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Math Expert V
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Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 64% (02:51) correct 36% (02:49) wrong based on 310 sessions

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Machine X can complete a job in half the time it takes Machine Y to complete the same job, and Machine Z takes 50% longer than Machine X to complete the job. If all three machines always work at their respective, constant rates, what is the ratio of the amount of time it will take Machines X and Z to complete the job to the ratio of the amount of time it will take Machines Y and Z to complete the job?

A. 5 to 1
B. 10 to 7
C. 1 to 5
D. 7 to 10
E. 9 to 10

Kudos for a correct solution.

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Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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Since rates are additive, using rates is usually easier.

"Machine X can complete a job in half the time it takes Machine Y to complete the same job,"
So machine X's rate is twice machine Y's rate. Rate of X:Rate of Y = 2:1

"Machine Z takes 50% longer than Machine X to complete the job"
Time taken by Z : Time taken by X = 3:2 so Rate of Z: Rate of X = 2:3

Rate of X : Rate of Y : Rate of Z = 6:3:4

Rate of X+Z : Rate of Y+Z = 10 : 7
Time taken by X+Z : Time taken by Y+Z = 7:10

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Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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Bunuel wrote:
Machine X can complete a job in half the time it takes Machine Y to complete the same job, and Machine Z takes 50% longer than Machine X to complete the job. If all three machines always work at their respective, constant rates, what is the ratio of the amount of time it will take Machines X and Z to complete the job to the ratio of the amount of time it will take Machines Y and Z to complete the job?

A. 5 to 1
B. 10 to 7
C. 1 to 5
D. 7 to 10
E. 9 to 10

Kudos for a correct solution.

Let time required by Y to complete the work = t
then time taken by X will be = $$\frac{t}{2}$$
and time taken by Z = $$t/2 + t/4$$ ----> $$\frac{t}{4}$$ as Z takes 50% longer then X
Now Rate for X = $$\frac{2}{t}$$
Rate for Y = $$\frac{1}{t}$$
Rate for Z = $$\frac{4}{3t}$$

amount of time it will take Machines X and Z to complete the job = $$1/[2/t + 4/3t]$$
i.e. $$\frac{3t}{10}$$
the amount of time it will take Machines Y and Z to complete the job = $$1/[1/t + 4/3t]$$
i.e. $$\frac{3t}{7}$$
Thus ratio = $$\frac{3t}{10}$$ / $$\frac{3t}{7}$$ = 7:10
[D]
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Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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Hi Bunuel,

Solutioning this exercise can be facilitated by using a R T W (rate time work) table:

We translate the exercise into the table:

R T W
X t/2 1
Y t 1
Z (t/2*3/2 =3t/4) 1

From this table we find the rates

Rx = 2/t
Ry = 1/t
Rz = 4/3t

The Q is what is the ratio of (Tx + Ty) / (Ty + Tz)

Rx + Ry = 2/t + 4/3t = 6/3t+4/3t = 10/3t
Ry+Rz = 1/t + 4/3t = 3/3t + 4/3t = 7/3t

The (10/3t)/(7/3t) = 10/7 then the work ratios is 10 to 7

Since Time Ratio is the inverse of work, the the answer is 7 to 10

Bunuel wrote:
Machine X can complete a job in half the time it takes Machine Y to complete the same job, and Machine Z takes 50% longer than Machine X to complete the job. If all three machines always work at their respective, constant rates, what is the ratio of the amount of time it will take Machines X and Z to complete the job to the ratio of the amount of time it will take Machines Y and Z to complete the job?

A. 5 to 1
B. 10 to 7
C. 1 to 5
D. 7 to 10
E. 9 to 10

Kudos for a correct solution.

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Originally posted by TudorM on 03 Feb 2015, 22:53.
Last edited by TudorM on 03 Feb 2015, 23:29, edited 2 times in total.
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Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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1
x=0,5t
y=t
z=0,75t

t=4hours so we get:
x=2
y=4
z=3

x+z=1/2+3/4=5/6 -> they need 6/5 hours
y+z=1/4+1/3=7/12 -> they need 12/7 hours

ratio is 6/5 divided by 12/7, or multiplied by 7/12 -> we get 7/10
Math Expert V
Joined: 02 Aug 2009
Posts: 7960
Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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ans D..
let time taken by X =x,
so by Y =2x,
and by z = 1.5x...
time taken by X and Z = 1/( 1/x+1/1.5x)... = 1.5 X^2/2.5X =3X^2/5X
time taken by Y and Z = 1/( 1/2x+1/1.5x)... = 3 X^2/3.5X...= 6X^2/7X
ratio = (3X^2/5X)/ (6X^2/7X)= 7/10
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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Hi All,

These types of rate questions can be solved in a few different ways. Since the prompt discusses how two machines will work together on a task, you can use the Work Formula. By TESTing VALUES, we can come up with a simple example that will answer the given question.

