A pump can drain a pool in 7 hrs. Another pump can drain it in 9 hrs. Both pumps together will take?

[@harrydonnison5343]

I think this method is simpler: 9*7=63. So in 63 hours the pumps could drain 7+9 =16 pools so it would take 63/16 hours to drain one which is 3 + 15/16ths hours.

[@PNEfc001]

Exactly same way I did it. 63/16=3.937 👍

[@StewsReplay]

Correct but who knows to get to that? Algebra

[@kenharpal773]

Easy to understand , first pump takes 7 hrs to empty pool, second pump takes 9 hrs:- in one hour first pump will empty 1/7 of pool:- second pump will empty pool 9 hrs, in one hour pump will empty 1/9 of pool, just add 1/7 + 1/9 which equals 16/63, simply put do long division of 63 divided by 16= 3.9375 hrs

[@tarajoyce3598]

Much better

[@Dragonsbrath]

I had a difficult time grasping algebra in school but watching your explanation just clicks. I thought it was me back then, but most likely it was the method of teaching. I retired as an aircraft mechanic 2 years ago. Enjoyed the video.

[@larrymeans7046]

Graduated high school in 1959 and enjoyed math in school. Thanks for the problem and logic in solving the problem. Have to keep the brain working....😊

[@plainsimple442]

Same year here, cool.

[@franhagaman6806]

me too amazed I still remembers how to do this

[@ZenWithKen]

After one hour, 1/7 + 1/9 = 16/63 is emptied or 0.2539 of the pool. Invert 0.2539 you get 3.9375. Three hours + .9375 of an hour. 0.9375 of an hour is 56.25 minutes. 0.25 of a minute is 15 seconds. Answer: 3hours 56 minutes 15 seconds.

[@Astrobrant2]

Your method was logical and correct, so if you were filling a pool and needed to know how long it would take, that method would be fine. Or if you're taking a standardized test with multiple choice and work doesn't have to be shown, that would also be fine. But if you're in an algebra class where work has to be shown, you would lose points.

[@paradiselost9946]

@@Astrobrant2 highlighting all the errors of the current system. "points" matter. the end result is unimportant but how you got there for some reason is.

[@Astrobrant2]

@@paradiselost9946 I don't know what you mean by the "current system". As for how you get to the result, yes, that is important for students of algebra. In algebra, the student is supposed to learn the algebraic method for solving certain kinds of problems -- thus the name of the course. It's all about the process. Finally, any kind of process in algebra may be a prerequisite for learning a more complex process later on: one which a clever thinker would be unable to solve just using logic. IOW, every step is a building block for higher steps.

[@ZenWithKen]

@@Astrobrant2 Lol.

[@gavindeane3670]

​@@Astrobrant2It seems to be that they showed their working just fine. What do you think was missing? How can you describe someone's method as logical and correct while simultaneously criticising them for not explaining their method?

[@luispnrf]

Without using any math I would say around 4 hours based on the fact that if you have pumps that empies the pool in 8 hours each and use two of tham it would take exactly 4 hour. Using one that does that in 7 hours and other in 9 hours will be something close to that.

[@Metheglyn]

But not exactly! It would be pretty close (3 hours 56 minutes 15 seconds), but the result would not be the Arithmetic mean, but the Harmonic mean. This can be understood by asking: "After 4 hours, which is the average of 7 and 9, divided by 2, how much water have each of the two pumps pumped?" You will see that the 9-hour pump has pumped less than half the pool, namely 4/9ths, while the 7-hour pump has pumped MORE than half, namely 4/7ths. Now, to compare these fractions, we must find a common denominator. Let's take 63. We now see that 4/9th is 28/63rds and 4/7th is 36/63rds the two pumps together, then has pumped (28+36)/63rds=64/63rds which is 1/63rds more water than is actually in the pool. Another way could be to say "What if it was a 15-hour pump and a 1-hour-pump?" In this case, the average would still be 8 hr and half of that still 4 hr, but the 1-hour-pump would have finished the job ALONE in a fifteenth of the time the 15-hour-pump would need, so the combined result would be LESS than an hour.

[@StephenWicks-jz8pw]

@@MetheglynBoth the 7 and 9 are one significant digit. Therefore, your answer should be rounded off to one significant digit. Therefore, I suggest the answer is 4 hours.

[@Metheglyn]

@@StephenWicks-jz8pw 7 and 9 should be read as exact, not approximate, values. the exercise is ment to teach about an aspect of algebra (well, sort of algebra), not about pump efficiencies. You would not, i suspect, claim that 5/9 is 1, because 5 and 9 are both one significant digit, would you? Or 7*9=6*10¹?

[@Metheglyn]

@@StephenWicks-jz8pw Even if we accept the premise, that significant digits are an issue here, you would still need to make the correct computations before rounding. And the arithmetic mean does not yield the correct answer.

