Machines X and V produced identical bottles at different

There are several important things you should know to solve work problems:

1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems.

\(time*speed=distance\) <--> \(time*rate=job \ done\). For example when we are told that a man can do a certain job in 3 hours we can write: \(3*rate=1\) --> \(rate=\frac{1}{3}\) job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then \(5*(2*rate)=1\) --> so rate of 1 printer is \(rate=\frac{1}{10}\) job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then \(3*(2*rate)=12\) --> so rate of 1 printer is \(rate=2\) pages per hour;

So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate).

2. We can sum the rates.

If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is \(rate_a=\frac{job}{time}=\frac{1}{2}\) job/hour and B's rate is \(rate_b=\frac{job}{time}=\frac{1}{3}\) job/hour. Combined rate of A and B working simultaneously would be \(rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\) job/hour, which means that they will complete \(\frac{5}{6}\) job in one hour working together.

3. For multiple entities: \(\frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T}\), where \(T\) is time needed for these entities to complete a given job working simultaneously.

For example if:
Time needed for A to complete the job is A hours;
Time needed for B to complete the job is B hours;
Time needed for C to complete the job is C hours;
...
Time needed for N to complete the job is N hours;

Then: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T}\), where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously.

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 \(t_1\) and \(t_2\) 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 \(T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2}\) hours, which is reciprocal of the sum of their respective rates (\(\frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T}\)).

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

\(T_{(A&B&C)}=\frac{t_1*t_2*t_3}{t_1*t_2+t_1*t_3+t_2*t_3}\) hours.

BACK TO THE ORIGINAL QUESTION:
Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken Machine X operating alone to fill the entire production lot?

You can solve this question as Karishma proposed in her post above or algebraically:

Let the rate of X be \(x\) bottle/hour and the rate of Y \(y\) bottle/hour.
Given: \(4x+3y=job\). Question: \(t_x=\frac{job}{rate}=\frac{job}{x}=?\)

(1) Machine X produced 30 bottles per minute --> \(x=30*60=1800\) bottle/hour, insufficient as we don't know how many bottles is in 1 lot (job).

(2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours --> \(4x=2*3y\), so \(3y=2x\) --> \(4x+3y=4x+2x=6x=job\) --> \(t_x=\frac{job}{rate}=\frac{job}{x}=\frac{6x}{x}=6\) hours. Sufficient.

Answer: B.

Some work problems with solutions:
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hours-to-type-pages-102407.html

Hope it helps.
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We know that Machine X worked for 4 hrs alone and then Machine Y worked for 3 hrs alone. This filled up the production lot.
Ques: If Machine X were operating alone, how many hours would it take?
The first thing that comes to mind is that it would take more than 4 hrs since it filled only a part of the lot in 4 hrs.

Statement I: I know the rate at which Machine X produces bottles. It is 30 bottles/min or 1800 bottles/hr. So Machine X must have made 7200 bottles in 4 hours. But I do not know how many bottles fills the lot since I do not know how many bottles were made by Machine Y in 3 hours. This statement alone is not sufficient.

Statement II: Machine X produced twice as many bottles in 4 hours as Machine Y did in 3 hrs. Then I can say that Machine X filled 2/3 rd of the lot in 4 hrs. (If this is unclear, think Machine Y made 'b' bottles in 3 hrs, then Machine X made '2b' bottles in 4 hrs and together the lot contained '3b' bottles. So Machine X filled 2/3 of the lot in 4 hours.) In how much time will Machine X fill the rest of the 1/3 rd lot? In 2 hours. Hence, it takes a total of 6 hours to fill the lot on its own. This is sufficient to answer the question.

Answer: B
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1) does not tell us anything about Y. INSUFFICIENT

2) machine X produced twice as much as Y did i.e. did \(\frac{2}{3}\) of the work.

\(\frac{2}{3}\) work required 4 hours

1 complete work requires \(\frac{4*3}{2} = 6\) hours

Answer: B

HTH

Word of caution in DS questions. One trick they use often is that they give you partial information in Statement (1), they give the rest in statement (II) so you think, "Of course, answer is an easy (C)." Mind you, if it seems to be an easy (C), go back to the question, read it again and then try and solve it using statement (II) alone, Try to 'wipe' statement (I) from your mind for the time being.
Here, I don't need to know how many bottles Machine A produced in total. I only need to know how many hours it will take to fill the lot. Since it filled 2/3rd in 4 hrs, it will the rest 1/3 in 2 hrs.
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Thank you that clears it up, I was timing myself so trying to get through the problem in 2min or less. When I read the two statements I came to the exact same conclusion you did on statement 1; I can determine exactly how many bottles machine X made, but I don't know how many bottles = 1 lot, Insufficient.

However in reading statement 2 I felt it wasn't enough to answer the question by itself (Machine X produced twice as many bottles as Y) I didn't have a rate or a total, However, from statement 1, I could get an exact answer of how many Machine X produced, then simply divide by 2 to get the amount Machine Y produced. Adding these two together will give me the total bottles in one lot. With both statements I have all the information I need to determine how long it would take Machine X to fill the lot by itself because now I have it's rate and the total number of bottles in one lot. So I chose "C" Both statements together are sufficient

How could this logic be wrong?

