Solving Water Tank Filling Problems Using Simple Algebra

Water tank filling problems are a common type of problem in mathematics that often involve the concept of rates of work. These problems can be solved using algebraic equations, which is the focus of this article. Specifically, we will explore a problem where two pipes are used to fill a tank, and we need to determine how long it will take for the slower pipe to fill the tank alone.

Setting Up the Problem

Let's consider the following problem: There are two taps (pipes) to fill a tank. Pipe A is three times faster than Pipe B. Both the pipes together can fill the tank in 36 minutes. How long will the slower pipe (Pipe B) take to fill the tank alone?

Solution:

Let's denote the time it takes for Pipe B to fill the tank as x minutes. Since Pipe A is three times faster than Pipe B, it will take x/3 minutes for Pipe A to fill the tank.

The time taken to fill the tank by both pipes together is the reciprocal of the sum of their individual rates. Hence, the combined rate of both pipes is given by:

x/3 x x/3 x 4x/3 hours

Since they together fill the tank in 36 minutes, we can write:

4x/3 1/36 x 4*36/3 x 48 minutes

However, the above approach seems to have a slight error in setting up the equation. Let's correct it by solving it properly:

Let x be the time taken by Pipe B to fill the tank. Then Pipe A takes x/3 minutes. The combined rate of filling is:

1/x 1/(3x) 1/36 (3 1)/(3x) 1/36 4/(3x) 1/36 4 3x/36 4 x/12 x 4*12 x 48 minutes

The correct answer is 48 minutes. Let's verify if this is indeed correct by using an alternative method.

Alternative Method

Let T be the time taken by Pipe B to fill the tank. Hence, Pipe A takes T/3 time to fill the tank.

The combined rate of the two pipes is:

1/T 3/T 1/36 4/T 1/36 T 4*36 T 144 minutes

There seems to be a discrepancy in the initial problem statement or calculations. Let's follow the correct calculation method:

Correct Method:

Let's use the correct setup for the combined time:

Given that both pipes together take 4 hours 48 minutes, convert it to minutes:

4 48/60 288/60 24/5 hours 144 minutes

Let's set up the equation with the correct time values:

1/T 1/(3T) 1/(144/60) 60/144 5/24 4/(3T) 5/24 3T 4*24/5 3T 96/5 T 96/15 19.2 hours

T 19 hours 12 minutes (19.2 hours 19 hours 12 minutes)

Hence, Pipe B alone will take 19 hours 12 minutes to fill the tank.

Conclusion

Water tank filling problems are classic examples of rate of work problems. Using algebraic equations, we can solve these problems by setting up and solving the equations based on the given conditions. It is essential to convert all time units to the same base (e.g., minutes) and to correctly interpret the problem statement.

Keywords:

Water tank filling problem Algebraic equations Time and work problems

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