[@StephenWicks-jz8pw]

@@MetheglynNo problem. You may want to go online and review how significant digits work. I understand the point of this question was obtaining the answer using algebra. In the real world standing by your pool, I would have used the average time of the two pumps. The average would be 8 hours. Since you are using both pumps the answer would be 8 hours divided by 2 pumps which would equal 4 hours.

[@laurencekonersmann639]

Another approach. t = time t/7 is the fraction of time for one pump t/9 is the faction of time for the other added together is the whole job (1) t/7 + t/9 = 1 multiply both sides by 63 9t + 7t = 63 16t = 63 t = 63/16 = 3 15/16 hours

[@heidicameron2015]

That’s the only way I knew to figure it out.

[@treborrelluf]

much better

[@jtperkins1948]

Very concise and helpful in visualizing the logic in reaching the solution.

[@gnomesanemann6705]

The primary problem I always had in math class was the annoying habit the teachers had of repeatedly stating the problem over and over and over again before they even began to explain the method of solution. When they finally got around to teaching the method, I was already bored to tears and nearly enraged at their redundancy to the point of not caring anymore.

[@lastxp]

On a hot day with kids standing around I can tell you now it takes 1 eternity to full a pool and no one cares how long to drain it.

[@MikeStallings2023]

Anyone who would need this formula would also need to know how to derive it. That was the only important aspect and that was totally skipped in this video. Yes, if you've seen one of these problems you've seen them all, but deriving it requires important insights into use of rates.

[@graemesmith5195]

Just assume the pool is 63 gallons. One pump works at 9 gallons per hour. The other pump works at 7 gallons per hour. So together they pump 16 gallons per hour. 63/16 = 3.94 hours.

[@danburch9989]

The answer to these types of problems was explained to me 60+ years ago that it is determined by the reciprocal of the sum of the reciprocals. A reciprocal of a number is 1 divided by that number. The reciprocal of 9 is 1÷9 = 0.111. The reciprocal of 7 is 1÷7 = 0.143. Their sum is 0.254. The reciprocal of 0.254 is 1÷0.254 = 3.938

[@wilmot0]

I graduated high school in 1978 and am 64 years old and I wish my teachers were as cool as you. I enjoy the mental work out. 👍

[@Poult100]

Adding the reciprocal of both pump rates to get the reciprocal of the combined pumps. Then revert by dividing by one. This is the same method used to fine the total resistance of resistors in parallel. Very useful!

[@andysymes8406]

I did this in my head in about 10 seconds. 1/7 + 1/9, and invert the result to get 63/16, or just under 4 hours.

[@vincentrobinette1507]

Product over the sum. 9X7=63 (put that product in the numerator) 9+7=16 (put that sum in the denominator) 63/16=3.9375.

[@rubencantu5067]

First you calculate how much water each pump is pumping by the hour. Smaller pump is pumping 11.11 percent per hour, large pump is pumping 14.28 percent per hour, they combine for a total of 25.39 percent each hour. Divided by 100 is 3.93 hours.

[@MrMousley]

I did it like this ... If they are left to drain pools one after another then the first time they will finish a pool at the same moment will be after 7 x 9 = 63 hours and that will be 7 + 9 = 16 pools and 16 pools in 63 hours is 63/16 3.9375 hours per pool just under 4 hours

[@gavindeane3670]

Yep. I prefer that method too. It seems more intuitive to me, and not hard to get from 63/16 hours to the exact time of 3 hours 56 minutes 15 seconds.

[@KM-gf8oy]

Start with the concept volume equals pump rate x time. V=R x T. The volume is 1 pool so for pump1 1=R1x7, so R1=1/7 or .143 for pump2, 1=R2x9 so R2=1/9 or .111 The two pumps working together will pump at a rate of 0.143 + .111 = .254. back to V=R x T. 1= .254 x T. T= 3.93 hours.

[@gavindeane3670]

@@KM-gf8oy Or you can do it the easy way like @MrMousley did :)

[@phantom4255]

That's just how I solved it - great minds think alike! 👍😂

[@btpcmsag]

Viewers' comments on this video are very interesting and helpful. I read them first, then watched the video, and the combined effect was twice as informative as one or the other would have been. Great teamwork!! 🙂 👌

[@kornami8678]

Equivalent formula is P1 x P2/ P1 + P2, 7x9 / 7+9 = 63/16 = 3.93

[@mylittlepitbull3143]

It better be less than 4 hours because we got lunch coming up.

[@tomtke7351]

Given: Two pumps P.1 = 1pool/7hrs = 0.1428pl/hr P.2 = 1pool/9hrs = 0.1111pl/hrs COMBINED they perform: P.1 + P.2 = 0.1428+0.1111=0.2539pl/hr inverse = 1÷ X pl/hr = hr/pl = 1/0.2539pl/hr = 3.9386 hr/pl Alternate: = 1÷(1/9+1/7) = 1÷((7+9)/(7×9)) = 63÷16 = 3.9375 hr/pl 0.938 × 60 = 56min Total = 3hr 56min

[@glennphelps9300]

You explained it the correct way. Always show units. A+

[@gavindeane3670]

You lost 15 seconds in your decimal approximations in the second approach.

