Logical Problem

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Filling a Tank Problem

Problem Statement:

Consider a tank that needs to be filled using two pipes, A and B. The rates of filling for Pipe A and Pipe B are denoted as (A) and (B) respectively, measured in units per hour. The goal is to determine the time ((T)) it takes to fill the tank when both pipes are open simultaneously.

Denoting the variables:

• (A) is the rate of Pipe A.
• (B) is the rate of Pipe B.
• (T) is the time taken to fill the tank.

The equation representing the filling of the tank is given by:

\[T = \frac{V}{A + B}\]

where:

• (V) is the total volume of the tank.

The formula calculates the time required by dividing the total volume ((V)) by the combined rate of both pipes ((A + B)).

Example:

Let’s consider a specific scenario:

• (A = 5) units per hour
• (B = 3) units per hour
• (V = 120) units (total volume of the tank)

Substituting these values into the formula:

\[T = \frac{120}{5 + 3} = \frac{120}{8} = 15 hours\]

Therefore, it takes 15 hours to fill the tank when both Pipe A and Pipe B are open simultaneously.

Work Completion Problem

Given Information:

Let’s denote the amount of work that A, B, and C can do in one day as (A), (B), and (C) respectively.

1. The equation representing the relationship between A, B, and C is: [ A = B + C ]

2. It’s given that B and A together can complete the work in 10 days. The combined rate of B and A is (B + A), and the time taken is 10 days. Using the formula (\text{Rate} = \frac{\text{Work}}{\text{Time}}), we can express this relationship as: $ B + A = \frac{1}{10} $

3. Additionally, C alone can complete the work in 50 days: $ C = \frac{1}{50} $

Task:

Now, the goal is to find the time it will take for B alone to complete the work. We can use the fact that (A = B + C):

[ A = B + C ]

Substitute the expressions for A, B, and C from the above equations into this equation and solve for B:

\[B + \frac{1}{50} = \frac{1}{10}\]

Combine like terms and solve for (B). Once you find (B), you can use it to calculate the time it takes for B alone to complete the work using the formula: \(\text{Time} = \frac{1}{\text{Rate}}\)