How Many Hours Do 2 Pipes Take to Fill a Tank? A Comprehensive Guide

When dealing with fluid transfer systems such as filling tanks, understanding the rate at which the tanks can be filled is crucial for efficient operation. In this article, we'll explore the problem of determining how many hours it takes for 2 pipes to fill a tank, given that 6 pipes can fill it in 4 hours. We will break down the solution step-by-step and discuss underlying principles.

Understanding the Problem

The problem involves a system where 6 water pipes can fill a tank in 4 hours. We need to determine how long it would take for 2 pipes to accomplish the same task.

Determining the Rate of Water Pipes

We start by defining the rate of water flow for the pipes. The rate can be thought of as the amount of tank filled per unit of time.

Rate of 6 pipes:

First, we calculate the rate at which 6 pipes can fill the tank.

Rate of 6 pipes 1 tank / 4 hours 1/4 tanks per hour

Rate of 1 pipe:

Next, we determine the rate of a single pipe.

Rate of 1 pipe (1/4 tanks per hour) / 6 1/24 tanks per hour

Rate of 2 pipes:

We now find the rate at which 2 pipes can fill the tank.

Rate of 2 pipes 2 × (1/24 tanks per hour) 2/24 1/12 tanks per hour

Calculating Time for 2 Pipes to Fill the Tank

Using the rate of 2 pipes, we can now calculate the time required to fill the tank.

Time 1 tank / (1/12 tanks per hour) 12 hours

Therefore, it would take 12 hours for 2 pipes to fill the tank.

Alternative Methods and Interpretations

1. Direct Proportionality

A second approach involves recognizing that the total capacity in pipe-hours remains constant. Since 6 pipes take 4 hours to fill the tank, the total capacity is 6 pipes × 4 hours 24 pipe-hours.

If we reduce the number of pipes to 2, the time required would be:

Time 24 pipe-hours / 2 pipes 12 hours

2. Inverse Proportionality

We can also interpret this as an inverse proportion problem. If the number of pipes is reduced, it will take longer to fill the tank. Since 6 pipes take 4 hours, 2 pipes would take:

Time (6 pipes / 2 pipes) × 4 hours 12 hours

Equation-Based Approach

We can further solidify our understanding using an equation. Let ( p ) be the number of pipes, and ( t ) be the time required. The relationship can be described as inverse proportion:

( p propto frac{1}{t} )

( p frac{k}{t} )

Given that 6 pipes take 4 hours, we find ( k ):

( 6 frac{k}{4} )

( k 24 )

( p frac{24}{t} )

Now, substitute ( p 2 ) to find ( t ):

( 2 frac{24}{t} )

( 2t 24 )

( t 12 )

Thus, it will take 12 hours for 2 pipes to fill the tank.

Conclusion

The calculations show that it would take 12 hours for 2 pipes to fill the tank, given that 6 pipes can fill it in 4 hours. This example demonstrates the principle of inverse proportionality and provides a comprehensive breakdown of the solution.

Keywords

Water pipe capacity, tank filling time, inverse proportion