[@jimuber9935]

John 🎉

[@simonfly1585]

Why complicate the problem with fractions just take the average of the two being 8 hrs and divide by 2 = 4 hrs job done , 👍

[@gavindeane3670]

@@simonfly1585 The main reason for not just taking the average of the two pump rates is that it doesn't give the right answer. It happens to be only a little bit wrong in this case because the pump rates aren't very different, but in general the average of the two rates has nothing to do with the correct answer.

[@larryparent7128]

Without using algebra, I estimated if they average 8 hours then 2 would be 4 hours, which is a good estimate.

[@Metheglyn]

Yes, Rather good. What if you had a 1-hour pump and a 15-hour pump?

[@dateland-twopalmspress755]

yeah, same here -- i guesstimated 4 hours -- took me about 5 seconds

[@leebrown6247]

I assigned an arbitrary value to the volume of the pool of 6300 gallons. At that volume, one pump is 900gal/hr. The other one is 700 gal/hr. Add them together and you can move 1600 gal/ hour. 6300/1600=3.9375 hours, which is 3 hours, 56 minutes, 15 seconds.

[@mojokahma6931]

I used 6300 divide by 9 = 700 units per hour, 6300 divide by 7 = 900 900+700 = 1600. 6300 divide by 1600 = 3.9375 hours

[@mojokahma6931]

used 6300 as both 7 and 9 divide into it evenly

[@mauricegold9377]

In one hour pump 1 alone pumps 1/7 of the pool; pump 2 alone pumps 1/9 of the pool. Together in one hour they pump 1/7 + 1/9 =16/63 of the pool. So to find how many hours, invert 16/63 to give you 63/16 hours=3.94 approx. Figuring out the combined rate per hour is the key to understanding it. Just positing 1/some number without clarification doesn't lead to understanding in my view.

[@brlyjo]

I set each pump to flow 1gal/hour. I imagined a 63 gallon tank. 7+9 is 16. 63/16=3.9375

[@Like_An_Eagle]

My logic, before seeing the answer... 2 pumps working together would divide the time in half ("IF" they drained at the same rate, that is). Half of 7 is 3.5 Half of 9 is 4.5 The average would be 4. My guess is 4 hours.

[@robertboyd3863]

That is how I did it

[@omanamohandas8385]

Less than 4 hours only.

[@JayaseelanPatchy]

There is no guessing and approximately in maths .

[@Metheglyn]

But if you look at it like this: First both pumps start on each their half of the pool. After 3 and a half hour, 7-hrp (seven-hour-pump) finishes its half, while 9-hrp still got an hour of work left. 7-hrp goes and helps 9-hrp finish up. Now, if they both worked at the same speed as 9-hrp, they would finish in half an hour. But 7-hrp works FASTER. So they finish in less than 4 hours.

[@carouselcoinleisure4141]

As an electronics engineer, it’s the same formulae for resistors in parallel, overall circuit resistance. The logic is the answer is always lower than the lowest resistance, so in this problem of two pumps, the answer is always going to be lower than 7hours.

[@MovieMakingMan]

You probably remember the color codes for electronic parts: The politically incorrect version: Bad Boys Rape Our Young Girls But Violet Gives Willingly for Gold and Silver.

[@theodoresweger4948]

I noticed the same thing >>

[@vincentrobinette1507]

Know also, that the exact same formula applies for putting capacitors of different values in series! The total capacitance will always be less than the smallest value.

[@Metheglyn]

@@MovieMakingMan Relevance?

[@MovieMakingMan]

@@Metheglyn It’s educational and interesting. Every day you learn something new is a good day.

[@MorgKev]

You offer solution by producing a formula with zero understanding. First pump: 1/7th of the pool per hour. Second pump: 1/9th of the pool per hour. Both together: 1/9th plus 1/7th of the pool per hour. 1/9 + 1/7 = (7+9)/63 of a pool per hour. That's 16/63 of the pool per hour. Same as 63/16 hours per pool.

[@ketoman4057]

Correct! This is the best way to explain it. Learning by rote sucks when the formulae can be easily explained.

[@ryancraig2795]

To make it easier to understand, I would first restate the capacity of the pumps in "pools per hour". Then, obviously, the total pump capacity is those two numbers added. Take the reciprocal of that sum and you've got "hours per pool", the number we're looking for.

[@topoftheleague8814]

Answer for me is far simpler! You have 9 and 7, so you imagine a pool amount in litres which is common to 9 and 7... Easy aye! 63! So I just divided 63 by the 9 + 7 (16) so the answer is 3,9375 hrs... Surely that is the simple maths for this!!

