If Machine X, working alone at its constant rate, performs half of a

Question

If Machine X, working alone at its constant rate, performs half of a certain task and then Machine Y, working alone at its constant rate, performs the rest of the task, the whole task will take a total of 16 hours. How many hours will it take machine X, working alone at its constant rate, to perform the task?

(1) Working together at their respective constant rates, machines X and Y perform the task in 6 hours.

(2) Machine X has a faster rate than machine Y.

(1) Working together at their respective constant rates, machines X and Y perform the task in 6 hours.

(2) Machine X has a faster rate than machine Y.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. Both statement TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Bunuel · Accepted Answer

If Machine X, working alone at its constant rate, performs half of a certain task and then Machine Y, working alone at its constant rate, performs the rest of the task, the whole task will take a total of 16 hours. How many hours will it take machine X, working alone at its constant rate, to perform the task?

(1) Working together at their respective constant rates, machines X and Y perform the task in 6 hours.

(2) Machine X has a faster rate than machine Y.

Assume Machine X and Machine Y take x and y hours, respectively, to complete the task alone. Thus, Machine X needs x/2 hours to perform half of the task, and Machine Y needs y/2 hours to complete the remaining half. Therefore, we are given that x/2 + y/2 = 16, leading to x + y = 32.

(1) Working together at their respective constant rates, machines X and Y perform the task in 6 hours.

This implies that 1/x + 1/y = 1/6. Substituting y = 32 - x, we get 1/x + 1/(32 - x) = 1/6, which leads to the quadratic equation x^2 - 32x + 192 = 0. Solving this results in x = 8 or x = 24. Therefore, either x = 8 and y = 24, or x = 24 and y = 8. Not sufficient.

We could avoid all the algebra above if you had spotted that Machines X and Y are indistinguishable in this scenario — there is not a single piece of information that differentiates Machine X from Machine Y. Consequently, we cannot differentiate between x and y. This means that the equation 1/x + 1/(32 - x) = 1/6 would yield two potential sets of values for x and y: x < y and x > y, because the machines are interchangeable in their roles. Without additional details, we cannot conclusively determine which completion time belongs to which machine — unless, of course, x equals y, which is clearly not the case here.

(2) Machine X has a faster rate than machine Y.

This statement only implies that x < y, which is clearly insufficient on its own.

(1)+(2) Since from (2) x < y, then x = 8 and y = 24. Sufficient.

Answer: C.

Hope it helps.

KarishmaB · Answer

Say machine X takes X hrs to perform the whole task alone and machine Y takes Y hrs. To perform half the task, they will take X/2 and Y/2 hrs respectively. Given that \frac{X}{2} + \frac{Y}{2} = 16 . We are asked the value of X. (1) Working together at their respective constant rates, machines X and Y perform the task in 6 hours. We can say for sure that X and Y are not equal because had they been equal, then working together, the machines would have simply taken 8 hrs to complete the work. Whether they did half the work simultaneously or one after the other, wouldn't have mattered. Each would have taken 8 hrs. \frac{1}{X} + \frac{1}{Y} = \frac{1}{6} We can eliminate Y using the given equation above and will get a quadratic in X which will give 2 distinct values for X (since X and Y are distinct). But which value will belong to X? The greater one or the smaller one? We don't know. Not sufficient alone. (2) Machine X has a faster rate than machine Y. This tells us that X < Y (because X is the number of hrs taken) Not sufficient alone. Using both we know that X will be the smaller value of the two that the quadratic will give us. Hence both statements together are sufficient. We don't need to actually solve the equations. Answer (C) Here are the basics of Work-Rate: https://youtu.be/88NFTttkJmA

VivekSri · Answer

OA IS C

Posted from my mobile device

vassivass · Answer

Hello could someone provide another solution please, I struggle to understand this one. Thank you

GMATGuruNY · Answer

Statement 2 is clearly INSUFFICIENT.

