How to Solve Complex Pipe-Filling Problems: A Comprehensive Guide
When dealing with problems involving pipes filling tanks, a common approach involves understanding the rates at which the pipes fill the tank and using algebraic methods to solve for unknown variables. This article will walk you through solving such problems step-by-step, ensuring you can apply these principles to a variety of scenarios.
Scenario 1: A Pipe Fills Three Times as Fast
Given a scenario where one pipe can fill a tank three times as fast as another, let's explore how to determine the time required for the slower pipe to fill the tank alone.
Problem Details:
The faster pipe is three times as fast as the slower pipe. Together, the pipes can fill the tank in 36 minutes.Solution:
Let's denote the rate of the slower pipe as r tanks per minute. Then the rate of the faster pipe is 3r.
Step 1: Combined Rate
The combined rate when both pipes are working together is r 3r 4r tanks per minute.
Step 2: Equation from Combined Rate
Given that the combined rate can fill the tank in 36 minutes, we have: $$ 4r times 36 1 quad text{since they fill one tank} $$
This simplifies to:
$$ 144r 1 $$Step 3: Solve for r
Dividing both sides by 144 gives:
$$ r frac{1}{144} quad text{tanks per minute} $$Step 4: Time Calculation for Slower Pipe
The slower pipe's rate is r, so the time t it takes for the slower pipe alone to fill the tank is:
$$ t frac{1}{r} frac{1}{frac{1}{144}} 144 text{ minutes} $$Thus, the slower pipe alone will take 144 minutes to fill the tank.
Scenario 2: Combined Time of 1.5 Hours
Another common problem involves determining the time each pipe takes alone when their combined working time is known.
Problem Details:
A section of pipe can fill a tank three times as fast as another. When both are used, it takes 1.5 hours to fill the tank.Solution:
Let's denote the faster pipe's rate as V/x and the slower pipe's rate as V/(3x), where V is the total capacity of the tank.
Step 1: Combined Rate Equation
The combined rate is given by:
$$ frac{V}{x} frac{V}{3x} frac{V}{1.5} quad text{since they fill one tank in 1.5 hours} $$Simplifying, we have:
$$ frac{3}{x} frac{1}{x} frac{2}{3} $$Further simplifying, this becomes:
$$ frac{4}{x} frac{2}{3} quad Rightarrow quad x 6 text{ hours} $$Thus, the faster pipe takes 2 hours, and the slower pipe takes 6 hours.
Step 2: Verification
Checking with the combined rate confirms the solution:
$$ frac{1}{2} frac{1}{6} frac{4}{6} frac{2}{3} quad text{which matches the given condition} $$Additional Scenario: Complex Combined Rate
For a more complex scenario involving rates that mix fractions, let's consider an example with a different time frame.
Problem Details:
One section takes x hours to fill the tank. The other section takes 3 x hours to fill the tank. Together, they fill the tank in 1.5 hours.Solution:
Let's denote the time taken by the slower pipe as x hours and the faster pipe as 3x hours.
Step 1: Combined Rate Calculation
The combined rate is given by:
$$ frac{1}{x} frac{1}{3x} frac{1}{1.5} quad text{since they fill one tank in 1.5 hours} $$This simplifies to:
$$ frac{3}{3x} frac{1}{3x} frac{2}{3} quad Rightarrow quad frac{4}{3x} frac{2}{3} quad Rightarrow quad x 2 text{ hours} $$Therefore, the slower pipe takes 2 hours, and the faster pipe takes 6 hours.
Conclusion
Understanding how to solve pipe-filling problems involves key algebraic techniques such as setting up the correct combined rates and solving for individual times. By practicing these types of problems, you can improve your problem-solving skills and ensure that you can tackle similar challenges with ease.