[@clambeaker17]

I got the correct answer by looking at the problem as 2 resistors in parallel. 1/(1/7)+(1/9).

[@tomtke7351]

YES YES YES. ... these problems are EXACTLY like resistors in parallel.

[@darkdelta]

Spot on!!

[@OhmMyGod1]

Yes. I used the product over sum formula.

[@keithterry2169]

Exactly. Take reciprocals of each time, add them and then take a reciprocal of the result. It doesn't take 16 minutes !

[@georgecattani7199]

1/7+1/9=1/x. X=63/16 =3.93 hrs

[@baseballsux2]

So I arrived at the 3:56:15 using math. But as an engineer, that is not reality. As the pool drained, the pumps would become less efficient, particularly once the pool is less than 5% full. But that’s a physics problem, which does include math, but more than this simple arithmetic.

[@djnixmixMTL]

Then it also depends if it's a centrifugal or positive pump😂

[@stevemcdonald1033]

The gradual lowering of pump efficiency would be built in to the time figure given for the two pumps and would not need to added to the calculation.

[@cnsdhar]

When we solve this mathematical problem, we make an assumption that each pump discharge rate is constant

[@nickbarton3191]

I'm engineer, 4 hrs is close enough. Is this really algebra? Isn't it about adding two fractions, then inverting?

[@AndyHullMcPenguin]

3hrs 56 mins 15 seconds ... if you stick to fractions, and remember that an hour consists of minutes and seconds you can work that out without a calculator. 3 hrs and 15/16ths of an hour, which is 56 1/4 minutes or 56 minutes and 15 seconds.

[@guypehaim1080]

Two 7 hour pumps can do it in 3-1/2 hours. Two 9 hour pumps can do it in 4-1/2 hours. Average the two times: (3-1/2 + 4-1/2)/2 = 4 hours.

[@JK-zl7vv]

3 hours 56 min would be the answer, since you have two pumps which doubles the volume being pumped, but one pumps a little slower by 2 hours, but since the other pump is still pumping during those two hours, that cuts the two hours in half, with one pump slightly larger than the other, which shortens the remaining hour is how you need to think about this mechanically, so mathematically it would be 3:56 hours.

[@therighthandmann]

I took the long way home. I assumed a 10k gal pool and figured the hourly rate of each pum, then added those products together to make the denominator for the 10k gal pool drain rate. I did it on a calc, but rounded things here and there and came up with just under 4hrs.

[@ordinanse]

One pump does 1/7 pl/h, the other does 1/9 pl/h. Or: one does 9/63 pl/hh, the other 7/63 pl/h. Combined capacity is 16/63 pl/hr. So they need 63/16 h/pl, or 3.938 h or 3:56:15

[@btpcmsag]

That's it in a nutshell. Thanks! 🙂👍

[@alphafountain]

Forgot to round up. Love your work!

[@TXMURF]

How about this… use an arbitrary pool size, let’s say 20,000 gallon pool. 20,000 divided by 7 hrs. Is 2,857 gallons an hour. Then same size pool, 20,000 divided by 9hr. Is 2,222 gallons an hour to drain. Add the two together and divide 20,000 by that number and there is your answer. 20,000 divided by 5,079=3.93 hrs. How’s that for common sense.

[@kelvinwilliams2908]

How about this - Pump A empties 1/7 of the pool in 1 hour while pump B empties 1/9 of the pool in 1 hour. Together they can empty 1/7 +1/9 of the pool in 1 hour which is roughly 1/4. So if two pumps working together can empy 1/4 of the pool then in 4 hours they can empty the whole pool!

[@davannaleah]

An analogy to working out this problem is to think of 2 resistors in parallel. The equivalent resistance 1/(1/7 + 1/9) gives 3.9375 hours

[@ramaamofarah8967]

Exactly 💯 x= 3 15/16 hours

[@h2onoah]

In reality, running two pumps on one circuit would probably blow the breaker...so the answer is going to 7 hours plus the time to trouble shoot the electrical issue.

[@robertrudick2492]

Thank you . Yiur tye nest Professor

[@irfankhoso4015]

Lets suppose the pool's storage capacity is 100 gallons of water The first pump empties the pool in 7 hours, means it sucks out 100÷7=14.5 gallons per hour The second pump takes 9 hour to complete the job, means it sucks 100÷9=11 gallons per hour so both pumps will suck 25.5 gallons per hour And both working ssimultaneously will take approximately 4 hours to empty the pool

[@OldFella547]

Hi This is The way an Engineer would work it out on site The Correct answer of mine was 3.93 Hrs I wouldn't use Algebra

[@Metheglyn]

@@OldFella547 If it was only about pool-pumping times, this would be fine, I suppose. But I don't feel a 1/63rd deviation in anything important engineers do elicits a lot of confidence.

