Two machines, X and Y, working together at their repsective constant r

Question

Two machines, X and Y, working together at their respective constant rates for 14 hours, and then machine Y, working alone at its constant rate for an additional 4 hours, can produce a total of m items. The time that it takes machine X, working alone at its constant rate, to produce m items is 8 hours less than the time that it takes machine Y, working alone at its constant rate, to produce m items. How many hours does it take machine Y, working alone at its constant rate, to produce m items?

A. 28
B. 30
C. 32
D. 34
E. 36

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Official Answer

E

A. 28
B. 30
C. 32
D. 34
E. 36

https://gmatclub.com/forum/download/file.php?id=83691

chetan2u · Accepted Answer

A straight forward logical way  
Y takes 8 hours more for entire work. So he will take 8/2 or 4 hrs extra to complete half work. 
Here too, he is working for 4 hours extra = X works for 14 hrs and Y works for 18 hours 🧐
So, this must be the time for doing half the work by each. Thus X would take 14*2 and Y would take 18*2 or 36 hours.

Another convenient way is to look at total time they work for.  
X works for 14 hrs while Y works for 14+4 or 17 hours. 
Total work done by X and Y is  \frac{14}{X}+\frac{18}{Y} . 
As X=Y-8 =>  \frac{14}{Y-8}+\frac{18}{Y}=1 
(I)  Now use options  => I would use the most friendly option, the one that would divide the numerator as finally we have to get 1.
36 would fit in perfectly as 36-8 or 28 would have common factors with 14 and 36 would have common factors with 18.
And 36 surely fits in properly. If it had not we would have gone for some other value. 
OR 
(II)  you could solve the equation.  
 \frac{14}{Y-8}+\frac{18}{Y}=1 
 14Y+18Y-8*18=Y(Y-8) 
 Y^2-40Y+8*18=0 
 (Y-4)(Y-36)=0 
Y cannot be less than 8, thus Y is 36

E

GradMentors · Answer

If we let x = the rate of Machine X and y = the rate of Machine Y. Machine X works for 14 hrs total, while Machine Y works for 18 hours total. So according to the information given in the problem:

14 ( x  +  y ) +  4 y  =  m

simplifying we get:

14x  +  18y  =  m

We also are told that Machine X takes 8 hrs less to produce m items. So with that information, we can write the following equation:

( x )( t-8 ) =  y ( t ) =  m

Now check out the first equation. The time difference between the work done by Machine X and Machine Y is 4 hrs, so let's multiple everything by two.

28x  +  36y  =  2m

If ( x )( t-8 ) =  y ( t ) =  m , then ( x )( t-8 ) +  y ( t ) =  2m

With this information, it's safe to say that  E) 36  is indeed the correct answer.

gmatophobia · Answer

IMO this a tough one if one tries to solve using an algebraic approach. Assume the rate of Machine X = x Assume the rate of Machine Y = y Two machines, X and Y, working together at their respective constant rates for 14 hours, and then machine Y, working alone at its constant rate for an additional 4 hours, can produce a total of m items. Inference → To produce m items ⇒ Machine X works for 14 hours, and Machine Y works for 18 hours 14x + 18y = Efforts required to produce m items The time that it takes machine X, working alone at its constant rate, to produce m items is 8 hours less than the time that it takes machine Y, working alone at its constant rate, to produce m items. Inference → X is faster than Y Time taken by machine X to produce m items = \frac{ ext{Efforts required to produce m items} }{ ext{Rate of machine X} } Time taken by machine X to produce m items = \frac{14x + 18y}{x} = 14 + 18 \frac{y}{x} Time taken by machine Y to produce m items = \frac{ ext{Efforts required to produce m items} }{ ext{Rate of machine Y} } Time taken by machine Y to produce m items = \frac{14x + 18y}{y} = 14\frac{x}{y} + 18 Given (14\frac{x}{y} + 18) - (14 + 18 \frac{y}{x}) = 8 14\frac{x}{y} + 4 - 18 \frac{y}{x} = 8 14\frac{x}{y} - 18 \frac{y}{x} = 4 Dividing by 2, we get 7\frac{x}{y} - 9\frac{y}{x} = 2 If, time taken by machine X = X ; Rate of machine X ⇒ x = \frac{1}{X} If, time taken by machine Y = Y ; Rate of machine Y ⇒ y = \frac{1}{Y} Transforming our equation in terms of X and Y 7\frac{\frac{1}{X} }{\frac{1}{Y} } - 9\frac{\frac{1}{Y} }{\frac{1}{X} } = 2 7\frac{Y}{X} - 9\frac{X}{Y} = 2 Now we can use the options to our advantage. The options provide us the value of Y ; we can calculate the value of X as shown below - Screenshot 2023-10-23 081303.png My initial hunch is to try Option E first because in that option X is a multiple of 7, and Y is a multiple of 9. Hence, the numerator and denominator cancel out when we substitute the values in our equation. No other option provides that advantage ( if I were running on a time crunch, I would choose this as my final answer ). LHS = 7\frac{36}{28} - 9\frac{28}{36} LHS = \frac{36}{4} - \frac{28}{4} LHS = 9 - 7 = 2 RHS = 2 Hence, option E matches. Option E avigutman - I would love to see your approach to this question. Can this be solved in a shorter way ?

