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Determining the Time to Fill Half of a Cistern Using Pipe A
Determining the Time to Fill Half of a Cistern Using Pipe A
Understanding the efficiency of pipes in filling containers is crucial in various applications, from industrial processes to everyday tasks. This article explores the problem of determining the time required for Pipe A to fill half of a cistern, given that it fills three-quarters of the cistern in 15 minutes. We will delve into the mathematical analysis and solve the problem step by step.
Making Use of Pipe A's Filling Rate
The problem at hand involves finding the time it takes for Pipe A to fill half of a cistern. To begin, we need to calculate the filling rate of Pipe A.
Calculating the Filling Rate of Pipe A
First, let's express the filling rate of Pipe A mathematically. Given that Pipe A fills three-quarters (3/4) of the cistern in 15 minutes, we can determine its rate as follows:
rate frac{frac{3}{4} , text{cistern}}{15 , text{minutes}} frac{3}{4 times 15} frac{3}{60} frac{1}{20} , text{cistern per minute}
Calculating the Time to Fill Half of the Cistern
Given that Pipe A fills 1/20 of the cistern in one minute, we can now determine the time it takes to fill half (1/2) of the cistern. The calculation is straightforward:
time frac{text{amount of cistern}}{text{rate}} frac{frac{1}{2}}{frac{1}{20}} frac{1}{2} times 20 10 , text{minutes}
Alternative Method: Proportional Reasoning
Another efficient way to solve this problem involves the application of proportional reasoning. By understanding the proportion between the part of the cistern filled and the corresponding time taken, we can derive the required time.
Using Proportions
We know that:
3/4 of the cistern is filled in 15 minutes. Determine how long it would take to fill 1/2 of the cistern.Using the method of proportions:
frac{15 , text{minutes}}{frac{3}{4} , text{cistern}} frac{x , text{minutes}}{frac{1}{2} , text{cistern}}
Multiplying both sides by the respective denominators:
15 , text{minutes} times frac{4}{3} x , text{minutes} times frac{2}{1}
frac{15 times 4}{3} 2x
20 , text{minutes} 2x , text{minutes}
x 10 , text{minutes}
Hence, the time required for Pipe A to fill half of the cistern is 10 minutes.
Additional Solution Methods
There are several additional ways to solve this problem. Here are a couple of the methods:
Method 1: Ratios
We know:
3/4 of the cistern is filled in 15 minutes. 1/4 of the cistern would be filled in 5 minutes (since 15 minutes / 3 5 minutes). Therefore, 1/2 of the cistern would be filled in 10 minutes (5 minutes times; 2).Method 2: Using Ratios and Proportions Directly
Using the method of setting up a proportion:
frac{frac{3}{4}}{15} frac{frac{1}{2}}{x}
By cross-multiplying:
frac{3}{4} times x frac{1}{2} times 15
x frac{15 times 4}{3 times 2}
x 10 , text{minutes}
Thus, the time required for Pipe A to fill half of the cistern is 10 minutes.
Conclusion
This solution clearly demonstrates the methodical approach to solving problems related to the filling rates of pipes or any container. The critical part of any problem of this type is to correctly set up the ratios and proportions. By understanding and applying these principles, you can solve similar problems with ease.