Machine X can complete a job in half the time it takes

[BTGmoderatorRO]

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
OA is d
What is the mathematical approach to use here in solving this problem? Can any expert help me out.
Thanks

[Jay@ManhattanReview]

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
OA is d
What is the mathematical approach to use here in solving this problem? Can any expert help me out.
Thanks

Say Machine Y takes 4 hours to complete the job, thus, Machine X would take 4/2 = 2 hours and Machine Z would take 2 + 50% of 2 = 3 hours to complete the job.

Time taken by Machines X and Z to complete the job = Reciprocal of (1/2 + 1/3) = Reciprocal of 5/6 = 6/5

Time taken by Machines Y and Z to complete the job = Reciprocal of (1/4 + 1/3) = Reciprocal of 7/12 = 12/7

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

= (6/5) / (12/7) = 7/10

The correct answer: D

Hope this helps!

-Jay
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[Scott@TargetTestPrep]

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 = (1/2)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/[(1/2)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.

Answer: D