Let x = X's rate alone and y = Y's rate alone, implying that the combined rate when X and Y work together = x+y

Statement 1: 
Case 1: task = 48 units
 Working together at their respective constant rates, machines X and Y perform the task in 6 hours. 
Combined rate for X and Y to complete the 48-unit task in 6 hours =  \frac{work}{time} = \frac{48}{6} = 8  units per hour, implying that  x+y = 8

If Machine X, working alone at its constant rate, performs half of a certain task and then Machine Y, working alone at its constant rate, performs the rest of the task, the whole task will take a total of 16 hours 
Time for X to produce half the 48-unit task at rate of x units per hour  = \frac{24}{x} 
Time for Y to produce half the 48-unit task at a rate of y units per hour  = \frac{24}{y} 
Since the total time is 16 hours, we get:
 \frac{24}{x} + \frac{24}{y} = 16

Since  \frac{24}{x} + \frac{24}{y} = 16  and  x+y=8 , look for two factors of 24 that sum to 8:
2 and 6
4 and 4
Only the first combination satisfies  \frac{24}{x} + \frac{24}{y} = 16 :
 \frac{24}{2 }+ \frac{24}{6 }= 12 + 4 = 16

If x=2 and y=6. then the time for X to complete the 48-unit job  = \frac{work}{rate} = \frac{48}{2} = 24  hours
If x=6 and y=2. then the time for X to complete the 48-unit job  = \frac{work}{rate} = \frac{48}{6} = 8  hours
Since the time for X can be different values, INSUFFICIENT.

Statements combined:
In Case 1, attributing the faster rate to X implies that x=6, with the result that X's time = 8 hours

Case 2: task = 96 units
 Working together at their respective constant rates, machines X and Y perform the task in 6 hours. 
Combined rate for X and Y to complete the 96-unit task in 6 hours =  \frac{work}{time} = \frac{96}{6} = 16  units per hour, implying that  x+y = 16

If Machine X, working alone at its constant rate, performs half of a certain task and then Machine Y, working alone at its constant rate, performs the rest of the task, the whole task will take a total of 16 hours 
Time for X to produce half the 96-unit task at rate of x units per hour  = \frac{48}{x} 
Time for Y to produce half the 96-unit task ar a rate of y units per hour  = \frac{48}{y} 
Since the total time is 16 hours, we get:
 \frac{48}{x} + \frac{48}{y} = 16

Since  \frac{48}{x} + \frac{48}{y} = 16  and  x+y=16 , look for two factors of 48 that sum to 16:
4 and 12
8 and 8
Only the first combination satisfies  \frac{48}{x} + \frac{48}{y} = 16 :
 \frac{48}{4}+ \frac{48}{12}= 12 + 4 = 16

Attributing the faster rate to X implies that x=12, with the result that X's time to complete the 96-unit job  = \frac{work}{rate} = \frac{96}{12 }= 8  hours

In each case, X's time is the same: 8 hours
Thus, the two statements combined are SUFFICIENT.

C

Dreamer24 · Answer

Assume there is a total 48u of work that need to be completed.
Machine X did 24u and Machine Y did 24u in a total of 16hrs

Statement1:
They took 6 hours to complete the 48u of work. So in 1 hr, they are doing 8u.
If Machine X did 'a' unit of work then Y did '8-a' in 1hr
So we can form the equation now:

24/a +24/8-a =16
Solving this we will get a=2,6

We don't know if Machine X is doing 2 or 6 unit of work in 1 hr

Statement 2:
Clearly Insufficient

Statement 1+2:
machine X rate > Y's rate
So Machine X is doing 6u of work and Y is doing 2u

Answer: C

kkannan2 · Answer

Hi  GMATGuruNY , how are you just plugging in 48 and 96 units for the values of the total amount of work done by the machines? We don't know enough in this problem to infer a number for the number of units of work completed by the machines. If I were to solve this problem, I wouldn't feel confident plugging in values, especially given that this is a DS problem. Can you explain what is driving that intuition here to just pick two cases for 48 and 96 units and conclude that that is sufficient?

Also, can someone please present a solution for this problem with variables instead of choosing numbers? That would feel much more thorough. I am struggling to understand this one.  Devvrata -your solution looks good, but u is still taken as an unknown (you haven't defined u as a specific value; u=1 or 2 or 3, etc.), so I don't understand how you arrived at sufficiency with the statements combined. From my interpretation of your solution, u could be anything. If you could elaborate on your approach, that would be helpful. Thanks in advance!

GMATGuruNY · Answer

Here, the question stem asks for a value: the number of hours it takes X to perform the task alone.
If we test two different options for the task and get a certain number of hours for X in the first case but a different number of hours for X in the second case, then we have insufficient information to determine the number of hours for X.
But if the time for X in both cases is the SAME, the implication is that the size of the task is irrelevant: the number of hours for X will remain constant, whether the task is 48 units, 96 units, or 48000 units.

