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The tank runneth over - How to solve word problems step by step

Two pipes "A" and "B" fill a tank in 15 hours and 20 hours respectively while a third pipe "C" can empty the full tank in 25 hours. All the three pipes are opened in the beginning, after 10 hours, "C" is closed. In how much time, will the tank be full ?

Here is a classic example of a word problem that deals with rate ( and fractions). It can appear quite daunting for someone who is out of school for some time, or even for someone who is in grade school! So lets figure out a method to turn the tables around and make this perilous problem into a pleasant one. Lets begin:

Break this problem up into 2 separate parts. You will soon see why it is so much easier to visualize the problem once you have done so.

Breaking it up gives you **Two pipes "A" and "B" fill a tank in 15 hours and 20 hours respectively. Now, lets treat this as one stand alone problem. In other words, how long will it take to fill the tank if both pipes are turned on?

Intuitively ( without retorting through with the rigors of math), we should (logically)think that if Pipe A alone can fill the tank in 15 hours and pipe B in 20, pipe A has a larger flow of the two. If we were to open both pipes, then the time it takes to fill the tank full should be the average flow of the two, correct? The answer is no. Why? Lets assume we take the average time of the two flows, 15 and 20 hours. This will give us an average time of 35/2 or 17.5 hours. Think for a moment why this answer is ludicrous. If opening Pipe A alone takes 15 hours to fill the tank, adding another pipe to the filling should shorten the time to less than 15 hours, so 17.5 hours is definitely a wrong answer,although it may seem intuitive that this is the solution at first thought.

One of the ways to solve this is to think of the flow rate per hour for each pipe. For example, if it takes 15 hours for pipe A to fill the tank, then it must have taken 1 hour to fill just 1/15th of the tank. Right? Similarly, it will take 1 hour for pipe B to fill just 1/20th of the tank. If we combine them both and open the pipes simultaneously, in 1 hour, both pipes would have filled 1/15+1/20 th of the tank, or 7/60th of the tank._ Take it one step further, how long will it take for both pipes to fill the tank to the brim? Well, in an hour, both pipes can fill 7/60th of the tank. How long will it take to fill the entire tank, or 60/60 th. ? Algebraically, 7/60 * x = 60/60. x= 60/7 or 8.57... hours. Now, this makes more sense, isn't it?

Step 2 . Putting it all together Two pipes "A" and "B" fill a tank in 15 hours and 20 hours respectively while a third pipe "C" can empty the full tank in 25 hours. All the three pipes are opened in the beginning, after 10 hours, "C" is closed. In how much time, will the tank be full ?

Since we've gone through some mathematical reasoning earlier, we can solve the rest of the question by putting it all together.

Again, lets think of the flows in terms of per hour. We have figured out in the last section how pipe A and B are factored into the mechanics of filling the tank. Now we add another pipe, C into the picture, but with a reverse role. Instead of adding, pipe C is capable of removing a full tank water and it can do so in 25 hours. This simply translates to pipe C is capable of removing 1/25th. of the water in the tank in 1 hour! Mathematically, if all pipes are open, we can conclude that Pipe A + Pipe B + Pipe C fills and empties the tank in ( 1/15 + 1/20 - 1/25) th in an hour. So, in 10 hours ( as per requested by the question), this much of the tank would have been filled, which is equal to (1/15+1/20-1/25)*10= 23/30 th. conversely, 7/30 empty.

Then, Pipe c is turned off. Be patient, the A-ha moment is near...

How long will it take to fill the rest of the tank ( 7/30th) if both pipes A and B are on?

Nothing has change for pipe A and B. They are both flowing at the same rates.

We know that in 1 hour, pipe A and B can fill 1/15 and 1/20 of the same tank. So now ask the question differently: How long will it take at this juncture to fill the rest of the tank or 7/30th of it?

(1/15+1/20) * x= 7/30. Solve for x and x comes out to 2 hours. Hence, it takes another 2 hours to fill the tank with just A and B after shutting off C. So the total time it takes to fill the tank, per request by the question, is 10 + 2= 12 hours.

Note by Venture Hi
1 year, 10 months ago

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Good explanation ... Archiet Dev · 1 year, 10 months ago

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@Archiet Dev thanks! Venture Hi · 1 year, 10 months ago

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