Question: Working at their respective constant rates, printing machine X, Y, and Z can finish a certain work in 9, 12, and 18 hours. If three machines work together to finish the work, what fraction of the work will be finished by the machine Z?

1. \(\frac{4}{9}\)
2. \(\frac{1}{3}\)
3. \(\frac{1}{4}\)
4. \(\frac{2}{9}\)
5. \(\frac{1}{9}\)

Answer:

Approach Solution (1):

X: 9 hours – 1 work
1 hour - \(\frac{1}{9}\)work

Y: 12 hours – 1 work
1 hour - \(\frac{1}{12}\)work

Z: 18 hours – 1 work
1 hour = \(\frac{1}{18}\)work

Total work done by all the machines in 1 hour:
\(\frac{1}{9}+\frac{1}{12}+\frac{1}{18}=\frac{(4+3+2)}{36}=\frac{9}{36}=\frac{1}{4}\) work

Jointly,
\(\frac{1}{4}\)work – 1 hour
1 work – \({\frac{\frac{1}{1}}{4}}\) = 4 hours (Jointly X, Y, Z all spent for 4 hours towards completing the job)

We know from before;
Z = 18 hours – 1 wok
1 hour = \(\frac{1}{18}\)work
If Z can perform \(\frac{1}{18}\) work in hour;
In 4 hours, he would have performed:

\(​\frac{1}{18}*4 work = \frac{2}{9} work\)

Correct Option: D

Approach Solution (2):

In 1 hour, X can finish \(\frac{1}{9} = \frac{4}{36}\)of the work, Y can finish \(\frac{1}{12} = \frac{1}{36}\) of the work, and Z can finish \(\frac{1}{18}= \frac{2}{36}\) of the work.
So in 1 hour, out of 4 + 3 + 2 = 9 parts of the work Z finished 2 parts or \(\frac{2}{9}\) of the work

Or if Z can do 1 part in some time interval then in the same time interval X can do 2 parts (as its rate is twice the rate of Z), and Y can do 1.5 parts (again as it rate is 1.5 times the rate of Z) so share of Z is 1 out of 1 + 2 + 1.5 = 4.5, or \(\frac{1}{4.5} = \frac{2}{9}\)

Correct Option: D

Approach Solution (3):

Let the total work be LCM of (9, 12, 18), i.e. 36 units
Thus, the total units are done by X in 1 hour = 4 units
Total units are done by Y in an hour = 4 units
Total units are done by Z in an hour = 2 units
If three machines work together:
Proportion of work done by Z = \(\frac{2}{(4+3+2)} = \frac{2}{9}\)

Correct Option: D

“Working at their respective constant rates, printing machine X, Y, and Z can finish a certain work in 9, 12, and 18 hours. If three machines work together to finish the work, what fraction of the work will be finished by the machine Z?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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