Thanks Bunuel! I like your work/rate resume. It is really helpful. +1

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE


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4R = X

3R* = Total - X

4R = 2(3R*)

R* = 4R/6 = 2/3R

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3R* = Total - 4R

2R = Total - 4R

6R = Total

Answer: 6. B.

Stmt 1 : Machine X produced 30 bottles per minute hence in 4 hrs i.e. 240 minutes how many did machine X produce?

(30 * 240) = 7200 bottles

But we don’t know the exact size of the production lot so while we know X’s work rate, we don’t know how many hrs it will take for X to fill up the production lot.

Hence this stmt is insufficient.



Stmt 2 : Let a be the number of bottle machine Y produces in 3 hrs

Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours

Then 2a is the number of bottles machine X produced in 4 hrs

Hence the total lot size = a + 2a = 3a

If X produces 2a bottles ibn 4 hrs then how many hrs will it take to produce 3a bottles?

(3a * 4)/2a = 6 hrs

Answer : B

I set up two rudimentary equations for each machine from the prompt before entering into determining the sufficiency of each statement:

\(x*4=n\) where x is the rate of Machine X and n is the amount of the job completed in four hours
\(y*3=m\) where y is the rate of Machine Y and m is the amount of the job completed in three hours
where \(n+m=1\) in order to preserve the fact that X and Y complete one job together (yet individually)

Statement 1: I won't beat a dead horse here, we know this is clearly insufficient

Statement 2: Here we see that n=2m or that Machine X completed 2/3 of the work and Machine Y completed 1/3 of the work. By substituting 2/3 for n and 1/3 for m in the equations above we are able to determine how long it would take Machine X to complete 1 (one complete unit) of work since we have \(x*4=\frac{2}{3}\) ....Sufficient

Answer B

If we know total Work then can find out total time take by machine A(Asked)
Major Goal: Total Work.
Machine A's contribution: 4/X
Machine B's contribution: 3/Y
Total Work: 4/X+3/Y. ------------Eq(1)

Statement 1: Machine X produced 30 bottles per minute.
nothing has been said about machine Y....So not sufficient.

Statement 2: Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.
4/X = 6/Y
Substitute in Eq(1) and get the raltionship between Total Work and X.
Finally total time....So Suffcient

Hence (B)

x = rate of machine X
y = rate of machine Y
one lot = 4x + 3y

1. we know x, but don't know y
not sufficient

2. 4x = 2*3y, so 3y = 2x
then, one lot = 4x + 2x = 6x -> 6 hours for machine x to complete the lot
sufficient

Easy to set up problems in hindsight when you can tailor the premise to the outcome. However, if you're working with some kind of framework, this would be the way to do it based on \(Rate * Time = Work\):

Given:
Machine X: \(Rx * 4 = p\), where Rx is rate of Machine X and p is part of the job
Machine Y: \(Ry * 3 = 1-p\)

Machine X would be working 4 + something to get the job done alone:
Find T+4: \(Rx * (T+4) = 1\)

(1) Clearly insufficient.

(2) Machine X did twice the job Machine Y:

\(p = 2(1-p)\)
\(p=2/3\)

Step 1:
Plug p back to given equation \(Rx * 4 = p\):
\(Rx * 4 = 2/3\)
\(Rx = 1/6\)

Step 2:
Plug Rx back to what you are looking for: \(Rx * (T+4) = 1\) :
\(1/6(T+4) = 1\) | Multiply by 6
\(T+4 = 6\)

Hi,

I got the question right (B), but, based on my estimation/calculation, the number of hours of X should be 5,5.
In fact, if X can fills part of the lot in 4 hours and in 3 hours X can produces twice the bottles that Y produces in 3 hours is possible to affirm that X in 1,5 hour can complete the task of Y,so, filling the whole lot in 5,5, hours.

Can you please show me why the hours X taks are 6 and not 5,5?

Thanks in advance!

(1) Machine X produced 30 bottles per minute.
Machine x worked for 4 hours so it made = 30*60*4 bottles
How many bottles Y made is unknown ; Total number of bottles unknown
Without knowing total production we cannot use X's rate to derive its time
INSUFFICIENT


(2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.
Machine x produces 2B bottles and rest of the remaining B bottles were made by Y
therefore total number of bottles = 3B
now X makes 2B bottles in 4 hour therefore it can make the remaining B bottles in 2 hours
total time X takes to complete the production is 4hour + 2 hour = 6 hours

SUFFICIENT

ANSWER IS B
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Hi Karisma, Thanks for the explaination. I wanted to know if we could solve this problem by assuming values for work? Is it possible?

We cannot assume a value for work here. Note that the data gives us values for time and rate. So work done can be found out. It's not possible to assume any random value for work.
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(1) No information is given about Y. Insufficient.
(2) Rates of X and Y are 1/4 and 1/3.
Let, Y produced 100 unitis
and so, X produced 200 units
Sufficient.
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Hi Karishma how did you directly reached to the conclusion that X filled 2/3 of the lot.