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

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.

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?

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...

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:

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

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

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

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

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 ?

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

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

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.

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