[@OldFella547]

@@MetheglynThat's what I like about free speech people can give their views however it seems some people don't realize Engineers too study & have a huge knowledge of Maths ,Algebra ,Physics & many other subjects. Common sense comes in handy also I quite like the Maths problems which John posts

[@Metheglyn]

@@OldFella547Well, I, for one, am well aware that engineers use a lot of advanced math in (almost) every field. I do believe that many get by fine by using standardized formulas, but then again a lot of them DO need a deeper understanding of subjects like statistics, probability and calculus in their everyday work. And - at least around here - they ALL have been taught these fields on quite an advanced level. My comment was directed against your original answer of 4 Hrs, which I took to mean that you had simply taken the arithmetic mean of the two pumping speeds, instead of the harmonic mean, as would be correct. But I now see that you have updated your answer, so my point about 1/63 error being excessive in some fields of engineering, is moot.

[@btpcmsag]

100÷7=14.286, not 14.5

[@tehatte]

Don’t turn this into an algebra problem. No letters (variables). Just use unit rate and logics to solve it. Explain 1/7 and 1/9 mean. Add them up. Flip it, explain why, and done. Students tend to forget algrebra after school but they’ll remember the logics, the meanings.

[@Metheglyn]

And the Algebra helps them do this. This problem, however, barely touches upon algebra, at least the way it is formulated. It is basically "take the harmonic mean of the two pump's pool-emptying times, and divide by number of pumps". Using variables as placeholders in a formula does not make it algebra, no more than saying "'there are 7 bricks on the left pallet and 9 bricks on the right pallet. How many bricks are there?' The formula for this type of problem is a+b=x Insert the two numbers at a and b, like this a=7, b=9; 7+9=x. There are 7+9=16 bricks." is algebra. Using algebra to solve the problem would entail deriving the formula (for geometric mean or pool-emptying time) from the stated premise. However, it will probably be a good exercise for getting used to working with variables.

[@tehatte]

@@MetheglynIt’s still Algebra because he came up with an equation with a variable in it and solve for it. Maybe because this problem was given in an Algebra class they had to learn how to use equation and variables. But my point is, truly this problem can be solved by 5th or 6th graders without using Algebra. Same problem: it takes A 4’ to eat a pizza and same for B. If they together eat 1 pizza how long will it takes? In one minute each of them can eat 1/4 of the pizza, so both can eat 1/4+1/4=1/2 of the pizza, so it’ll take them 2 (invert 1/2) minutes. Exact same problem. No Algebra.

[@Ayelmar]

Solved in my head at the thumbnail, I'm getting around 3 hr, 56 minutes. The first pump can remove 1/7 of the water per hour. The second pump can remove 1/9 of the water per hour. Tpgetjer, adding 1/9 + 1/7 = 16/63 of the water pumped per hour. In 4 hours, the two pumps would remove 64/63 of the water, so we take away 1/16 of an hour, just under 4 minutes, gives us 63/63 of the water removed in about 3:56. (fixed typo in the last line)

[@Astrobrant2]

As I mentioned to another commenter, that method is logical and correct, but if you're in an algebra class and have to show your work, you would lose points.

[@thejashorianguyofficial]

Very simple: 7*9/7+9=63/16=3.9375 hrs(Ans)

[@thomasjohnson1959]

Solving the equation for two 9 hour pumps working together giving 4.5 hours would prove the equation's logic. Then simply replacing one 1/9 with 1/7 and solve again. That should help a student understand how and why the equation works.

[@disc999golfer999]

If you have a 1,000 pool, the first pump can drain in 7 hours at 142.85 gallons per hour. The second pump can drain in 9 hours at 111.11 gallons per hour. Combined they do 253.96 gallons per hour. So a 1,000 gallon pool will drain in 3.9375 hours.

[@ferozenoordeen3711]

Very simple arithmetic = 1/7 + 1/9 =16/63 . So the answer is 63÷16 = 3.937..hours.

[@CaptainDangeax]

14175 seconds, 236 minutes 15 seconds, 3 hours 56 minutes 15 seconds. Simple parallel resistor formula

[@Kleermaker1000]

1/7 (first pump) +1/9 (second pump) = 1/X (both pumps together) => 1/x = 16/63 => 16X = 63 => X = 63/16 = about 3,94 (hours). So very close to 4 hours.

[@RaviShankar-1028]

Instead of by hearting formulae it would be better to explain in the following manner: Let v the volume of the tank, then Rate of pump p1=v/7 Rate of pump p2=v/9 Therefore when working together their rate will be= v/7+v/9=(9v+7v)/63=16v/63, they will hence empty the tank together in v/(16v/63)=63v/16v=63/16=3.93hrs

[@midnightwatchman1]

Ra= pump 1 unit volume per hr Rb = pump 2-unit volume per hr. V = total pool volume = V. Therefore Ra = V/7 and Rb = V/9 also V/ (Ra +Rb ) = Tn where Tn is Time with both pumps working. Solving for Tn . Tn= (V)/(V/7 +V/9) = (V)/(16V/63)= 49/16* (V/V)= 63/16= 3.9375 hrs

[@MovieMakingMan]

The total time to drain the pool is 4 hours. They each can drain the pool an average of 8 hours. Divide by 2 to get the total time or 8/2=4. I calculated this in 5 seconds.