We're told that:
1) Machine X can do a job in HALF the time that it takes Machine Y
2) Machine Z takes 50% longer to the do the job than Machine X

Since Machine X appears in both 'facts', we want to TEST a VALUE for X that is easily doubled AND halved....

X = 2 hours to complete the job

With that value in place, we can figure out the other 2 values....
Y = 4 hours to complete the job
Z = 3 hours to complete the job

We're asked for the ratio of the time it takes (X and Z) working together vs. the time it takes (Y and Z) working together.

The Work Formula: (A)(B)/(A+B)

For Machines X and Z, we have: (2)(3)/(2+3) = 6/5 hours to complete the job
For Machines Y and Z, we have: (4)(3)/(4+3) = 12/7 hours to complete the job

The ratio 6/5 : 12/7 needs to be "clean up" a bit. Since we're dealing with fractions, we can use common denominators....

6/5 : 12/7

42/35 : 60/35

Now we can eliminate the denominators...

42 : 60

And reduce the ratio....

7 : 10

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Math Expert V
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Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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Bunuel wrote:
Machine X can complete a job in half the time it takes Machine Y to complete the same job, and Machine Z takes 50% longer than Machine X to complete the job. If all three machines always work at their respective, constant rates, what is the ratio of the amount of time it will take Machines X and Z to complete the job to the ratio of the amount of time it will take Machines Y and Z to complete the job?

A. 5 to 1
B. 10 to 7
C. 1 to 5
D. 7 to 10
E. 9 to 10

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

In this ratio problem, it's helpful to begin with the ratio of each machine's RATES as opposed to times, as rates are additive when machines are working together. So if Machine X as a rate that's twice as fast as Machine Y, their rates have the ratio X:Y = 2:1. And the ratio of X to Z is 3:2, as X works 50% faster (and therefore would accomplish 3 jobs in the time that it takes for Z to complete 2). So to find a common three-way ratio, you can use X as the "anchor", and make the ratio of rates:

X : Y : Z = 6 : 3 : 4

So X and Y working together would complete 10 jobs in the time that it would take Y and Z working together to complete 7 jobs, for a ratio of 10 : 7 in their respective rates. But since the question asks for times, not rates, you'll need to invert the ratio to 7 : 10, answer choice D.
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Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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let 2,4,3=respective times of X,Y,Z
1/2,1/4,1/3=respective rates of X,Y,Z
X+Z rate/Y+Z rate=(5/6)/(7/12)=10:7
inverting, X+Z time/Y+Z time=(6/5)/(12/7)=7:10
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Re: Machine X can complete a job in half the time it takes Machine Y to co  [#permalink]

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Bunuel wrote:
Machine X can complete a job in half the time it takes Machine Y to complete the same job, and Machine Z takes 50% longer than Machine X to complete the job. If all three machines always work at their respective, constant rates, what is the ratio of the amount of time it will take Machines X and Z to complete the job to the ratio of the amount of time it will take Machines Y and Z to complete the job?

A. 5 to 1
B. 10 to 7
C. 1 to 5
D. 7 to 10
E. 9 to 10

We can let the time of machine Y to complete the job = y, so the time of machine X to complete the job is 0.5y = (½)y and the time of machine Z to complete the job = (1.5)(0.5y) = 0.75y = (3/4)y. .

Since rate is inverse of time, the rate of Y = 1/y, the rate of X = 1/[(½)y] = 2/y and the rate of Z = 1/[(3/4)y] = 4/(3y).

Thus, the amount of time it will take Machine X and Z to complete the job is

1/[2/y + 4/(3y)] = 1/[6/(3y) + 4/(3y)] = 1/[10/(3y)] = 3y/10

Similarly, the amount of time it will take Machine Y and Z to complete the job is

1/[1/y + 4/(3y)] = 1/[3/(3y) + 4/(3y)] = 1/[7/(3y)] = 3y/7

Therefore, the ratio is (3y/10)/(3y/7) = (1/10)/(1/7) = 7/10.

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