GMAT Changed on April 16th - Read about the latest changes here

It is currently 25 Apr 2018, 07:41

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Running at their respective constant rates, machine X takes

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Intern
Intern
avatar
B
Joined: 17 May 2016
Posts: 31
GMAT 1: 740 Q46 V46
Reviews Badge
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 08 Jun 2017, 01:09
Bunuel wrote:
Please post full questions with answer choices.

Running at their respective constant rate, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.
A. 4
B. 6
C. 8
D. 10
E. 12

For work problems one of the most important thin to know is \(rate*time=job \ done\).

Let the time needed for machine X to produce \(w\) widgets be \(t\) days, so the rate of X would be \(rate=\frac{job \ done}{time}=\frac{w}{t}\);

As "machine X takes 2 days longer to produce \(w\) widgets than machines Y" then time needed for machine Y to produce \(w\) widgets would be \(t-2\) days, so the rate of Y would be \(rate=\frac{job \ done}{time}=\frac{w}{t-2}\);

Combined rate of machines X and Y in 1 day would be \(\frac{w}{t}+\frac{w}{t-2}\) (remember we can sum the rates). In 3 days two machines together produce 5w/4 widgets so: \(3(\frac{w}{t}+\frac{w}{t-2})=\frac{5w}{4}\) --> \(\frac{w}{t}+\frac{w}{t-2}=\frac{5w}{12}\).

\(\frac{w}{t}+\frac{w}{t-2}=\frac{5w}{12}\) --> reduce by \(w\) --> \(\frac{1}{t}+\frac{1}{t-2}=\frac{5}{12}\).

At this point we can either solve quadratic equation: \(5t^2-34t+24=0\) --> \((t-6)(5t-4)=0\) --> \(t=6\) or \(t=\frac{4}{5}\) (which is not a valid solution as in this case \(t-2=-\frac{6}{5}\), the time needed for machine Y to ptoduce \(w\) widgets will be negatrive value and it's not possible). So \(t=6\) days is needed for machine X to produce \(w\) widgets, hence time needed for machine X to produce \(2w\) widgets will be \(2t=12\) days.

Answer: E.



Hi Bunuel,

Could you please explain how do you solve the quadratic in one step ? I don't understand how can we factorize it with a coefficient greater than 1 on the \(x^2\).
When trying to solve with the quadratic formula, it involves big numbers with lot of calculation and requires lot of time.

Thank you so much for your help,
Sincerly yours,
Expert Post
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2298
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 10 Jun 2017, 08:10
heyholetsgo wrote:
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.

A. 4
B. 6
C. 8
D. 10
E. 12


We are given that running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machine Y does. If we let t = the time in days that it takes machine Y to produce w widgets, then t + 2 = the time in days that it takes machine X to produce w widgets. Furthermore, we can say the following:

rate of machine Y = w/t

rate of machine X = w/(t + 2)

We are given that the machines produce 5w/4 widgets in 3 days. Since work = rate x time and each machine works for 3 days, we first calculate the work done by machine X and machine Y individually.

work of machine Y = (w/t) x 3 = 3w/t

work of machine X = w/(t + 2) x 3 = 3w/(t + 2)

Since the machines work together to produce 5w/4 widgets, we can sum their work and set that sum to 5w/4.

3w/t + 3w/(t + 2) = 5w/4

Divide the entire equation by w and we have:

3/t + 3/(t + 2) = 5/4

Now, multiplying the entire equation by 4t(t + 2) to eliminate the denominators in the equation, we obtain:

3[4(t + 2)] + 3(4t) = 5[t(t + 2)]

12t + 24 + 12t = 5t^2 + 10t

5t^2 - 14t - 24 = 0

(5t + 6)(t - 4) = 0

t = -6/5 or t = 4

Since t cannot be negative, t must equal 4. That is, it takes machine Y 4 days to produce w widgets. Thus, it will take machine X 6 days to produce w widgets and 12 days to produce 2w widgets.

Answer: E
_________________

Jeffery Miller
Head of GMAT Instruction

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Manager
Manager
User avatar
B
Joined: 30 Apr 2017
Posts: 75
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 03 Oct 2017, 18:25
Bunuel wrote:
heyholetsgo wrote:
Thanks a lot, this was very helpful! I know how to factorize but as soon as it looks like this I can"t handle it anymore 5t^2-34t+24=0 After I found the factors for 24, there is no way getting around trial and error, is there?

