Advanced Rates Problem: QR #173 PS

[tonebeeze]

I usually do relatively well with rate problems, but this problem tripped me up. I would appreciate help mapping this problem out. Thanks!

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 5/4w 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

[anshumishra]

I usually do relatively well with rate problems, but this problem tripped me up. I would appreciate help mapping this problem out. Thanks!

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 5/4w 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

Total work = 2w
Lets say , y takes "t" time
y -> w work -> t time -> rate of y = w/t
x -> w work -> t+2 time -> rate of x = w/(t+2)

So,
w/(t+2) + w/t = 5w/(4*3) = 5w/12
=> (wt+wt+2w)/[t*(t+2)] = 5w/12
=> 2w(t+1)/[t*(t+2)] = 5w/12
=> 5t^2 + 10t -24t -24 = 0
=> 5t^2 - 14t - 24 = 0
=> t = 14+-√676/(10) = (14+26)/10 (discard -ve value) = 4

So, rate of x = w/6
So, time taken to complete 2w work = 2w/(w/6) = 12

E

[ankur.agrawal]

Is dere any other method?

Can we do it by picking answer options. ?

[AIM GMAT]

Thanks anshumishra for a wonderful explanation of the problem .

[GMATGuruNY]

I usually do relatively well with rate problems, but this problem tripped me up. I would appreciate help mapping this problem out. Thanks!

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 5/4w 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

An efficient way to solve this problem is to plug in a value for w and then plug in the answer choices.

Let w=12.
In 3 days, X and Y need to produce 5/4*w = 5/4*12 = 15 widgets.
The answer choices represent the time for X to produce 2w=24 widgets.

Answer choice C: 8 days for X to produce 24 widgets
Thus, X produces 12 widgets in 4 days.
Since X takes 2 days longer, Y produces 12 widgets in 4-2=2 days.
Rate for X = w/t = 12/4 = 3/day.
Rate for Y = w/t = 12/2 = 6/day.
Combined rate for X+Y = 3+6 = 9/day .
Work completed in 3 days = r*t = 9*3 = 27 widgets.
Incorrect. We need much less work to get done, so the time for X must be much longer.

Answer choice E: 12 days for X to produce 24 widgets
Thus, X produces 12 widgets in 6 days.
Since X takes 2 days longer, Y produces 12 widgets in 6-2=4 days.
Rate for X = w/t = 12/6 = 2/day.
Rate for Y = w/t = 12/4 = 3/day.
Combined rate for X+Y = 2+3 = 5/day.
Work completed in 3 days = r*t = 5*3 = 15 widgets. Success!

The correct answer is E.

[ankur.agrawal]

I usually do relatively well with rate problems, but this problem tripped me up. I would appreciate help mapping this problem out. Thanks!

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 5/4w 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

An efficient way to solve this problem is to plug in a value for w and then plug in the answer choices.

Let w=12.
In 3 days, X and Y need to produce 5/4*w = 5/4*12 = 15 widgets.
The answer choices represent the time for X to produce 2w=24 widgets.

Answer choice C: 8 days for X to produce 24 widgets
Thus, X produces 12 widgets in 4 days.
Since X takes 2 days longer, Y produces 12 widgets in 4-2=2 days.
Rate for X = w/t = 12/4 = 3/day.
Rate for Y = w/t = 12/2 = 6/day.
Combined rate for X+Y = 3+6 = 9/day .
Work completed in 3 days = r*t = 9*3 = 27 widgets.
Incorrect. We need much less work to get done, so the time for X must be much longer.

Answer choice E: 12 days for X to produce 24 widgets
Thus, X produces 12 widgets in 6 days.
Since X takes 2 days longer, Y produces 12 widgets in 6-2=4 days.
Rate for X = w/t = 12/6 = 2/day.
Rate for Y = w/t = 12/4 = 3/day.
Combined rate for X+Y = 2+3 = 5/day.
Work completed in 3 days = r*t = 5*3 = 15 widgets. Success!