When implementing this approach, we should aim to test values that make the math easy.
For a work/rate problem, it is generally wise to select values divisible by the given rates and/or times.
Here, we are given two times -- 16 hours in the prompt, 6 hours in Statement 1 -- so I first tested a task of 48 units (the LCM of 16 and 6) and then a task of 96 units (the next greatest multiple of 16 and 6).

kkannan2 · Answer

Thank you  Bunuel  for the explanation, it makes more sense now. @ GMATGuruNY -makes sense on your approach too with getting two different values for X and concluding it's insufficient, I guess I'm just a bit hesitant to plug in numbers for value-based DS questions in general, and am still not understanding how I'd arrive at sufficiency with your approach when combining the statements together. Regardless, I think Bunuel's explanation cleared up my doubts, thanks!

GMATGuruNY · Answer

The logic behind the testing-cases approach is as follows:
When the statements are combined, the time for X alone is 8 hours,  no matter what value we assign to the task .

In my earlier post, Case 1 (task=48) and Case 2 (task=96) both yielded a time of 8 hours for X alone.

Case 3: task = 240 units

Again, let x = the rate for X and y = the rate for Y, implying that their combined rate = x+y.

Working together at their respective constant rates, machines X and Y perform the task in 6 hours. 
Combined rate for X and Y to complete the 240-unit task in 6 hours =  \frac{work}{time} = \frac{240}{6} = 40  units per hour, implying that  x+y = 40 .

If Machine X, working alone at its constant rate, performs half of a certain task and then Machine Y, working alone at its constant rate, performs the rest of the task, the whole task will take a total of 16 hours. 
Time for X to produce half the 240-unit task at rate of x units per hour  = \frac{120}{x} 
Time for Y to produce half the 240-unit task at a rate of y units per hour  = \frac{120}{y} 
Since the total time is 16 hours, we get:
 \frac{120}{x} + \frac{120}{y} = 16

Since  \frac{120}{x} + \frac{120}{y} = 16  and  x+y=40 , look for two factors of 120 that sum to 40:
10 and 30
This combination satisfies both of the equations above.
Since X has the faster rate, x=30 and y=10.
At a rate of 30 units per hour, the time for X alone to complete the entire 240-unit task =  \frac{work}{rate} = \frac{240}{30} = 8  hours.

When the statements are combined, the time for X alone is 8 hours, no matter value is assigned to the task.
If the task = 48, X's time alone is 8 hours.
If the task = 96, X's time alone is 8 hours.
If the task = 240, X's time alone is 8 hours.
Since the time is THE SAME in every case, the two statements combined are SUFFICIENT.
Regardless of the size of the task, X's time alone will be 8 hours.

Venu01298 · Answer

Think might have found an easier solution

Q - rephrased

Assume, T is the portion of work completed by machine X and Y separately when they work one after the other. Machine X's rate is x, Y's rate is y

T/x + T/y = 16 (Question)

(Tx + Ty) / (xy) = 16

T(x+y)/xy = 16

Statement 1 - Both machines working together complete it in 6 hours

Total work = T + T = 2T (from above)

2T / (x+y) = 6
T = 3(x+y)

Plugging T = 3(x+y) in above equation from Q

3(x+y)(x+y) / (xy) = 16

3(x+y)^2 / (xy) = 16

(x+y)^2 / (xy) = 16/3

only values that satisfy are 3 and 1. However, this is still not sufficient as we don't know which is which (x could be 3 or 1)

Statement 1 = Not sufficient

Statement 2 - Machine X is faster

This information alone is clearly not sufficient

Statement 2 = Not sufficient

Combining Statement 1 and 2,

x = 3 and y =1. Both statements together are sufficient

mbaprepavi · Answer

Let the time taken by MX be x and MY be y
Clearly, both statements individually are insufficient

We are given that x/2 + y/2 = 16 in the question
Which means x+y = 32 - (a)

Using both statements
1. 1/x+1/y = 1/6

Which gives us xy = 192

We also know "rate" of x is > "rate" of y
Which means time taken by x is less than time taken by y
Therefore we establish that x < y

Now, here comes the observation (Which can help and save time, instead of making a quadratic etc)
192 = 2^6 * 3

Possible products 
64*3
32*6
12*16
24*8 - Voila => This sums up to 32 hours, and is the only valid solution

Therefore, C and not E
Great question

Cheers.

