Solving Tank Filling Problems Using Combined Rates

Tank filling problems are a common type of mathematical question that often requires an understanding of rates and the concept of combined rates. In this article, we will solve a specific problem involving two pipes filling a tank at different rates. We will then discuss how to approach similar problems in a general way.

Problem Statement

Two pipes, A and B, can fill a tank in 5 minutes and 10 minutes, respectively. Both pipes are opened together, but after 2 minutes, pipe A is turned off. What is the total time required to fill the tank?

Solution

To solve this problem, we first need to determine the filling rates of each pipe and then calculate the total time required to fill the tank by considering both the combined and individual rates.

Step 1: Determine the Rates

The rate is defined as the reciprocal of the time it takes to fill the tank. Therefore:

Rate of A 1/5 tank per minute

Rate of B 1/10 tank per minute

Step 2: Calculate the Combined Rate

When both pipes are open, the combined rate is the sum of their individual rates:

Combined Rate Rate of A Rate of B 1/5 1/10

To add these fractions, we need a common denominator:

1/5 2/10, so Combined Rate 2/10 1/10 3/10 tank per minute

Step 3: Calculate the Amount Filled in the First 2 Minutes

In the first 2 minutes, both pipes are open, so:

Amount filled in 2 minutes Combined Rate × 2 3/10 × 2 6/10 3/5 of the tank

Step 4: Calculate the Remaining Tank to Fill

After 2 minutes, the remaining amount of the tank to be filled is:

Remaining amount 1 - 3/5 2/5 of the tank

Step 5: Calculate the Time Required for Pipe B to Fill the Remaining Tank

Since pipe A is turned off, only pipe B is now filling the tank at a rate of 1/10 tank per minute:

Time required by B Amount remaining / Rate of B (2/5) / (1/10) (2/5) × 10 4 minutes

Step 6: Calculate the Total Time to Fill the Tank

The total time required to fill the tank is the sum of the time both pipes were open and the time pipe B was open alone:

Total time 2 minutes 4 minutes 6 minutes

Thus, the total time required to fill the tank is 6 minutes.

General Approach to Tank Filling Problems

When solving tank filling rate problems, the key steps are:

Define the individual rates of each pipe. Calculate the combined rate when pipes are opened together. Determine the amount filled in a given period when all pipes are open. Calculate the remaining amount to be filled. Calculate the time required for the pipe(s) that are still open to fill the remaining amount. Add the time both pipes were open to the time only one pipe was open to find the total time.

Understanding these basic steps and practicing with different scenarios will help you master tank filling rate problems effectively.

Related Keywords

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