[@sam-y8t]

You ESTIMATED this in 5 seconds

[@MovieMakingMan]

@@sam-y8t Yeah. It took me that long to read the question, then calculate it and finally recheck my work. I could’ve done it faster :)

[@sam-y8t]

@MovieMakingMan 4 hours is a good estimate quickly. In that instance, it doesn't need to be closer than that, but if you want accuracy, you need to do it out.

[@MovieMakingMan]

@@sam-y8t Yeah, that’s true. Ballpark figures are good for most things.

[@JamesSutton-ip2oi]

3 15/16 or 3.94 hours

[@MH-qy6ri]

4 hours. 7 hours and 9 hours are each only one significant figure, so the answer can only have 1 significant figure.

[@rjwiechman]

False assumption

[@allent4021]

Without getting into algebra. 1 hour total pumped 1/7 + 1/9. 4 hours pumped 4/7 + 4/9 which is 36/63 + 28/63 which is 64/63. Divide 4 hours into 64 parts and take only 63 parts to fully pump the tank. Hence, answer is 63/64 of 4 hours which is 3 + 15/16 hours.

[@vswr10]

At first glance, less than 7 h and more than 3.5 h, the answer is 3.9375 h. One of the first things you learn in electrical engineering is how to calculate the value of resistances in parallel. R=1/(1/R1+1/R2+1/R3+....+1/Rn), for two resistors can be simplified as R1xR2/(R1+R2) Note that you can solve this problem with any number of pumps or resistors.

[@sutediheriyonoBaladMaUng]

Divide pool in 2, P1= 7/2 = 3½, P2=9/2=4½. , gather P1+P2 = 8 Divide pool in 4, P1.1= P1.2=7/4= 1¾, P2.1= P2.2=9/4=2¼, P1.1+P2.1= P1.2+P2.2= 1¾ + 2¼ = 4. etc,. than lim (dP) --> 0. t = { sygma lim (dP)}/n, If n = 4 than (1 + 2 + 4 + 8)/4 = 15/4 = 3¾ hour.

[@GoofballFlyer]

Based on arbitrary pool capacity calculate flow rate for both, add rates, and divide into pool capacity.

[@panlomito]

Nice but there is variant on this puzzle that is really interesting: We call the pumps A, B en C... A and B can do the job in 2 hours A and C in 3 hours B and C in 4 hour How fast if all pumps work together? The answer will surpise you!

[@gavindeane3670]

I’m going to strap together the A and B and call it an AB pump. My AB pump can drain the pool in 2 hours. I’ll do the same to make an AC pump (3 hours) and a BC pump (4 hours). So in 12 hours, AB can drain 6 pools, AC can drain 4 pools, and BC can drain 3 pools. So working together, AB and AC and BC can drain 13 pools in 12 hours, or 1 pool in 12/13 hours. But I’ve actually got two As, two Bs, and two Cs all working together here, so one each of A, B, and C can drain 1 pool in 24/13 hours, or 1 hour 50 minutes 46.15 seconds. Assuming I haven’t gone wrong somewhere, what’s the surprise?

[@panlomito]

@@gavindeane3670 Your answer is right, the surprise is that AB can do the work in only 2 hours, so you would expect the ABC would do it much faster, but it is only a 9 minutes gain adding pump C to AB.

[@gavindeane3670]

​@@panlomito Yeah, I thought about that after I replied. It turns out that adding C isn't much help. I think the reason I wasn't surprised is that I had no prior expectation (beyond obviously knowing that ABC was going to be some amount faster than AB). The problem is completely opaque to me in terms of being able to estimate before I start, despite the fact that I can immediately see how I need to go about solving it. I genuinely can't look at that problem statement and have any intuitive feel for whether ABC is going to be 9 minutes faster or 90 minutes faster. I think that's quite unusual: "I know exactly what I need to do here, and I have no idea where I'm going to get to when I do it".

[@davedeatherage4902]

Thanks John for the help.👍

[@Ullion404]

Just by reading the question I knew the time would have to be little below 4h, which is the average of half the both times each pump needs, because too the faster pump would "steal" a Little water from the slower pump.

[@davidbrown8763]

Thanks for the challenge, which I nailed in three steps.

[@tjvaliant]

I started thinking about using a real pool and calculating how many gallons per hour (gph) each one would be pumping. Then I would just need to at the gph or each pump together and then divide the number of gallons in the the pool by the total gph. At first, I thought I would use a 10,000 gallon pool, but that wasn't easily divisible, so I chose a 63000 gallon pool. So the 7hour pump would be pumping 9000 gph and the 9hour pump would be pumping at 7000 gph. Combined they would be pumping 16000 gph. So, for a 63,000 gallon pool the answer would be 63,000gallons/16,000gph = 3.9375 hours.