Factoring Quadratics: http://www.purplemath.com/modules/factquad.htm

For our original question you can also use substitution method: we \(\frac{1}{t}+\frac{1}{t-2}=\frac{5}{12}\) and we know that answer would be \(2t\). Try to substitute half of the values listed in the answer choices and you'll see that answer choice E will work.

heyholetsgo wrote:
And is this a formula to compute the times 2 machines need to finish the same job? I always thought the times of the machines do NOT add up in work problems..
1/T = 1/T1 + 1/T2 or T=(T1*T2)/(T1+T2)


If:
Time needed for A to complete the job =A hours;
Time needed for B to complete the job =B hours;
Time needed for C to complete the job =C hours;
...
Time needed for N to complete the job =N hours;

Then if time needed for all of them working simultaneously to complete the job is \(T\), then: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+..+\frac{1}{N}=\frac{1}{T}\) (General formula).

For two and three entities (workers, pumps, ...):

General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:

Given that \(a\) and \(b\) are the respective individual times needed for \(A\) and \(B\) workers (pumps, ...) to complete the job, then time needed for \(A\) and \(B\) working simultaneously to complete the job equals to \(T_{(A&B)}=\frac{a*b}{a+b}\) hours, which is reciprocal of the sum of their respective rates (\(\frac{1}{a}+\frac{1}{b}=\frac{1}{t}\)).

General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:

\(T_{(A&B&C)}=\frac{a*b*c}{ab+ac+bc}\) hours.

Hope it helps.






thank you for your help
but
shouldn't we use the same formula for rate??!!!
the reciprocal of rates shouldn't be added together?
Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 44655
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 03 Oct 2017, 20:56
soodia wrote:
Bunuel wrote:
heyholetsgo wrote:
Thanks a lot, this was very helpful! I know how to factorize but as soon as it looks like this I can"t handle it anymore 5t^2-34t+24=0 After I found the factors for 24, there is no way getting around trial and error, is there?

Factoring Quadratics: http://www.purplemath.com/modules/factquad.htm

For our original question you can also use substitution method: we \(\frac{1}{t}+\frac{1}{t-2}=\frac{5}{12}\) and we know that answer would be \(2t\). Try to substitute half of the values listed in the answer choices and you'll see that answer choice E will work.

heyholetsgo wrote:
And is this a formula to compute the times 2 machines need to finish the same job? I always thought the times of the machines do NOT add up in work problems..
1/T = 1/T1 + 1/T2 or T=(T1*T2)/(T1+T2)


If:
Time needed for A to complete the job =A hours;
Time needed for B to complete the job =B hours;
Time needed for C to complete the job =C hours;
...
Time needed for N to complete the job =N hours;

Then if time needed for all of them working simultaneously to complete the job is \(T\), then: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+..+\frac{1}{N}=\frac{1}{T}\) (General formula).

For two and three entities (workers, pumps, ...):

General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:

Given that \(a\) and \(b\) are the respective individual times needed for \(A\) and \(B\) workers (pumps, ...) to complete the job, then time needed for \(A\) and \(B\) working simultaneously to complete the job equals to \(T_{(A&B)}=\frac{a*b}{a+b}\) hours, which is reciprocal of the sum of their respective rates (\(\frac{1}{a}+\frac{1}{b}=\frac{1}{t}\)).

General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:

\(T_{(A&B&C)}=\frac{a*b*c}{ab+ac+bc}\) hours.

Hope it helps.






thank you for your help
but
shouldn't we use the same formula for rate??!!!
the reciprocal of rates shouldn't be added together?


We ARE adding the rates. A, B and C above are times. The rate is reciprocal of time, so 1/A, 1/B and 1/C are rates.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Manager
Manager
avatar
B
Joined: 19 Aug 2016
Posts: 73
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 04 Oct 2017, 19:59
nickimonckom wrote:
Bunuel wrote:
Please post full questions with answer choices.

Running at their respective constant rate, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.
A. 4
B. 6
C. 8
D. 10
E. 12

For work problems one of the most important thin to know is \(rate*time=job \ done\).

Let the time needed for machine X to produce \(w\) widgets be \(t\) days, so the rate of X would be \(rate=\frac{job \ done}{time}=\frac{w}{t}\);

As "machine X takes 2 days longer to produce \(w\) widgets than machines Y" then time needed for machine Y to produce \(w\) widgets would be \(t-2\) days, so the rate of Y would be \(rate=\frac{job \ done}{time}=\frac{w}{t-2}\);

Combined rate of machines X and Y in 1 day would be \(\frac{w}{t}+\frac{w}{t-2}\) (remember we can sum the rates). In 3 days two machines together produce 5w/4 widgets so: \(3(\frac{w}{t}+\frac{w}{t-2})=\frac{5w}{4}\) --> \(\frac{w}{t}+\frac{w}{t-2}=\frac{5w}{12}\).

