P can lay a railway track between two stations in 16 days. Q can do the same job in 12 days. With the help of R, they complete the job in 4 days. How many days does it take for R alone to complete the work? 9(3/5) days 9(1/5) days 9(2/5) days 10 daysSee answer

P can lay a railway track between two stations in 16 days Q can do the same job in 12 days With the help of R they complete the job in 4 days How many days does it take for R alone to complete the…

Question

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Basic Answer

Step 1: Determine the work rates of P and Q

P can complete the job in 16 days, so P’s work rate is $\frac{1}{16}$ of the job per day.
Q can complete the job in 12 days, so Q’s work rate is $\frac{1}{12}$ of the job per day.

Step 2: Calculate the combined work rate of P, Q, and R

Together, P, Q, and R complete the job in 4 days, so their combined work rate is $\frac{1}{4}$ of the job per day.

Step 3: Determine the work rate of R

Let R’s work rate be $\frac{1}{R}$ of the job per day.
The combined work rate of P, Q, and R is:
$\frac{1}{16} + \frac{1}{12} + \frac{1}{R} = \frac{1}{4}$

Step 4: Solve for R

First, find a common denominator for $\frac{1}{16}$ and $\frac{1}{12}$ :
$\frac{1}{16} = \frac{3}{48}, \frac{1}{12} = \frac{4}{48}$
Combine these rates:
$\frac{3}{48} + \frac{4}{48} = \frac{7}{48}$
Now, set up the equation with the combined rate:
$\frac{7}{48} + \frac{1}{R} = \frac{1}{4}$
Subtract $\frac{7}{48}$ from both sides:
$\frac{1}{R} = \frac{1}{4} - \frac{7}{48}$
Convert $\frac{1}{4}$ to a common denominator:
$\frac{1}{4} = \frac{12}{48}$
Perform the subtraction:
$\frac{1}{R} = \frac{12}{48} - \frac{7}{48} = \frac{5}{48}$
Invert to find R:
$R = \frac{48}{5} = 9 \frac{3}{5} days$

Final Answer

It takes R alone $9 \frac{3}{5}$ days to complete the work.