Question

Pushkin, Qutub, and Ravinder together can complete a piece of work in 4 days. Ravinder and Samrat can complete the same work in 6 days. If Qutub and Ravinder take 20 and 10 days respectively to finish the work on their own, in how many days can Pushkin and Samrat complete the same work?

• A. 4
• B. 5
• C. 6
• D. 8
• E. 9

Hint:

To solve work problems involving multiple people, find their individual work rates and combine them. Then, use the formula \( \text{Time} = \frac{1}{\text{Rate}} \).

Answer 1

The Correct Option is C

Step 1: Finding the work rates.
Let the work rates of Pushkin, Qutub, Ravinder, and Samrat be \( P \), \( Q \), \( R \), and \( S \) respectively, where the work is measured in terms of "work per day." - Pushkin, Qutub, and Ravinder together can complete the work in 4 days, so: \[ P + Q + R = \frac{1}{4} \] - Ravinder and Samrat together can complete the work in 6 days, so: \[ R + S = \frac{1}{6} \] - Qutub can complete the work in 20 days, so: \[ Q = \frac{1}{20} \] - Ravinder can complete the work in 10 days, so: \[ R = \frac{1}{10} \] Step 2: Solving for Pushkin and Samrat's rates.
We can substitute the values of \( Q \) and \( R \) into the equations. From \( P + Q + R = \frac{1}{4} \), we get: \[ P + \frac{1}{20} + \frac{1}{10} = \frac{1}{4} \] Simplifying: \[ P + \frac{1}{20} + \frac{2}{20} = \frac{1}{4} \] \[ P + \frac{3}{20} = \frac{1}{4} \] Subtract \( \frac{3}{20} \) from both sides: \[ P = \frac{1}{4} - \frac{3}{20} = \frac{5}{20} - \frac{3}{20} = \frac{2}{20} = \frac{1}{10} \] Now, from \( R + S = \frac{1}{6} \), we know \( R = \frac{1}{10} \), so: \[ \frac{1}{10} + S = \frac{1}{6} \] Solving for \( S \): \[ S = \frac{1}{6} - \frac{1}{10} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15} \] Step 3: Finding the total rate for Pushkin and Samrat.
The rate at which Pushkin and Samrat work together is: \[ P + S = \frac{1}{10} + \frac{1}{15} \] Finding the LCM of 10 and 15: \[ P + S = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \] Step 4: Finding the time to complete the work.
The time taken for Pushkin and Samrat to complete the work is the reciprocal of their combined rate: \[ \text{Time} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \] Final Answer: \[ \boxed{6} \]