The correct answer is E.

" We need much less work to get done, so the time for X must be much longer."

How do we know whether to move up or down from C OPTION. Pls explain.

[GMATGuruNY]

I usually do relatively well with rate problems, but this problem tripped me up. I would appreciate help mapping this problem out. Thanks!

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 5/4w 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

An efficient way to solve this problem is to plug in a value for w and then plug in the answer choices.

Let w=12.
In 3 days, X and Y need to produce 5/4*w = 5/4*12 = 15 widgets.
The answer choices represent the time for X to produce 2w=24 widgets.

Answer choice C: 8 days for X to produce 24 widgets
Thus, X produces 12 widgets in 4 days.
Since X takes 2 days longer, Y produces 12 widgets in 4-2=2 days.
Rate for X = w/t = 12/4 = 3/day.
Rate for Y = w/t = 12/2 = 6/day.
Combined rate for X+Y = 3+6 = 9/day .
Work completed in 3 days = r*t = 9*3 = 27 widgets.
Incorrect. We need much less work to get done, so the time for X must be much longer.

Answer choice E: 12 days for X to produce 24 widgets
Thus, X produces 12 widgets in 6 days.
Since X takes 2 days longer, Y produces 12 widgets in 6-2=4 days.
Rate for X = w/t = 12/6 = 2/day.
Rate for Y = w/t = 12/4 = 3/day.
Combined rate for X+Y = 2+3 = 5/day.
Work completed in 3 days = r*t = 5*3 = 15 widgets. Success!

The correct answer is E.

" We need much less work to get done, so the time for X must be much longer."

How do we know whether to move up or down from C OPTION. Pls explain.

When w=12, X and Y must produce in 3 days 5/4*w = 5/4*12 = 15 widgets.
Answer choice C resulted in the production of 27 widgets, a much higher number.
To reduce the number of widgets produced, we need X to work more slowly.
To work more slowly means to take more time, so we need a bigger answer choice.

[anshumishra]

Thanks anshumishra for a wonderful explanation of the problem .

Thanks AIM GMAT!

[ankur.agrawal]

I usually do relatively well with rate problems, but this problem tripped me up. I would appreciate help mapping this problem out. Thanks!

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 5/4w 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

An efficient way to solve this problem is to plug in a value for w and then plug in the answer choices.

Let w=12.
In 3 days, X and Y need to produce 5/4*w = 5/4*12 = 15 widgets.
The answer choices represent the time for X to produce 2w=24 widgets.

Answer choice C: 8 days for X to produce 24 widgets
Thus, X produces 12 widgets in 4 days.
Since X takes 2 days longer, Y produces 12 widgets in 4-2=2 days.
Rate for X = w/t = 12/4 = 3/day.
Rate for Y = w/t = 12/2 = 6/day.
Combined rate for X+Y = 3+6 = 9/day .
Work completed in 3 days = r*t = 9*3 = 27 widgets.
Incorrect. We need much less work to get done, so the time for X must be much longer.

Answer choice E: 12 days for X to produce 24 widgets
Thus, X produces 12 widgets in 6 days.
Since X takes 2 days longer, Y produces 12 widgets in 6-2=4 days.
Rate for X = w/t = 12/6 = 2/day.
Rate for Y = w/t = 12/4 = 3/day.
Combined rate for X+Y = 2+3 = 5/day.
Work completed in 3 days = r*t = 5*3 = 15 widgets. Success!

The correct answer is E.

" We need much less work to get done, so the time for X must be much longer."

How do we know whether to move up or down from C OPTION. Pls explain.

When w=12, X and Y must produce in 3 days 5/4*w = 5/4*12 = 15 widgets.
Answer choice resulted in the production of 27 widgets, a much higher number.
To reduce the number of widgets produced, we need X to work more slowly.
To work more slowly means to take more time, so we need a bigger answer choice.[/quote

Bingo! thax man.