KarishmaB · Answer

Two machines, X and Y, working together at their respective constant rates for 14 hours, and then machine Y, working alone at its constant rate for an additional 4 hours, can produce a total of m items. The time that it takes machine X, working alone at its constant rate, to produce m items is 8 hours less than the time that it takes machine Y, working alone at its constant rate, to produce m items. How many hours does it take machine Y, working alone at its constant rate, to produce m items?

All the given information and the question are about m items. So, we can consider m items 1 completed job.

Since X takes 8 fewer hours than Y takes, X is faster than Y.

Working together, they take 14 + 18 = 32 hours to complete the job. So, Y must take more than 32 hours since, working alone, Y would be slower than X and Y working together.

A. 28
B. 30
C. 32
D. 34
E. 36

So, of the choices, only 34 and 36 are possible values for the number of hours Y takes.

Test 34:

If Y takes 34 hours, X takes 34 - 8 = 26 hours to complete 1 job alone.

So, X's rate is 1/26 per hour and Y's rate is 1/34 per hour.

Using the given information on X and Y working together for 14 and 18 hours to complete 1 job, we get the following:

14/26 + 18/34 = more than 1 completed job since 14/26 and 18/34 are both greater than 1/2.

So, Y must go slower than 1/34 per hour and take more than 34 hours.

We're now done with this question after testing just one answer choice because 36 is the only remaining possible value. So, 36 must be the correct answer.

Just to see how 36 works, though, let's test it.

Test 36:

If Y takes 36 hours, X takes 36 - 8 = 28 hours to complete one job alone.

So, X's rate is 1/28 per hour and Y's rate is 1/36 per hour.

14/28 + 18/36 = 1

28 hours for X and 36 hours for Y work.

Correct answer: E

Kinshook · Answer

GIven: Two machines, X and Y, working together at their respective constant rates for 14 hours, and then machine Y, working alone at its constant rate for an additional 4 hours, can produce a total of m items. The time that it takes machine X, working alone at its constant rate, to produce m items is 8 hours less than the time that it takes machine Y, working alone at its constant rate, to produce m items.

Asked: How many hours does it take machine Y, working alone at its constant rate, to produce m items?

Let the time taken by Machine X & Machine Y to produce m items be x & y hours respectively.

14/x + 18/y = 1

y - x = 8
x = y - 8

y = ?

14/(y-8) + 18/y = 1

A. 28
B. 30
C. 32
D. 34
E. 36: 14/28 + 18/36 = 1

IMO E

Kinshook · Answer

nick13 
Given: Two machines, X and Y, working together at their respective constant rates for 14 hours, and then machine Y, working alone at its constant rate for an additional 4 hours, can produce a total of m items. The time that it takes machine X, working alone at its constant rate, to produce m items is 8 hours less than the time that it takes machine Y, working alone at its constant rate, to produce m items.

Asked: How many hours does it take machine Y, working alone at its constant rate, to produce m items?
Let the time taken by machine X & machine Y to produce m items be x & y hours respectively

14/x + 14/y + 4/y = 1

x = y - 8

y = ?