\(\frac{w}{t}+\frac{w}{t-2}=\frac{5w}{12}\) --> reduce by \(w\) --> \(\frac{1}{t}+\frac{1}{t-2}=\frac{5}{12}\).

At this point we can either solve quadratic equation: \(5t^2-34t+24=0\) --> \((t-6)(5t-4)=0\) --> \(t=6\) or \(t=\frac{4}{5}\) (which is not a valid solution as in this case \(t-2=-\frac{6}{5}\), the time needed for machine Y to ptoduce \(w\) widgets will be negatrive value and it's not possible). So \(t=6\) days is needed for machine X to produce \(w\) widgets, hence time needed for machine X to produce \(2w\) widgets will be \(2t=12\) days.

Answer: E.



Hi Bunuel,

Could you please explain how do you solve the quadratic in one step ? I don't understand how can we factorize it with a coefficient greater than 1 on the \(x^2\).
When trying to solve with the quadratic formula, it involves big numbers with lot of calculation and requires lot of time.

Thank you so much for your help,
Sincerly yours,



Hi...

we can factorize by taking the product of a and c...so 5*24=120

So 30*4=120

5x^2-30x-4x+24=0
Intern
Intern
avatar
S
Joined: 01 Sep 2016
Posts: 5
CAT Tests
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 29 Oct 2017, 22:14
I solved this way.
What am i making mistake?
Let x,y be the time for X and Y machines.
(w/x)-(w/y)=w/2 ---------1 eqn.

(w/x)+(w/y)=5w/12----2 eqn

solve 1 and 2 we get.
2/x=11/12
x=24/11

I think iam making mistake in eqn 1.Can someone please elaborate the mistake,which I made.
Intern
Intern
avatar
Joined: 25 Apr 2011
Posts: 5
Location: Pakistan
Concentration: Entrepreneurship, International Business
WE: Other (Commercial Banking)
Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 12 Nov 2017, 09:49
balagevr wrote:
I solved this way.
What am i making mistake?
Let x,y be the time for X and Y machines.
(w/x)-(w/y)=w/2 ---------1 eqn.

(w/x)+(w/y)=5w/12----2 eqn

solve 1 and 2 we get.
2/x=11/12
x=24/11

I think iam making mistake in eqn 1.Can someone please elaborate the mistake,which I made.


Don't get bogged down in algebra. As others above have already shown suppose numbers for variables like w. For instance we can suppose w=4 here, therefore 5w/4=5 and 2w=8. Now when the two machines run together, they produce 5 widgets in 3 days. Therefore 4 widgets will require 12/5 days. This is our target OR goal. Start plugging in answer choices always starting with C and see which choice matches our target or goal. Choice E says Machine X alone takes 12 days to produce 2w OR 8 widgets. Therefore Machine X will take 6 days to produce w OR 4 widgets. Machine Y takes 2 days short to produce 4 widgets i.e. it takes 4 days to produce 4 widgets. Both will take 6X4/(6+4) days or 12/5 days to produce 4 widgets. This matches our target or goal we calculated earlier and is therefore the right answer.
Intern
Intern
avatar
B
Joined: 15 Dec 2016
Posts: 9
GMAT 1: 530 Q42 V20
Reviews Badge CAT Tests
Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 16 Jan 2018, 13:01
heyholetsgo wrote:
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.

A. 4
B. 6
C. 8
D. 10
E. 12



My Method to solve - let w = 12 widgets
Given - x takes two more days than y
assume x takes 6 days
Therefore y will take 4 days

Both together produce 5w/4 = (5 * 12)/4
Total 15 widgets.

x takes 6 days to produce w(12) widgets so x alone will take 12 days to produce (2w) 24 widgets.
Intern
Intern
avatar
B
Joined: 01 Nov 2017
Posts: 11
CAT Tests
Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 22 Jan 2018, 02:18
I solved by applying substitution:
1)
X=Y+2
X+Y=2Y+2=3 <=> Y=0.5 X=2.5=5/2

2)
(5/4)w=3
5w=12
w=12/5

To answer question: 2*w*X=2*(12/5)*(5/2)=120/10=12
Intern
Intern
avatar
B
Joined: 30 Nov 2017
Posts: 25
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 21 Feb 2018, 00:12
yyz5028 wrote:
Hi,

I tried a different method to solve the problem, but somehow I couldn't. This was what I came up with so far:

I was trying to apply:Time = job done / rate

Let X = X widgets/day,
Y = Y widgets/day

1)w/X = (w/Y) + 2
2) 5w/4 = (X + Y)*3
From 1), Y = wX/(w - 2X) 3)
I plugged 3) into 2), but I came out with something really ugly and got stuck.

