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

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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

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 days

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:
  $$R=\frac{48}{5}=9\frac{3}{5}\text{ days}$$

Final Answer

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