Working alone at its constant rate, machine K took 3 hours to produce

Question

Working alone at its constant rate, machine K took 3 hours to produce 1/4 of the units produced last Friday. Then machine M started working and the two machines, working simultaneously at their respective constant rates, took 6 hours to produce the rest of the units produced last Friday. How many hours would it have taken machine M, working alone at its constant rate, to produce all of the units produced last Friday?

A. 8
B. 12
C. 16
D. 24
E. 30

A. 8
B. 12
C. 16
D. 24
E. 30

Jp27 · Accepted Answer

1st 2nd and 3 columns are Rate, Time and Work done.

R * 3 = 1/4
so rate of K = 1/12

Now 3/4 of the job is done by both K and M in 6 hrs

(1/12 + rate of M ) * 6 = 3/4
solving we get rate of M = 1/24

Finally

R * t = W

1/24 * t = 1

there t = 24 hrs

HTH.

Cheers

Bunuel · Answer

Given that machines M and K, working together, can produce 3/4 of units in 6 hours. Since also given that machine K can produce 1/4 of units in 3 hours, then in 6 hours it can produce 2/4 of units. Which means that when working together, machine M produced 3/4-2/4=1/4 of units in 6 hours, thus it needs 6*4=24 hours to produce all of the units.

Answer: D.

KarishmaB · Answer

Try using Ratios. It needs some effort in the beginning but is very rewarding. I solve all these questions orally using ratios. I will tell you how.

Working alone at its constant rate, machine K took 3 hours to produce ¼ of the units produced last Friday.

Ok. K takes 3 hrs for 1/4 of units so it takes 12 hrs for all units.

Then machine M started working and the two machines, working simultaneously at their respective constant rates, took 6 hours to produce the rest of the unites produced last Friday.

In 6 hrs, they both together produced 3/4 of units. K alone would have taken 9 hrs to produce 3/4 of units (since it takes 3 hrs for 1/4 of units)
Ratio of time taken together : time taken by K = 6:9 = 2:3
Ratio of speed together : speed of K = 3:2
Since speed together is 3 and speed of K is 2, speed of M alone is 1 i.e. speed of K is twice the speed of M. So M will take twice the time taken by K. Since K takes 12 hrs to produce all the units alone, M will take 24 hrs to produce all the units alone.

samdarsh88 · Answer

the explanation sounds logical and the correct answer to this is  D .Though,that's the (pick an answer an work upon) approach.how can we approach the same question through the conventional method (conceptual), by taking out the respective rates etc.can any one explain?

ratinarace · Answer

rate of machine k = (1/4)/3 (i.e it takes 3 hours to complete 1/4 of the work) ----> 1/12

Now both M and K working together at their respective rate complete 3/4 (the remaining work i.e 1- 1/4) of the work in 6 hours.

1/12 + 1/M = (3/4)/6 ------M = 24

hence the answer is D...

EMPOWERgmatRichC · Answer

Hi All,

As wordy and complex as this question might first appear, you can break it down into pieces and solve with very little in the way of tough calculations. We're told that Machine K took 3 hours to produce 1/4 of the units. From this, we have a rate (every 3 hours another 1/4 of the units are produced).

Machines K and M took 6 more hours to finish producing the rest of the units. Let's break this into two pieces:

Machine K worked for 6 more hours 
Machine M worked for 6 hours

We know Machine K's rate: 3 hours to complete 1/4 of the units. Since it worked 6 more hours, Machine K completed 2(1/4) = 1/2 of the units during that time. Counting the initial 3 hours (and 1/4 of the units) that Machine K worked, it completed 1/4 + 1/2 = 3/4 of the units.

The remaining 1/4 of the units must have been produced by Machine M over the course of those 6 hours….

Machine M takes 6 hours to produce 1/4 of the units.

We're asked how long it would take Machine M to produce ALL of the units…

6 hours to produce 1/4 of the units = 
24 hours to produce all of the units.

Final Answer:  D

GMAT assassins aren't born, they're made, 
Rich

BrentGMATPrepNow · Answer

Let's  assign a nice value  to the TOTAL number of units produced on Friday.
We want a number that works well with the given numbers in the question (1/4 and 6)
So let's say a TOTAL of   24   units were produced

Working alone at its constant rate, machine K took 3 hours to produce 1/4 of the units produced last Friday.   
1/4 of   24   = 6
In other words, machine K took 3 hours to produce 6 units
 Rate = output/time  = 6/3 = 2
So, machine K produces 2 units PER HOUR

Then machine M started working and the two machines, working simultaneously at their respective constant rates, took  6  hours to produce the rest of the units produced last Friday.  
24 - 6 =  18 
So, when machine M starts helping, the two machines have  18  units to produce
 Rate = output/time  =  18 / 6  = 3
So, the COMBINED rate of the two machines is 3 units PER HOUR

We already know that machine K produces 2 units PER HOUR
3 - 2 =   1  , so machine M produces   1   unit PER HOUR

How many hours would it have taken machine M, working alone at its constant rate, to produce all of the units produced last Friday?  
Time = output/rate =   24  /  1   = 24

Answer: D

Cheers, 
Brent

JanStracke · Answer

K works 9 hours in total and produces 3/4 of the units (1/4 * 3) as mentioned before.
So machine M produces 1/4 in 6 hours.

Since the question asks for the time M needs to produce the units of last Friday (1/4 as stated in the task) it should have take M 6 hours to produce this amount of units. Thus; the previous answers really confuse me since they do not answer the actual question and my solution is not stated in the options

EMPOWERgmatRichC · Answer

Hi JanStracke,

The question asks how long it would have taken Machine M to produce ALL of the units from last Friday (not only the ones that Machine M produced, but also the ones that Machine K produced). Since Machine M can produce 1/4 of the units in 6 hours, then it would take 4(6) = 24 hours to produce ALL of the units.
 
