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Machine P can produce x widgets in 10 hours, Machine Q can produce x

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Machine P can produce x widgets in 10 hours, Machine Q can produce x  [#permalink]

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New post 21 Feb 2018, 01:06
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Machine P can produce x widgets in 10 hours, Machine Q can produce x widgets in 6 hours, and Machine R can produce 2x widgets in 15 hours. If the three machines work together but independently, without interruption, how much time, expressed in hours, will be needed for them to produce 5x widgets?

A. 7 2/3
B. 8
C. 10 2/3
D. 12 1/2
E. 23 1/2

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Re: Machine P can produce x widgets in 10 hours, Machine Q can produce x  [#permalink]

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New post 21 Feb 2018, 03:37
Bunuel wrote:
Machine P can produce x widgets in 10 hours, Machine Q can produce x widgets in 6 hours, and Machine R can produce 2x widgets in 15 hours. If the three machines work together but independently, without interruption, how much time, expressed in hours, will be needed for them to produce 5x widgets?

A. 7 2/3
B. 8
C. 10 2/3
D. 12 1/2
E. 23 1/2



x in 10 hours i.e in 1 hour x/10
x in 6 hours i.e in 1 hour x/6
x in 7.5 hours i.e in 1 hour x/7.5

widgets produced by machines combined in one hour = x/10 + x/6 + x/7.5 = 6+10+8/60 x = 24/60x = 2/5x

In one hour 2/5 x of widgets done

so in 5 hours 2x of widgets done

for 5x widgets 5x5/2 = 12 1/2

(D) imo
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Machine P can produce x widgets in 10 hours, Machine Q can produce x  [#permalink]

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New post 21 Feb 2018, 12:03
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Bunuel wrote:
Machine P can produce x widgets in 10 hours, Machine Q can produce x widgets in 6 hours, and Machine R can produce 2x widgets in 15 hours. If the three machines work together but independently, without interruption, how much time, expressed in hours, will be needed for them to produce 5x widgets?

A. 7 2/3
B. 8
C. 10 2/3
D. 12 1/2
E. 23 1/2

Find combined rate (add rates), then divide work by rate to get time.
Rate of P: \(\frac{x}{10}\)
Rate of Q: \(\frac{x}{6}\)
Rate of R: \(\frac{2x}{15}\)

All units are in hours. No need to write "units/hour." Time will be in hours.

Combined rate
= \(\frac{x}{10}+\frac{x}{6}+\frac{2x}{15}\)

Use LCM = 30
\(\frac{x}{10}+\frac{x}{6}+\frac{2x}{15}\)

\((\frac{3x}{30}+\frac{5x}{30}+\frac{4x}{30})=\frac{12x}{30}=\frac{6x}{15}=\)
Combined rate

Time needed to produce 5x widgets?

\(W= 5x\), \(R*T = W\), and \(T= \frac{W}{R}\)

\(T =\frac{5x}{\frac{6x}{15}}=(5x*\frac{15}{6x})=\frac{75}{6}=\frac{25}{2}=12\frac{1}{2}\)
hrs

Answer D
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Machine P can produce x widgets in 10 hours, Machine Q can produce x  [#permalink]

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New post 21 Feb 2018, 12:14
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Bunuel wrote:
Machine P can produce x widgets in 10 hours, Machine Q can produce x widgets in 6 hours, and Machine R can produce 2x widgets in 15 hours. If the three machines work together but independently, without interruption, how much time, expressed in hours, will be needed for them to produce 5x widgets?

A. 7 2/3
B. 8
C. 10 2/3
D. 12 1/2
E. 23 1/2


If machine P can produce x widgets in 10 hours, Machine Q can produce x widgets in 6 hours, and
Machine R can produce 2x widgets in 15 hours, machine R can produce x widgets in \(\frac{15}{2} = 7.5\) hours.

Let's assume the total units of work(to produce x widgets) to be 30 units.
Now, machine P does the 3 units/hr, machine Q does 5 units/hr and machine R does 4 units/hr

Since we need to produce 5x widgets, the total units of work must be 150 units.
Together, they would do \(3+5+4 = 12\) units of work in an hour

Therefore, the three machines would take \(\frac{150}{12} = 12\frac{1}{2}\) hours to produce 5x widgets(Option D)
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Machine P can produce x widgets in 10 hours, Machine Q can produce x  [#permalink]

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New post 22 Feb 2018, 02:26
Bunuel wrote:
Machine P can produce x widgets in 10 hours, Machine Q can produce x widgets in 6 hours, and Machine R can produce 2x widgets in 15 hours. If the three machines work together but independently, without interruption, how much time, expressed in hours, will be needed for them to produce 5x widgets?

A. 7 2/3
B. 8
C. 10 2/3
D. 12 1/2
E. 23 1/2


Time of machine P to produce = 10 hrs

Time of machine Q to produce = 6 hrs

Time of machine R to produce = \(\frac{15}{2}\) hrs...........Note: You will find later this easier than putting it 7.5 hrs

You can apply times fro above directly in this equation:

Total combined time = \(\frac{P*Q*R}{(P*Q)+(P*R)+(R*Q)}\)

It may look intimidating but you will get directly combines time for x = \(\frac{5}{2}\)

Time for 5x = \(\frac{5*5}{2}\) = 12.5

Answer: D
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Re: Machine P can produce x widgets in 10 hours, Machine Q can produce x  [#permalink]

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New post 26 Feb 2018, 10:44
Bunuel wrote:
Machine P can produce x widgets in 10 hours, Machine Q can produce x widgets in 6 hours, and Machine R can produce 2x widgets in 15 hours. If the three machines work together but independently, without interruption, how much time, expressed in hours, will be needed for them to produce 5x widgets?

A. 7 2/3
B. 8
C. 10 2/3
D. 12 1/2
E. 23 1/2


The combined rate of P, Q and R is:

x/10 + x/6 + 2x/15 = 3x/30 + 5x/30 + 4x/30 = 12x/30 = 2x/5.

We know that rate x time = total work. If we let t = the number of hours to produce 5x widgets, we can create the following equation;

(2x/5)(t) = 5x

2xt = 25x

2t = 25

t = 12.5

Answer: D
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Re: Machine P can produce x widgets in 10 hours, Machine Q can produce x   [#permalink] 26 Feb 2018, 10:44
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