Question

Two pipes X and Y together can fill a cistern in 4 hours. Had they been opened separately, then Y would have taken 6 hours more than X to fill the cistern. How much time (in hours) will be taken by X alone to fill the cistern?

• A. 4
• B. 5
• C. 6
• D. 8

Hint:

Instead of solving the quadratic equation, we can substitute the options directly into the equation:
Test option (C) $t = 6$: \[ \frac{1}{6} + \frac{1}{6 + 6} = \frac{1}{6} + \frac{1}{12} = \frac{2 + 1}{12} = \frac{3}{12} = \frac{1}{4} \] Since this matches the required combined rate of $\frac{1}{4}$, $t = 6$ is correct.
Substituting options is often faster in multiple-choice exams.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This problem belongs to the topic of Pipes and Cisterns (which is mathematically identical to Time and Work).
We are given the combined rate of two pipes filling a cistern, along with a relative difference in the time they take to fill the cistern individually.
We need to set up a quadratic equation and solve for the individual time taken by pipe X.

Step 2: Key Formula or Approach: Let the time taken by pipe X alone to fill the cistern be $t$ hours.
Since Y takes 6 hours more than X, the time taken by Y alone is $t + 6$ hours.
The work done in one hour by X is $\frac{1}{t}$ and by Y is $\frac{1}{t + 6}$.
Together, they fill the cistern in 4 hours, so their combined one-hour work is $\frac{1}{4}$.
The rate equation is: \[ \frac{1}{t} + \frac{1}{t + 6} = \frac{1}{4} \]

Step 3: Detailed Explanation:

• Formulate the Equation: Combine the fractions on the left-hand side: \[ \frac{(t + 6) + t}{t(t + 6)} = \frac{1}{4} \] \[ \frac{2t + 6}{t^2 + 6t} = \frac{1}{4} \] Cross-multiply to remove the denominators: \[ 4(2t + 6) = t^2 + 6t \] \[ 8t + 24 = t^2 + 6t \]
• Rearrange into a Standard Quadratic Equation: Move all terms to one side: \[ t^2 + 6t - 8t - 24 = 0 \] \[ t^2 - 2t - 24 = 0 \]
• Factorize the Quadratic Equation: We look for two numbers that multiply to $-24$ and add up to $-2$. These numbers are $-6$ and $+4$: \[ t^2 - 6t + 4t - 24 = 0 \] \[ t(t - 6) + 4(t - 6) = 0 \] \[ (t - 6)(t + 4) = 0 \] This gives two possible values for $t$: \[ t = 6 \quad \text{or} \quad t = -4 \] Since time cannot be negative, we reject $t = -4$.
Thus, $t = 6$ hours.

Step 4: Final Answer:
Pipe X alone takes 6 hours to fill the cistern.
Therefore, the correct option is (C).