Ram can do a piece of work in 20 days and Shyam in 30 days. They work together for 10 days. After that Shyam leaves and rest of the work is completed by Ram alone. How long does it take Ram to finish the remaining work?

Options

A. 3 days
B. `2(1)/(3)` days
C. `3(1)/(3)` days
D. `4(1)/(3)` days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the work done by Ram and Shyam in one day
- Ram can complete the work in 20 days, so his work rate (efficiency) is:
\[ \text{Ram's efficiency} = \frac{1}{20} \text{ of the work per day} \]
- Shyam can complete the work in 30 days, so his work rate (efficiency) is:
\[ \text{Shyam's efficiency} = \frac{1}{30} \text{ of the work per day} \]

### Step 2: Find a common time frame to compare their efficiencies
To make calculations easier, we can find a common time frame. The least common multiple (LCM) of 20 and 30 is 60 days. Therefore, we can express their efficiencies in terms of 60 days:
- In 60 days, Ram would complete:
\[ \text{Work done by Ram} = 3 \text{ (since } 60/20 = 3\text{)} \]
- In 60 days, Shyam would complete:
\[ \text{Work done by Shyam} = 2 \text{ (since } 60/30 = 2\text{)} \]

### Step 3: Calculate the total work done together in one day
When Ram and Shyam work together, their combined efficiency is:
\[ \text{Combined efficiency} = \text{Ram's efficiency} + \text{Shyam's efficiency} = \frac{1}{20} + \frac{1}{30} \]
To add these fractions, we find a common denominator (which is 60):
\[ \text{Combined efficiency} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]
This means together they complete \(\frac{1}{12}\) of the work in one day.

### Step 4: Calculate the work done in 10 days
In 10 days, the work done by both Ram and Shyam is:
\[ \text{Work done in 10 days} = 10 \times \frac{1}{12} = \frac{10}{12} = \frac{5}{6} \]

### Step 5: Determine the remaining work
The total work is considered as 1 (or 100%). After 10 days, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{5}{6} = \frac{1}{6} \]

### Step 6: Calculate how long it takes Ram to finish the remaining work
Now, Ram will complete the remaining \(\frac{1}{6}\) of the work alone. Since Ram's efficiency is \(\frac{1}{20}\) of the work per day, we can find the time taken by Ram to finish the remaining work:
\[ \text{Time taken by Ram} = \frac{\text{Remaining work}}{\text{Ram's efficiency}} = \frac{\frac{1}{6}}{\frac{1}{20}} = \frac{1}{6} \times 20 = \frac{20}{6} = \frac{10}{3} \text{ days} \]
This simplifies to approximately 3.33 days.

### Final Answer
Ram takes \(\frac{10}{3}\) days, or approximately 3 days and 8 hours, to finish the remaining work.

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