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There are two inlets and one outlet to a cistern. One of the [#permalink]

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09 Mar 2014, 01:40

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There are two inlets and one outlet to a cistern. One of the inlets takes 3 hours to fill up the cistern and the other inlet takes twice as much time to fill up the same cistern. Both of the inlets are turned on at 9:00 AM with the cistern completely empty, and at 10:30AM, the outlet is turned on and it takes 1 more hour to fill the cistern completely. How much time does the outlet working alone takes to empty the cistern when the cistern is full?

There are two inlets and one outlet to a cistern. One of the inlets takes 3 hours to fill up the cistern and the other inlet takes twice as much time to fill up the same cistern. Both of the inlets are turned on at 9:00 AM with the cistern completely empty, and at 10:30AM, the outlet is turned on and it takes 1 more hour to fill the cistern completely. How much time does the outlet working alone takes to empty the cistern when the cistern is full?

The combined inflow rate of the two inlets is 1/3 + 1/6 = 1/2 cistern/hour. Thus, working together, it takes 2 hours (time is reciprocal of rate) to fill the cistern.

From 9:00 AM to 10:30 AM, so in 1.5=3/2 hours, the inlet pipes will fill (time)*(rate) = 3/2*1/2 = 3/4 th of the cistern .

Then the outlet is turned on and the remaining 1/4 th of the cistern is filled in 1 hour.

Letting x to be the rate of the outlet, we would have: 1/2 - x = 1/4 --> x = 1/4 cistern/hour, which means that it takes 4 hours the outlet working alone to empty the cistern.

Re: There are two inlets and one outlet to a cistern. One of the [#permalink]

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09 Mar 2014, 10:11

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Great explanation from Bunuel as usual.

Related alternative approach 1: The first inlet fills up the cistern in 3 hours. Assume that the capacity of the cistern is 6L and therefore the rate of the first inlet is 2L/hr. The rate of the second inlet therefore becomes 1L/hr. The water contributed by both inlets in 2.5 hours (9:00 to 11:30am) = 2.5 (2+1) = 7.5L. As the cistern is just full in this time, the 1.5L over the 6L capacity must have been drained away by the outlet in one hour (10:30 to 110:30am). So the outlet will take 6/1.5 = 4 hours to drain the full cistern. Option (E).

Related alternative approach 2: Lets calculate the time taken by the two inlets to fill the cistern. This comes out to be (6*3)/(6+3) = 2 hours. With the outlet open for one hour, the inlets take 2.5 hours to fill the cistern. This means Volume of water flowing through the outlet in one hour = Volume of water flowing in through both the inlets in 0.5 hours => Vol of water flowing through the outlet in one hour = 0.5 (1/3 + 1/6) = 1/4 of the capacity of the cistern => Time taken to empty the cistern by the outlet working alone = 4 hours. Option (E).
_________________

Re: There are two inlets and one outlet to a cistern. One of the [#permalink]

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09 Mar 2014, 19:45

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Bunuel wrote:

Then the outlet is turned on and the remaining 1/4 th of the cistern is filled in 1 hour.

Letting x to be the rate of the outlet, we would have: 1/2 - x = 1/4 --> x = 1/4 cistern/hour, which means that it takes 4 hours the outlet working alone to empty the cistern.

Answer: E.

Hope it's clear.

This is right where I got stuck. I knew that it should have taken only 1/2 an hour to fill the remaining 1/4 of the cistern. Instead it took double the time.

Can you explain your last equation? I'm confused about exactly why 1/4 and 1/2 have this relationship. Thank you.

Edit:

Is this a good explanation? If it takes double the time, that means that really only 1/4 per hour is filled. The other 1/4 is "stolen" by the outlet. So that must be the rate that the outlet pumps. Therefore, it would take 4 hours.

Similarly, if instead we had inlets pumping at a rate of 1/2 per hour and they were slowed down to 1/3 per hour, the outlet would be stealing 1/6, so the full time it would take the outlet is 6 hours? Is that correct?

Then the outlet is turned on and the remaining 1/4 th of the cistern is filled in 1 hour.

Letting x to be the rate of the outlet, we would have: 1/2 - x = 1/4 --> x = 1/4 cistern/hour, which means that it takes 4 hours the outlet working alone to empty the cistern.

