M06-21

Official Solution:


Machine A can complete a certain job in 12 days, working 2 full shifts per day, while Machine B can complete the same job in 15 days, also working 2 full shifts per day. If Machine A works during the first shift each day, Machine B works during the second shift, and both machines work half of the third shift, approximately how many days will it take for Machines A and B to complete the job under this work schedule?


A. 14
B. 13
C. 11
D. 9
E. 7


Machine A requires 12 days * 2 shifts = 24 shifts to complete the entire job;

Machine B requires 15 days * 2 shifts = 30 shifts to complete the entire job;

Each day, both machines work 1.5 shifts (\(\frac{3}{2}\) shifts). Together, in one day, they complete \(\frac{(\frac{3}{2})}{24}+\frac{(\frac{3}{2})}{30}=\frac{9}{80}\) of the whole job. Therefore, under the current work schedule, they will need approximately \(\frac{80}{9} \approx 9\) days to finish the entire job.


Answer: D
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Alternative approach.
Machine A can do a job in 12 days * 2 shifts = 24 shifts.
Machine B can do a job in 15 days * 2 shifts = 30 shifts.

\(\frac{1}{24} + \frac{1}{30} + (\frac{1}{24} + \frac{1}{30})/2\);

\(\frac{18}{240} + \frac{9}{240} = \frac{27}{240}\);

\(\frac{240}{27} ≈ 9\)

Hello
I understand your point up to the point that each machine will work 1.5 shifts in a day as per the question. But I am unable to grasp the last point that you made in the equation \(\frac{(\frac{3}{2})}{24}+\frac{(\frac{3}{2})}{30}=\frac{9}{80}\)th of the whole. Can you please explain that how you arrive at this equation.

Thanks

Sure.

In one day Machine A works of \(\frac{3}{2}\) shifts, and since Machine A needs 24 shifts to do the whole job, then in one day Machine A does (3/2)/24th of the whole job.

In one day Machine B works of \(\frac{3}{2}\) shifts, and since Machine B needs 30 shifts to do the whole job, then in one day Machine B does (3/2)/30th of the whole job.

Together they do \(\frac{(\frac{3}{2})}{24}+\frac{(\frac{3}{2})}{30}=\frac{9}{80}\)th of the whole job.

Hope it's clear.
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Hi,

I understand 3/2=1.5. How did you deduce "In one day each machine works 1.5 shifts (3/2 shifts)"

Each day machine A is working during the first shift (1 shift), machine B is working during the second shift (1 shift) and both machines A and B are working half of the third shift (0.5 shifts each). Therefore, each work for 1 + 0.5 shifts per day.

Hope it's clear.
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another straight approach:

rate=============time======work
2Sa=============12=======1 machine A working 2 shifts completes the job in 12 days. => 1/12=2Sa => Sa=1/24
2Sb=============15=======1 machine B working 2 shifts completes the job in 15 days. => 1/15=2Sb => Sb=1/30
Sa+Sb+.5*(Sa+Sb)==x========1

the combined rate will be Sa+Sb (each machine works for 1 shift) + .5Sa+.5Sb (each machine works half of the third shift)

x=240/27=80/9=approx less than 9

A better way to solve this question is to take LCM of 24 and 30=120.
It means 120 work to be done.
A does 1/24 work each day=>5 work each shift
B does 1/30 work each day=>4 work each shift.
In third shift both work half of the time,so work done=2.5+2=4.5 work
Total work done in a day=13.5 work.For doing 120 work,it will take 9 days approx.

Hey,

Please help me out.

A 12 d for 2 shifts so for 1.5 shifts, it is 9 d
B 15 d for 2 shifts so for 1.5 shifts it is 15/4 d

How to move from here? I am stuck and unable to understand how to move forward?

Hi, you must first calculate how many days it takes each machine working just ONE shift:

Machine A: If it takes 12 days working 2 shifts per day, then it takes 24 days working 1 shift per day.
Machine B: If it takes 15 days working 2 shifts per day, then it takes 30 days working 1 shift per day.

We are told that each machine is going to be working 1.5 shifts each day, which is 3/2 as a fraction, therefore:

Rate of Machine A + Rate of Machine B = [(3/2)/24] + [(3/2)/30]
The least common denominator of 24 & 30 is 120

{[(3/2)/24]x(5/5)} + {[(3/2)/30]x(4/4)} = [(15/2)/120] + (6/120) = [(15/2)+6]/120

Let's solve the numerator:
(15/2) + (6/1) = (15/2) + (12/2) = 27/2

Plug the numerator back into the equation:
(27/2)/120 = (27/2)/(120/1) = (27/2)x(1/120) = (9/2)x(1/40) = 9/80

Therefore, each machine working 1.5 shifts would complete 9 jobs in 80 days
But we want to know how many days it would take to complete just 1 job at this rate, so we need to divide 80 by 9

80/9 = 8.88 which is roughly 9 days

The correct answer is choice (D)

A takes 24 shifts to complete the job.
B takes 30 shifts to complete the job.

Let's say the work is 240.

A is doing 10 units per shift.
B is doing 8 units per shift.

Now they are doing 10 in the first shift, 8 in the second and (10+8)/2=9 in the third shifts.

Which means total is 27 in a day.

We have work of 240. Days taken will be 240/27 = 8.8888888 Or you just know it is less than 10 and around 9.

My way:
machine A: 12 days to complete a job with 2 shifts/day - a total of 24 days with 1 shift per day to complete 1 job.
machine B: 15 days to complete a job with 2 shifts/day - a total of 30 days with 1 shift per day to complete 1 job.

RATE * TIME = W
A 1/24 24 days 1
B 1/30 30 days 1
A+B=1/24+1/30 13.3days 1


13.3 days = 2 shifts per day -- to complete the job by both machines working together with a total of 2 shifts /day.question asks us for 3 shifts/day
x days= 3 shifts per day

13.3*2=x*3
x=26.6/3
x~9days

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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