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# Three machines X, Y, and Z operate independently of one another.

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Three machines X, Y, and Z operate independently of one another. [#permalink]
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OCDianaOC wrote:
Three machines X, Y, and Z operate independently of one another. Machine X working alone can complete a job in 12 hours, machine Y working alone can complete the job in y hours, machine Z working alone can complete the job in 4 hours, and all three machines working together and independently can complete the job in 1 3/5 hours. What is the value of y?

A) 1 3/5
B) 3
C) 3 3/7
D) 5 1/4
E) 17 3/5

One approach, $$R*T = W$$: find combined rate; add rates to solve for Y's rate; and use Y's rate to find Y's time

Given
Rates are in $$\frac{Jobs}{hour}$$
X's rate = $$\frac{1}{12}$$
Y's rate = $$\frac{1}{Y}$$
Z's rate = $$\frac{1}{4}$$
TIME all three machines take, working together, to finish the job = $$1\frac{3}{5}hrs=\frac{8}{5}$$ hours

Combined RATE?
$$R*T = W$$, so $$R=\frac{W}{T}$$
All three, RATE = $$\frac{1}{(\frac{8}{5})}=(1*\frac{5}{8})=\frac{5}{8}$$

Add rates, solve for Y's rate:
($$\frac{1}{X}+\frac{1}{Y}+\frac{1}{Z})=(\frac{1}{12}+\frac{1}{Y}+\frac{1}{4})=\frac{5}{8}$$

$$(\frac{5}{8}-\frac{1}{4}-\frac{1}{12})=\frac{1}{Y}=(\frac{15}{24}-\frac{6}{24}-\frac{2}{24})=\frac{7}{24}$$

Y's TIME?
$$\frac{7}{24}$$ is Y's rate.
The job is 1
Rate and time are inversely proportional. Flip Y's rate fraction to get Y's time*:
$$\frac{24}{7}hrs=3\frac{3}{7}hrs$$

*$$R*T=W$$, so $$T=\frac{W}{R}$$
$$W = 1$$
$$T =\frac{1}{(\frac{7}{24})}=(1 * \frac{24}{7})=(\frac{24}{7})=3\frac{3}{7}$$hrs
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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
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The Total work can be assumed to be 120 (LCM of (time taken to complete the job) 12,4,8/5 (ignore the denominator )) =120 units
This will make the calculation quite easier.
Now No of units of works done by X per hour = $$120/12 =10$$
No of units of works done by Z per hour = $$120/4 =30$$
No of units of works done per hour by all three of them = $$120/(8/5) =75$$
Thus work done by Y per hour = work done by all three - (work done by X +work done by Z) =75-(10+30) =35
Hence the time taken by Y to complete the work = 120/35 = 24/7 = 3 3/7 = Answer = C

OCDianaOC wrote:
Three machines X, Y, and Z operate independently of one another. Machine X working alone can complete a job in 12 hours, machine Y working alone can complete the job in y hours, machine Z working alone can complete the job in 4 hours, and all three machines working together and independently can complete the job in 1 3/5 hours. What is the value of y?

A) 1 3/5
B) 3
C) 3 3/7
D) 5 1/4
E) 17 3/5
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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
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OCDianaOC wrote:
Three machines X, Y, and Z operate independently of one another. Machine X working alone can complete a job in 12 hours, machine Y working alone can complete the job in y hours, machine Z working alone can complete the job in 4 hours, and all three machines working together and independently can complete the job in 1 3/5 hours. What is the value of y?

A) 1 3/5
B) 3
C) 3 3/7
D) 5 1/4
E) 17 3/5

1/y=[1/(8/5)-(1/4+1/12)]
C
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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
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Bunuel wrote:
OCDianaOC wrote:
Three machines X, Y, and Z operate independently of one another. Machine X working alone can complete a job in 12 hours, machine Y working alone can complete the job in y hours, machine Z working alone can complete the job in 4 hours, and all three machines working together and independently can complete the job in hours. What is the value of y?

A) 1 3/5
B) 3
C) 3 3/7
D) 5 1/4
E) 17 3/5

Please edit the highlighted part: 8/5 hours?

Thank you.

X can complete the work in 1/12 hours
Y can complete the work in 1/ x hours
Z can complete the work in 1/4 hours
Together they can complete the work in 5/8 hours

So 1/12+1/x+1/4=5/8
1/x=5/8-1/3
1/x=7/24

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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
OCDianaOC wrote:
Three machines X, Y, and Z operate independently of one another. Machine X working alone can complete a job in 12 hours, machine Y working alone can complete the job in y hours, machine Z working alone can complete the job in 4 hours, and all three machines working together and independently can complete the job in 1 3/5 hours. What is the value of y?

A) 1 3/5
B) 3
C) 3 3/7
D) 5 1/4
E) 17 3/5

The rate of machine X is 1/12, the rate of machine Y is 1/y, and the rate of machine Z is 1/4.

Since the 3 machines can complete the job in 1 3/5 = 8/5 hours, we can create the following equation for the combined rate of the 3 machines:

1/12 + 1/y + 1/4 = 1/(8/5)

1/12 + 1/y + 1/4 = 5/8

Multiplying by 24y, we have:

2y + 24 + 6y = 15y

24 = 7y

24/7 = 3 3/7 = y

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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
For me the easiest way to solve it would be:

X does 1/12 of the work in 1 hour
Y does 1/y of the work in 1 hour
Z does 1/4 of the work in 1 hour

When they work together they can do the work in 1 3/5 hours -> 8/5 hours

Hence in 1 hour only 1/(8/5) of the total work is done, what means that (1/12+1/y+1/4)=5/8

it follows that 4/12+1/y=5/8
so, (5/8)-(4/12)=1/y
finding the common denominator -> (15/24)-(8/24)=1/y
7/24=1/y
y= 3 and 3/7
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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
1/Y= (1/X + 1/Y + 1/Z) - (1/X + 1/Z) = 5/8 - (1/12 + 1/4) = 5/8 - 1/3 = 7/24 Y's Rate.
Y's time = 24/7 = 3 3/7
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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
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Re: Three machines X, Y, and Z operate independently of one another. [#permalink]
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