Question

Two pipes A and B can fill a tank in 24 hours and 30 hours respectively. If both the pipes are open simultaneously in the empty tank, how much time will be taken by them to fill it?

• A. 14 Hours
• B. 13 Hours 20 minutes
• C. 10 Hours 5 minutes
• D. 12 Hours 10 minutes

Hint:

To solve \( \frac{xy}{x+y} \) questions rapidly in exams, use the direct formula:
\[ \text{Time} = \frac{24 \times 30}{24 + 30} = \frac{720}{54} = \frac{40}{3} = 13\frac{1}{3} \text{ hours} = 13\text{ hrs } 20\text{ mins} \]

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a standard time and work problem applied to pipes and cisterns.
The rate of work of a pipe is defined as the fraction of the tank filled by it in one hour.
When multiple pipes are opened together, their individual rates of filling are added to find the combined rate.
Key Formula or Approach:
If Pipe A fills the tank in \( T_A \) hours and Pipe B fills it in \( T_B \) hours, their combined rate per hour is:
\[ \text{Combined Rate} = \frac{1}{T_A} + \frac{1}{T_B} \] The total time taken to fill the tank together is the reciprocal of this combined rate:
\[ \text{Total Time} = \frac{1}{\text{Combined Rate}} = \frac{T_A \times T_B}{T_A + T_B} \]

Step 2: Detailed Explanation:
Let us perform the calculation step-by-step:
1. The time taken by Pipe A is \( T_A = 24 \text{ hours} \).
The rate of Pipe A is \( \frac{1}{24} \text{ of the tank per hour} \).
2. The time taken by Pipe B is \( T_B = 30 \text{ hours} \).
The rate of Pipe B is \( \frac{1}{30} \text{ of the tank per hour} \).
3. Calculate the combined rate when both are open:
\[ \text{Combined Rate} = \frac{1}{24} + \frac{1}{30} \] Find the Least Common Multiple (LCM) of 24 and 30:
The prime factorization is:
\( 24 = 2^3 \times 3 \)
\( 30 = 2 \times 3 \times 5 \)
\( \text{LCM} = 2^3 \times 3 \times 5 = 120 \)
Now, rewrite the fractions with the common denominator:
\[ \text{Combined Rate} = \frac{5}{120} + \frac{4}{120} = \frac{9}{120} = \frac{3}{40} \text{ of the tank per hour} \] 4. Find the total time taken to fill the empty tank:
\[ \text{Total Time} = \frac{40}{3} \text{ hours} \] Convert this fraction into hours and minutes:
\[ \frac{40}{3} = 13 \frac{1}{3} \text{ hours} \] Since \( 1 \text{ hour} = 60 \text{ minutes} \), we have:
\[ \frac{1}{3} \text{ of an hour} = \frac{1}{3} \times 60 \text{ minutes} = 20 \text{ minutes} \] Therefore, the total time required is 13 hours and 20 minutes.

Step 3: Final Answer:
The correct option is (B).