Answer: E.

Hope it's clear.

This is right where I got stuck. I knew that it should have taken only 1/2 an hour to fill the remaining 1/4 of the cistern. Instead it took double the time.

Can you explain your last equation? I'm confused about exactly why 1/4 and 1/2 have this relationship. Thank you.

Edit:

Is this a good explanation? If it takes double the time, that means that really only 1/4 per hour is filled. The other 1/4 is "stolen" by the outlet. So that must be the rate that the outlet pumps. Therefore, it would take 4 hours.

Similarly, if instead we had inlets pumping at a rate of 1/2 per hour and they were slowed down to 1/3 per hour, the outlet would be stealing 1/6, so the full time it would take the outlet is 6 hours? Is that correct?

Recall that we can sum/subtract the rates.

Inflow rate of the two inlets is 1/2 cistern/hour; Outflow rate of the outlet is x cistern/hour;

Net inflow is 1/2 - x cistern/hour, hence 1/2 - x = 1/4.

Hope it helps.

P.S. Yes, your approach is correct.
_________________

Re: There are two inlets and one outlet to a cistern. One of the [#permalink]

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10 Mar 2014, 10:17

Bunuel wrote:

Recall that we can sum/subtract the rates.

Inflow rate of the two inlets is 1/2 cistern/hour; Outflow rate of the outlet is x cistern/hour;

Net inflow is 1/2 - x cistern/hour, hence 1/2 - x = 1/4.

Hope it helps.

P.S. Yes, your approach is correct.

Thank you so much! I have seen others use this approach on other Rate/Work problems, but I'm not sure I find all the formulas intuitive. I get this one now. Is there a comprehensive explanation of why we can use these shortcuts? I find that I have to reason through each individual problem, which is certainly inefficient.

Inflow rate of the two inlets is 1/2 cistern/hour; Outflow rate of the outlet is x cistern/hour;

Net inflow is 1/2 - x cistern/hour, hence 1/2 - x = 1/4.

Hope it helps.

P.S. Yes, your approach is correct.

Thank you so much! I have seen others use this approach on other Rate/Work problems, but I'm not sure I find all the formulas intuitive. I get this one now. Is there a comprehensive explanation of why we can use these shortcuts? I find that I have to reason through each individual problem, which is certainly inefficient.

THEORY 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:

In the case of tank eventually filling, the formula is 1/ f1 + 1/f2 - 1/d = 1 /F where f1 and f2 are the time taken by pipes 1 and 2 resp to fill, d is the time taken to drain and F is the total time taken to Fill.

Here f1 and f2 happen for 2.5 hrs and d happens for 1 hr. So the formula becomes,

Re: There are two inlets and one outlet to a cistern. One of the [#permalink]

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12 Sep 2015, 20:25

I approached it this way. at 10.30am the inlets had completed 3/4 of the work of filling the cistern. 1/4 cistern is empty. they were supposed to finish it by 11.00am.

However at 10.30am the outlet is turned on.

the result is that the cistern will now be full by 11.30am. So the inlets end up working 30 mins more to nullify the effect of outlet working for 1 hour from 10.30am to 11.30am.

so the work done by outlet in 1 hr = work done by both inlets in 30 mins. = emptying / filling 1/4 of the cistern. work done by outlet in 1 hr = 1/4 of the cistern. hence outlet takes 4 hours to complete the work

Re: There are two inlets and one outlet to a cistern. One of the [#permalink]

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There are two inlets and one outlet to a cistern. One of the [#permalink]

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20 Oct 2017, 09:12

HarveyS wrote:

There are two inlets and one outlet to a cistern. One of the inlets takes 3 hours to fill up the cistern and the other inlet takes twice as much time to fill up the same cistern. Both of the inlets are turned on at 9:00 AM with the cistern completely empty, and at 10:30AM, the outlet is turned on and it takes 1 more hour to fill the cistern completely. How much time does the outlet working alone takes to empty the cistern when the cistern is full?

3/2 hour*1/2 combined rate=3/4 of cistern filled by 2 inlets together by 10:30 with no outlet running, 2 inlets would need another half hour to fill last 1/4 with outlet running from 10:30, 2 inlets need another hour to fill last 1/4 thus, outlet empties 1/4 of cistern in one hour, and entire cistern in 4 hours E