As it stands, you should have been a bit 'suspicious' of your original answer since it did not require ANY work to come up with. The prompt tells us that Machine M worked for 6 hours on Friday, so it's unlikely that after reading through the prompt - and doing all of the necessary math - that the answer would be the exact number that's listed in the prompt itself.

GMAT assassins aren't born, they're made,
Rich

KavyaGanta · Answer

Everything looks fine till 24 hrs. 
M takes 24 hrs to complete whole task.
Question asks about the time it takes to complete the number of units produced last Friday.
Only 1/4th of units are produced last friday 
So shouldn't the answer be 24/4 = 6 hrs ?

EMPOWERgmatRichC · Answer

Hi KavyaGanta,

The prompt defines how ALL of the various units were produced last Friday (some were produced when Machine K was working by itself and some were produced when both Machine K and Machine M were working). The question specifically asks us to figure how long Machine M would have taken - by itself - to produce ALL of those units (the 1/4 produced by Machine K by itself AND the remaining 3/4 that were produced when Machine K and Machine M were working together). 
 
GMAT assassins aren't born, they're made,
Rich

ScottTargetTestPrep · Answer

Solution:

Since it takes machine K 3 hours to produce 1/4 of the units, machine K’s rate is (1/4)/3 = 1/12. If we let m be the number of hours it takes machine M to produce all the units by itself, then machine M’s rate is 1/m. Since it takes both machines 6 hours to produce the rest of the units (i.e., 3/4 of the units), we can create the equation:

(1/12 + 1/m) x 6 = 3/4

1/12 + 1/m = 1/8

1/m = 1/24

m = 24

Answer: D

Poojans · Answer

Machine K completes 25% of the work in 3 hours
Machine K working simultaneously with Machine M completes the rest of the 75% of the work in 6 hours
In that 6 hours Machine K completes => 25% in first 3 hours and 25% in the next 3 hours => So a total of 50%
The rest 25% is completed by Machine M in 6 hours ( as the machines are working for 6 hours simultaneously) 
Therefore,  Machine M takes 24 hours to complete the entire work  alone.

Suhas1101 · Answer

consider total work = 120 units

Machine A working alone takes 3 hour to complete 1/4 of work

i.e. (1/4)*120 = 30, 30/3 = 10 unit / hr.

A=10 ....(1)

Now machine A & B will do remaining 3/4 of work in 6 hours

i.e. (3/4)*120 = 90, 90/6 = 15 unit/hr.

A+B = 15....(2)

Substituting A=10 we get B = 5

work done by B is 120/5 = 24

Ans : 24

Sakha · Answer

Bunuel  can you give us similar questions to practice?

Bunuel · Answer

Work and Rate Problems

Theory 
 
  Work Rate Problems - All in One Topic!  
  Work Word Problems Made Easy  
  How to Solve Relative Rate of Work Questions on the GMAT

Questions 
 
  DS Questions  
  PS Questions

For more check below: 
 
  ALL YOU NEED FOR QUANT ! ! !  
  Ultimate GMAT Quantitative Megathread

egmat · Answer

This is a classic work-rate problem with a twist - you've got two machines working in sequence, and then together. Let me walk you through how to think about this systematically.

The Setup:

First, let's understand what happened on Friday:
 
 Machine K worked alone for 3 hours → completed  \frac{1}{4}  of Friday's total units 
 Then Machine M joined in → both machines worked together for 6 hours → completed the remaining  \frac{3}{4}  of the units 
 Question asks: How long would Machine M alone take to produce ALL the units?

Step 1: Find Machine K's rate

Let's call the total work "1 unit" (representing all of Friday's production).

Machine K produced  \frac{1}{4}  unit in 3 hours, so:

Machine K's rate =  \frac{1/4}{3} = \frac{1}{12}  units per hour

Notice how we're dividing work by time to get rate.

Step 2: Find the combined rate

When K and M worked together, they completed  \frac{3}{4}  of the work in 6 hours:

Combined rate =  \frac{3/4}{6} = \frac{3}{4} 	imes \frac{1}{6} = \frac{1}{8}  units per hour

Step 3: Calculate Machine M's individual rate

Here's the key principle you need to remember: When machines work together, their rates add up.

ext{K's rate} + 	ext{M's rate} = 	ext{Combined rate}

\frac{1}{12} + 	ext{M's rate} = \frac{1}{8}

Solving for M's rate:
 	ext{M's rate} = \frac{1}{8} - \frac{1}{12}

To subtract these fractions, find a common denominator (24):
 \frac{1}{8} = \frac{3}{24}  and  \frac{1}{12} = \frac{2}{24}

ext{M's rate} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24}  units per hour

Step 4: Find time for M to complete all units

If M produces  \frac{1}{24}  of the work each hour, how long to complete 1 whole unit?

Time =  \frac{	ext{Work}}{	ext{Rate}} = \frac{1}{1/24} = 24  hours

Answer: D. 24 hours

Want to master this faster?

This problem tests a fundamental work-rate principle, but there's actually a much faster approach using "smart numbers" that can save you valuable time on test day. You can check out the  complete solution on Neuron by e-GMAT  to learn this alternative method and understand the systematic framework that applies to all sequential work-rate problems. You can also explore detailed solutions for  other official GMAT questions on Neuron  to build pattern recognition and avoid common traps like fraction errors and timing confusion.

Working alone at its constant rate, machine K took 3 hours to produce

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Working alone at its constant rate, machine K took 3 hours to produce

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