14/x + 18/y = 1
14/(y+8) + 18/y = 1
14y + 18(y+8) = y(y+8)
42y + 18*8 = yˆ2 + 8y
yˆ2 - 34y - 2ˆ4*3ˆ2 = 0
yˆ2 - 36y - 4y - 2ˆ4*3ˆ2 = 0
(y-36) - 4(y-36) = 0
(y-4)(y-36) = 0
y = 4 or 36
Since x = y-4 > 0 ; y = 4 is NOT POSSIBLE

y = 36

IMO E

einstein801 · Answer

Hi,

Don't think your first way is right. Half work will take (t+8)/2 less hours which = t/2 + 4.

chetan2u · Answer

unicornilove 
It is perfectly fine. 
Full work: X= t and Y=t+8
hakf work: X=t/2 and Y=(t+8)/2 or t/2 + 4. You are making a mistake by taking t/2 + 4 as the less time. 
X takes (t/2 + 4) - t/2 = 4 hours less and that is exactly what has been mentioned.

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=82925 Harsha GMAT-Club-Forum-6fk39cl2.png

MartyMurray · Answer

Two machines, X and Y, working together at their respective constant rates for 14 hours, and then machine Y, working alone at its constant rate for an additional 4 hours, can produce a total of m items. The time that it takes machine X, working alone at its constant rate, to produce m items is 8 hours less than the time that it takes machine Y, working alone at its constant rate, to produce m items. How many hours does it take machine Y, working alone at its constant rate, to produce m items?

A. 28
B. 30
C. 32
D. 34
E. 36

Show Spoiler

Show Answer

Official Answer

E

All the given information and the question are about m items. So, we can consider m items 1 completed job.

Since X takes 8 fewer hours than Y takes, X is faster than Y.

Working together, they take 14 + 18 = 32 hours to complete the job. So, Y must take more than 32 hours since, working alone, Y would be slower than X and Y working together.

A. 28
B. 30
C. 32
D. 34
E. 36

So, of the choices, only 34 and 36 are possible values for the number of hours Y takes.

Test 34:

If Y takes 34 hours, X takes 34 - 8 = 26 hours to complete 1 job alone.

So, X's rate is 1/26 per hour and Y's rate is 1/34 per hour.

Using the given information on X and Y working together for 14 and 18 hours to complete 1 job, we get the following:

14/26 + 18/34 = more than 1 completed job since 14/26 and 18/34 are both greater than 1/2.

So, Y must go slower than 1/34 per hour and take more than 34 hours.

We're now done with this question after testing just one answer choice because 36 is the only remaining possible value. So, 36 must be the correct answer.

Just to see how 36 works, though, let's test it.

Test 36:

If Y takes 36 hours, X takes 36 - 8 = 28 hours to complete one job alone.

So, X's rate is 1/28 per hour and Y's rate is 1/36 per hour.

14/28 + 18/36 = 1

28 hours for X and 36 hours for Y work.

Correct answer:  E

KarishmaB · Answer

Ratios help you see the answer here quickly. X works for 14 hrs and Y works for 18 hrs. Difference between the hours is 4. They produce m units. So if X works for 28 hrs and Y works for 36 hrs, they will produce 2m units. Difference between the hours is 8. We are given that X working alone to produce m items takes 8 hours less than does Y, working alone to produce m items. Since the difference we obtained between the hrs is 8 for total 2m units, it fits. X working for 28 hrs will produce m units and Y working for 36 hrs will produce m units. Hence answer would be 36. Answer (A) If the values were a bit different, you might have needed to use algebra and equations etc. But knowing that GMAT would not expect too many calculations from us, I would always look at the given numbers and whether they fit to give the answer quickly. Some other interesting work rate discussion: https://anaprep.com/arithmetic-avoiding ... work-rate/ https://youtu.be/6CT3AXW-QzA https://anaprep.com/arithmetic-work-rat ... s-and-all/

wtang123 · Answer

This is probably a slower approach but this was the method that initially came to my mind.