[@flashtheoriginal]

Factorised 9-7 = 2 Pump availability = 2 2x2 = 4 ANS 4 (Approx) NOT! 😢😂😂😅

[@matthewsynnott2371]

I worked on the basis that the pool was 63,000 litres, the product of 9 and 7 hours(1,000). The 9 hour pump's capacity is 7,000 litres per hour and the 7 hour pump's capacity is 9,000 litres per hour. Working together, they pump 16,000 litres per hour. Divide 63,000 litres by 16,000, answer, 3.9375 hours.

[@jeffreywhitmoyer860]

I did exactly the same thing and got the same result. Sometimes, it only matters that you answer correctly and regardless of how may gallons/litres you assume, the ratio ultimately will remain the same.

[@luillierstephane1463]

I did pretty much the same thing.

[@bigdog3628]

A is the first pump (7 hrs) B is the Second pump (9 hrs) y is the combined time for both working together (Solve for y) Y = (a * b) / (a + b) y = 63/16 or 3.9375 Hours

[@thaido3220]

Formula for this type of problem is as follows: Together Time = (Product of all Separate Times) / (Sum of all Separate Times) So simple isn’t it? In this case : Time = (7x9)/(7+9)= 63/16=3.9375

[@Metheglyn]

Indeed. It might not be as easy to remember or to translate to other situations, though.

[@garoldcounts8777]

without using algebra, I figured 4 hours. my reason was 8 being the average time and it will take half the time equaling 4. my question is which pump was able to pump water faster when working together

[@Metheglyn]

Try looking at it this way. One pump can drain the pool in 7 hours. We call it 7-hrp (for 7-hour-pump) the other can drain the pool in 9 hours. We call it 9-hrp. They start at the same time, and begin draining half a pool each. After 3 1/2 hour, 7-hrp is done with its half, while 9-hrp still got an hour of work left. 7-hrp goes to help 9-hrp finish up. If they both worked as fast as 9-hrp, they would finish in half an hour, bringing the total time to 4 hours. But 7-hrp works FASTER, so the end result is less than 4 hours.

[@PaulDeCamp]

3.9375 rounds to 3.94. Your explanation was way more complicated than necessary.

[@ladymoon425]

Nice. But were you when I was in Algebra class ?

[@butdidwedie5664]

Just shy of 4hrs. If a pool drains in 7hrs that means in 3.5 50% is drained. If it takes 9hrs that's 4.5 hrs 50% is drained. The difference is 1hr. So if both pumps run together it will be faster than 4.5 hrs but longer than 3.5. So 4.

[@tinpony9424]

Didn't realize I was using algebra.... Giving the pumps a g/h rating, using both would produce 16 g/h. Working for a common lowest volume in the pool, multiplying the rate of the pumps gives a lowest volume of 63 gallons. Dividing 63 gallons by 16 g/h gives us 3.9375 hours.

[@Syesavage]

You calculate average time for the pump (9+7)/2 = 8hrs per machine to clear pool If the work together they take half the time 8/2 = 4hrs

[@samzach2057]

About 4 hours, and unless I’m taking medication every time I drain the poll, that’s close enough.

[@pas6295]

Please note first does it in 7 hours. Meaning in 1hour it can empty 1/7th of the pool. Similarly the second does itin one hour1/9th. Together 1/9+1/7=9+7the whole divided by 63. In one hour they do 16/63.for full tank. They take 63/16. Hours. Hours .

[@geoffmcrorie90]

16 minutes for a simple equation that could be explained in 2 minutes. It's baby stuff. So much waffle until you got to the end.

[@btpcmsag]

I knew a man who attempted to be a school teacher, but he consistently explained the same problems and how to solve them by using the exact same words, over and over, getting more emphatic each time. He lost his teaching job. Here at minute 7:00 John repeats his explanation of a formula calmly, using a different method the second time. I have always admired professors who were wont to do this.

[@LarryMaddux]

Thanks😊

[@btpcmsag]

3:56 testing . . . When taking a test, like the SAT, reading each question 3 times will ensure you RUN OUT OF TIME!!

[@lc79tourer26]

3 hrs 59minutes and 15 seconds

[@willyspinney1959]

One pump empties the pool in 7 hours and the other pump takes 9 hours. Therefore, the average time taken for each pump is 8 hours and half of that for both pumps working together is 4 hours.

[@btpcmsag]

Okay, that's it. YOU can teach the class. But you have to draw mad faces well.