Mannish · Answer

Here is a solution without solving the quadratic.

Simplifying the question stem:

1/(2X) + 1/(2Y) = 16
Do not stop, simplify even more, as said by our dearest GMATNinja
1/X + 1/Y = 32 ---- (I)

We need to find 1/X

Simplifying Statement 1:
1/(X+Y) = 6 ------- (II)

It is quite evident that putting (I) and (II) and solving quadratic, we will get the values of X and thus 1/X. But we DO NOT want to solve quadratic and without solving quadratic we will not know if we get only ONE positive value of X or TWO. Many of us will get trapped by our minds and mark A and move on.

Here is the smart way of looking at both the equations (I) & (II) above.
When I look at equation (II), I know the value of X+Y which is 1/6. When I look at equation (I), it looks that I can substitute X+Y into equation (I), it look like

(X+Y)/XY = 32
1/XY = 32*6 - this looks to a bit easy to find X and X without solving. Do not forget our dearest and simplify (do not take the meaning for the word seriously

) more

1/XY = 32*6 = 8*4*6 = 8*24
(1/X)*(1/Y) = 8*24

1/X = 8 or 24; 1/Y = 24 or 8 ----- INSUFFICIENT

Statement 2: Alone it is non-sense.

Statement 1 + 2: Bingo!

) more 1/XY = 32*6 = 8*4*6 = 8*24 (1/X)*(1/Y) = 8*24 1/X = 8 or 24; 1/Y = 24 or 8 ----- INSUFFICIENT Statement 2 : Alone it is non-sense. Statement 1 + 2 : Bingo!

Aboyhasnoname · Answer

Time taken by X = x hours, so half job done in x/2 hours 
Time taken by Y = y hours, so half job done in y/2 hours 
Now, x/2 + y/2 = 16.... x+y = 32

Statement 1. 
1/x + 1/y = 1/6

Means?......xy/x+y = 6 
Now x + y = 32 So, xy/32 = 6 
xy = 196...

Now, we'll try substituting y, by (32 - x)..... so x(32-x) = 196

32x - x^2= 196 
or 
x^2 - 32x + 196 = 0

Solving Quadratic, 24 and 8 ..The Values of X..So.... NS..There are 2 values

Statement 2 alone, is NS

Combine them. ....Out of 24 and 8, we know it will be 8 Since X is faster.

Combined. C Answer.

mimundertaker · Answer

Yes, this is how I tried solving - thanks for putting this solution, big ups!

dhruvie · Answer

Here's how I approached this; a little less complicated than other solutions IMO:

Machines 
  Time 
 
 Efficiency 
 
  Machine X 
  a  hrs 
  b  units/hr 
 
  Machine Y 
  b  hrs 
  a  units/hr

Explanation for the above table:

Assume  a hrs and b hrs  are the times taken by the machines.  LCM  of a and b is  ab,  which we assume as the  total work . Therefore, the efficiency of Machine X is  \frac{Total Work}{Work Per Hour Of X} , which is  \frac{ab}{a} , i.e.  b units/ hr . Same for Machine Y.

Back to the question:

Equation 1: Individual work

If X is doing half the work, then so is Y for the 16 hour time frame. That means both doing half work takes 16 hours.

Therefore time taken by both to complete the full work is  32 hours  (16*2).

Time of X + Time of Y = 32, i.e.,   a + b = 32

Equation 2: Work done together

Total work is  ab . And combined efficiency is  (a+b) . Hence, as per the question,  \frac{ab}{(a+b)}=6 hrs

We already know a + b = 32, hence   ab = 32*6   (No need to calculate it).

Finding the answer:

Solving both equations, we get the following two answers: 24 and 8. But we don't know which is which.

Statement II completes that gap.

Hence, the answer is C.

Why I didn't calculate a*b:  32*6 can be written as 8*4*6, which is 8*24 and 8+24=32 too, so why should you complicate things for yourself?

If Machine X, working alone at its constant rate, performs half of a

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Machines	Time	Efficiency
Machine X	a hrs	b units/hr
Machine Y	b hrs	a units/hr

GMAT Data Sufficiency (DS) Questions

If Machine X, working alone at its constant rate, performs half of a

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