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Running at their respective constant rates, machine X takes

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Running at their respective constant rates, machine X takes

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Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines 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 is E
Experts, may you help me here? I am confused. What formulas should I set here?

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Roland2rule wrote:
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines 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 is E
Experts, may you help me here? I am confused. What formulas should I set here?
Work = Rate * Time

Machine X:
w = Rx * T
Rx = w/T

Machine Y:
w = Ry * (T-2)
Ry = w/(T-2)

Situation from question:

5w/4 = (Rx + Ry)*3
5w/4 = [w/T + w/(T-2)]*3 --> we can divide through by W
5/4 = [1/T + 1/(T-2)]*3
5/12 = 1/T + 1/(T-2)
5(T)(T-2) = 12(T-2) + 12(T)
5T^2 - 10T = 24T - 24
5T^2 - 34T + 24 = 0
(T-6)(5T-4) = 0

T must be > 2 so the only solution is T = 6

Machine X can produce w widgets in T so it must be able to produce 2w widgets in 2T; given T= 6 then the answer is 2*6 = 12

Answer. E) 12

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Roland2rule wrote:
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines 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
Another approach is to assign a nice value to the job (w)

Let's say that w = 12.

GIVEN: Running at their respective constant rates, machine X takes 2 days longer to produce 12 widgets than machine Y
Let t = time for machine Y to produce 12 widgets
So, t+2 = time for machine X to produce 12 widgets

RATE = output/time

So, machine X's RATE = 12 widgets/(t + 2 days) = 12/(t+2) widgets per day
And machine Y's RATE = 12 widgets/(t days) = 12/t widgets per day

The two machines together produce 5w/4 widgets in 3 days
In other words, The two machines together produce 5(12)/4 widgets in 3 days
Or the two machines together produce 15 widgets in 3 days
This means the COMBINED RATE = 5 widgets per day

So, we can write: 12/(t+2) + 12/t = 5
Multiply both sides by (t+2)(t) to get: 12t + 12t + 24 = 5(t+2)(t)
Simplify: 24t + 24 = 5t² + 10t
Rearrange: 5t² - 14t - 24 = 0
Factor to get: (5t + 6)(t - 4) = 0
So, EITHER t = -6/5 OR t = 4
Since the time cannot be negative, it must be the case that t = 4

If t = 4, then it takes Machine Y 4 days to produce 12 widgets
And it takes Machine X 6 days to produce 12 widgets

How many days would it take machine X alone to produce 2w widgets?
In other words, how many days would it take machine X alone to produce 24 widgets? (since w = 12)

If it takes Machine X 6 days to produce 12 widgets, then it will take Machine X 12 days to produce 24 widgets

Answer: E

Cheers,
Brent

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Quote:
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.

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Roland2rule wrote:
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines 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
We can let x = the number of days it takes Machine X to produce w widgets and thus x - 2 = the number of days it takes Machine Y to produce w widgets. Furthermore, the rate of Machine X is w/x and the rate of Machine Y is w/(x - 2). We are given that they can produce 5w/4 widgets in 3 days. Thus, we have:

3(w/x) + 3[w/(x - 2)] = 5w/4

Dividing both sides by w, we have:

3/x + 3/(x - 2) = 5/4

Multiplying both sides by 4x(x - 2), we have:

12(x - 2) + 12x = 5x(x - 2)

12(x - 2) + 12x = 5x(x - 2)

12x - 24 + 12x = 5x^2 - 10x

5x^2 - 34x + 24 = 0

(5x - 4)(x - 6) = 0

x = 4/5 or x = 6

However, x can’t be 4/5; if it were, y would be negative. Thus, x must be 6. Since it takes Machine X 6 days to produce w widgets, it will take 12 days to produce 2w widgets.

Answer: E

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