[@davannaleah]

This is a physical analogy of 2 resistors in parallel

[@thorbjrnhellehaven5766]

I think about it slightly different. I think about units, and inversion of units. The math is the same, but I don't think about the work formula. Pump A use 7 hours : 1/7 pool/hour Pump B use 9 hours : 1/9 pool/hour Together: (1/7 + 1/9) pool/hour = 16/63 pool/hour : 63/16 hour/pool 63/16 h = 3 h + 15/16 h = 3h + ... 15*60/16 m = 56 m + 1/4 m = 56 m + 25 s Answer: 3 h 56 m 25 s

[@Metheglyn]

How do you use an ant to invert a unit?

[@thorbjrnhellehaven5766]

@@Metheglyn sorry, "and".(Edited text to fix it) I appreciate that "ant" didn't make much sense.

[@Metheglyn]

@@thorbjrnhellehaven5766 👍

[@willybecker7521]

The line through (a,a^2) and (b,b^2) hits the x-axis at ab/(a+b). Is that just a coincidence, or is something deeper going on?

[@willybecker7521]

Btw, if we define a multiplication * on the positive real numbers by a*b = ab/(a+b), then * is commutative and associative, but does not distribute over addition.

[@sih9696]

Is it the same pool or two different pools?

[@gronver7254]

An easier method is to take the middle between the 2 numbers (8) and divide that by 2 which equals 4.

[@gavindeane3670]

That's very easy, but it doesn't give you the right answer. It happens to be close to the right answer in this case because both pumps have fairly similar rates. But in a more extreme case, e.g. one pump can drain the pool in 1 hour and the other can drain it in 15 hours, your method would still give 4 hours for the answer but that way off being correct.

[@raya.pawley3563]

Thank you

[@chrisdissanayake6979]

3 hrs, 56 minutes and 15 seconds

[@kulturfreund6631]

12 minutes into the video and you’re still repeating what the "prom“ is.

[@torreyvent9429]

You like four hours. Area times velocity equals volume.

[@luillierstephane1463]

There something I don't really understand, it's the little difference between the result we got using algebra and the result we can get using much more basic calculation. There the thing : If you drain the first half of the pool with P1, it will take 7h÷2 = 3h30m Then you drain the second half with P2, which will take 9h÷2 = 4h30mn. So, using one pump after another will take 3h30 + 4h30 = 8h. Therefore using both at the same time will take 8h÷2 = 4h My question is about the little difference between 3h56mn, obtained with algebra, and 4h, obtained by very basic calculus.

[@gavindeane3670]

How long it takes for each pump to drain half a pool is irrelevant, because that's not the amount that either pump drains. The fact that your simple method nearly gets the correct answer here is just coincidence, because we happen to be dealing with two pumps that pump at similar rates. If the pump rates were less similar, your simple calculation would be further from the correct answer. For example, we could have a 6 hour pump and a 10 hour pump, or a 4 hour pump and a 12 hour pump, or a really extreme example with a 1 hour pump and a 15 hour pump. In all three of those cases, the simple calculation would suggest that both pumps working together would drain the pool in 4 hours, but the actual answers are, respectively: 3 hours 45 minutes 3 hours 56.25 minutes

[@luillierstephane1463]

@@gavindeane3670 That's a good point 👍, now I see clearly where I made an error, and I feel very much like a jackass

[@Metheglyn]

Try looking at it this way. One pump can drain the pool in 7 hours. We call it 7-hrp (for 7-hour-pump) the other can drain the pool in 9 hours. We call it 9-hrp. They start at the same time, and begin draining half a pool each. After 3 1/2 hour, 7-hrp is done with its half, while 9-hrp still got an hour of work left. 7-hrp goes to help 9-hrp finish up. If they both worked as fast as 9-hrp, they would finish in half an hour, bringing the total time to 4 hour. But 7-hrp works FASTER, so the end result is less than 4 hours.

[@Metheglyn]

@@gavindeane3670 Especially the 4-hrp & 12-hrp example is hilarious, since it suggests that the 12-hrp does exactly nothing. And the 15-hrp actually pumps extra water INTO the pool!

[@Metheglyn]

@@luillierstephane1463 Don't feel that way. The purpose of these videos (and the commentary) is to help people understand stuff that is difficult for some. The fact that you was identify the problem you percieved, and ask about it, shows that you are capable of critical thinking and an understanding of the matter at hand. And your question was held in a respectful tone, which is not always the case.

[@gilliankingston8259]

Pool 1 7 hours Pool 2 9 hours Total 16 hours ÷ 2 = 8 hours (average) ÷ 2 = 4 (3.93) hours Calculated 4 hours in my head. or 7 + 9 = 16 and 7× 9 = 63 16 ---- 63 ÷ 16 = 3.93 hours 63

[@johnvriezen4696]

I assumed the pool was 63 gallons so the 7 pump would pump 9gal/hr. Similarly the 9 pump pumps 7gal/hr. Combined they pump 16gal/hour. So 63 gal/ (16gal/hr)= 63/16 hr.

[@Mod-rw9cw]

The pool would be a bath with only 63 gallons of water.