation above, so let's plug in:

1/(6) + 1/(6-2) = 5/12

1/6 + 1/4 = 5/12

2/12 + 3/12 = 5/12 --> Good. E is our answer.


More on the method used above here: https://www.youtube.com/watch?v=GAOel-n ... TbTHvt40S0