14(x + y) + 4y = m → 14x + 18y = m

The time it takes to produce m units is 8 hours less for Machine X
m/x = m/y - 8 → 8 = m/y - mx → 8 = m(x - y)/xy → m = 8xy/(x-y)

Substituting for m
14x + 18y = 8xy/(x-y) → 14x2 - 4xy - 18y2 = 0 → 7x2 - 2xy - 9y2 = 0 → (7x - 9y)(x + y) = 0

Rate cannot be negative so the only solution is 7x = 9y or (7/9)x = y

Since we know X takes 8 hours less to produce m units, an equation can be formed 
x(t - 8) = yt → x(t - 8) = (7/9)xt → t - 8 = (7/9)t → (2/9)t = 8 → t = 36

hemanthPA · Answer

Hey guys, we all know this is a time consuming, mad problem but here is the simplest solution for you. it cannot get any simpler than this. First visualise and know the basics of work, rate and time. i.e., w = r.t or r = w/t, and consider the below terms Work done by X – Wx Work done by Y – Wy Time taken by X – Tx Time taken by Y – Ty VISUALIZE - VISUALIZATION IS THE KEY https://gmatclub.com/forum/download/file.php?id=83224 it is given that in the beginning X and Y work together work for 14 hrs, then x is no more there and y work for the remaining 4 hrs From this, we can derive our main equation To do 1 unit of work, i.e., M as they said x works for 14hrs, Y works for 14hrs and Y works an additional of 4 hrs So work done by X and Y individually to complete 1 unit (or M amount) of work would be Wx + Wy = 1 14/Tx + 14/Ty + 4/Ty = 1 14/Tx + 18/Ty = 1 Next they have given the relationship, Tx = Ty – 8 Now coming until here must take you about a min, at max. now lets just plug in the options and see for what values of Tx and Ty we get 1, this must be relatively simple and error free. We need to follow the AEBDC method. Plug options and check in the order AEBDC in the equation 14/tx + 16/ty = 1 work units, with the relation tx = ty – 8 Given options are for ty Option A. Ty = 28, tx = 20, this will be 14/20 + 18/28 = 7/10 + 9/14 = (98 + 90) / 140 which is not equal to 1 Option D. Ty = 36, tx = 28, this will be 14/28 + 18/36 = 1⁄2 + 1⁄2 = 1 units of work, hence ty = 36hrs Viola! as simple as that, under 2 mins. THANKS TO MY NEW TUTOR SAM, i learnt these tricks!! GMAT-Club-Forum-ymf07q97.png

KPBAN · Answer

For simplicity lets say the rate for machine X and machine Y is x and y and the units of work required to produce m items be M

Then 14(x+y) + 4y = M implies 14x + 18y = M (Eqn 1)

Let the time taken by X alone to produce m items be T1 and time taken by Y alone to produce m items be T2

Then (T1-8) * x = T2 * y = M

Which can be re written as (T1-8) * x + T2 * y = 2*M implies that (T1-8) * x + T2 * y = 28x + 36 y (From eqn 1). We can directly conclude that T2 is 36

sarthak1701 · Answer

LCM of 14 and 18 is 126, so M is a multiple of 126. But the LCM allows us to get the rates of X and Y

9*14 = 126 (X's rate) and 7*18 = 126 (Y's rate)

Now, 9*T-8 = M and 7*T = M

Equating we get, 9T-72 = 7T = 2T = 72, T = 36.

Ans. E

EDIT: The solution is wrong but somehow I still got the answer right, keeping this up so others don't make the same mistake.

Dream009 · Answer

time of x is y-8 not effort - how can you write x= y-8

Bunuel · Answer

In that setup, Y represents the number of hours machine Y takes alone to produce m items, and X represents the number of hours machine X takes alone to produce the same m items.

So:
• X’s rate = 1/X
 • Y’s rate = 1/Y

X worked for 14 hours, and Y worked for 18 hours, producing a total of m items (one complete job). Hence, the total work equation is:
14/X + 18/Y = 1

Since X = Y - 8, we get:
 14/(Y - 8) + 18/Y = 1

nihilre · Answer

god bless you  chetan2u

Two machines, X and Y, working together at their repsective constant r

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Two machines, X and Y, working together at their repsective constant r

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