Is there anyone able to solve these two equations, or is there anything wrong with my method? Thank you.

I approached it in exactly the same manner and am stuck at exactly the same place :?
Intern
Intern
avatar
B
Joined: 08 Dec 2017
Posts: 17
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 27 Feb 2018, 05:29
You have three variables and two equations. Your method would work if you think of w widgets as 1 job. Then you would have two equations in two variables.

Job per day (days)= job

X (1/x)= 1

Y. (1/y)=. 1

3(x+y)=5/12
(1/x)=(1/y)+2

Posted from my mobile device
Intern
Intern
avatar
B
Joined: 16 Jan 2018
Posts: 11
Location: Canada
Concentration: Strategy, General Management
GMAT 1: 630 Q48 V28
GPA: 3.5
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 27 Feb 2018, 05:47
E. 12
Let Y does in y days
X does in y+2 days
1/y + 1/y+2 = 5/12
Putting y=4 solve this equation, hence X will take (4+2)2 days to make 2w widgets

Sent from my [device_name] using [url]GMAT Club Forum mobile app[/url]
Expert Post
Top Contributor
SVP
SVP
User avatar
P
Joined: 12 Sep 2015
Posts: 2317
Location: Canada
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 05 Mar 2018, 10:14
Expert's post
Top Contributor
heyholetsgo wrote:
Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.

A. 4
B. 6
C. 8
D. 10
E. 12


Another approach is to assign a nice value to the job (w)

Let's say that w = 12.

GIVEN: Running at their respective constant rates, machine X takes 2 days longer to produce 12 widgets than machine Y
Let t = time for machine Y to produce 12 widgets
So, t+2 = time for machine X to produce 12 widgets

RATE = output/time

So, machine X's RATE = 12 widgets/(t + 2 days) = 12/(t+2) widgets per day
And machine Y's RATE = 12 widgets/(t days) = 12/t widgets per day

The two machines together produce 5w/4 widgets in 3 days
In other words, The two machines together produce 5(12)/4 widgets in 3 days
Or the two machines together produce 15 widgets in 3 days
This means the COMBINED RATE = 5 widgets per day

So, we can write: 12/(t+2) + 12/t = 5
Multiply both sides by (t+2)(t) to get: 12t + 12t + 24 = 5(t+2)(t)
Simplify: 24t + 24 = 5t² + 10t
Rearrange: 5t² - 14t - 24 = 0
Factor to get: (5t + 6)(t - 4) = 0
So, EITHER t = -6/5 OR t = 4
Since the time cannot be negative, it must be the case that t = 4

If t = 4, then it takes Machine Y 4 days to produce 12 widgets
And it takes Machine X 6 days to produce 12 widgets

How many days would it take machine X alone to produce 2w widgets?
In other words, how many days would it take machine X alone to produce 24 widgets? (since w = 12)

If it takes Machine X 6 days to produce 12 widgets, then it will take Machine X 12 days to produce 24 widgets

Answer: E

Cheers,
Brent
_________________

Brent Hanneson – Founder of gmatprepnow.com

Image

Manager
Manager
avatar
B
Joined: 30 Jul 2014
Posts: 137
GPA: 3.72
Reviews Badge
Re: Running at their respective constant rates, machine X takes [#permalink]

Show Tags

New post 06 Mar 2018, 05:41
The equation from first two conditions would be -
3/(P+2) + 3/(P) = 5/4
Now we need to solve for P to get 2*(P+2)?
The trial and error method and a manipulation of P+2=x saves the time.
Answer comes out to be E.
_________________

A lot needs to be learned from all of you.

Re: Running at their respective constant rates, machine X takes   [#permalink] 06 Mar 2018, 05:41

Go to page   Previous    1   2   3   [ 54 posts ] 

Display posts from previous: Sort by

Running at their respective constant rates, machine X takes

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.