If Hemant works for 20 days and Manoj works for 15 days then `(3)/(5)`th of the work has been completed. If Manoj works for 6 days and Hemant works for 16 days then `(2)/(5)`th of the work has been completed. Find in how many days both can complete the work together.

Options

A. A)30 days
B. B)50 days
C. C)70days
D. D)60 days

Answer

To solve the problem step by step, we will follow the information provided and derive the necessary equations to find the required solution.

### Step 1: Define Variables
Let:
- \( H \) = Efficiency of Hemant (work done per day)
- \( M \) = Efficiency of Manoj (work done per day)

### Step 2: Set Up the First Equation
According to the first condition:
- Hemant works for 20 days and Manoj works for 15 days, completing \( \frac{3}{5} \) of the total work.

This can be expressed as:
\[ 20H + 15M = \frac{3}{5} K \]
where \( K \) is the total work.

### Step 3: Set Up the Second Equation
According to the second condition:
- Manoj works for 6 days and Hemant works for 16 days, completing \( \frac{2}{5} \) of the total work.

This can be expressed as:
\[ 16H + 6M = \frac{2}{5} K \]

### Step 4: Express Total Work \( K \)
From the first equation, we can express \( K \):
\[ K = \frac{5}{3}(20H + 15M) \]
From the second equation, we can also express \( K \):
\[ K = \frac{5}{2}(16H + 6M) \]

### Step 5: Equate the Two Expressions for \( K \)
Set the two expressions for \( K \) equal to each other:
\[ \frac{5}{3}(20H + 15M) = \frac{5}{2}(16H + 6M) \]

### Step 6: Simplify the Equation
We can eliminate \( \frac{5}{3} \) and \( \frac{5}{2} \) by cross-multiplying:
\[ 2(20H + 15M) = 3(16H + 6M) \]
Expanding both sides:
\[ 40H + 30M = 48H + 18M \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 40H - 48H + 30M - 18M = 0 \]
\[ -8H + 12M = 0 \]
This simplifies to:
\[ 2H = 3M \quad \text{or} \quad \frac{H}{M} = \frac{3}{2} \]

### Step 8: Express \( H \) in Terms of \( M \)
Let \( H = \frac{3}{2}M \).

### Step 9: Substitute Back to Find Total Work \( K \)
Substituting \( H \) back into one of the equations for \( K \):
Using the first equation:
\[ K = \frac{5}{3}(20H + 15M) \]
Substituting \( H \):
\[ K = \frac{5}{3}(20 \cdot \frac{3}{2}M + 15M) = \frac{5}{3}(30M + 15M) = \frac{5}{3}(45M) = 75M \]

### Step 10: Find Combined Efficiency
Now, the combined efficiency of Hemant and Manoj:
\[ H + M = \frac{3}{2}M + M = \frac{5}{2}M \]

### Step 11: Calculate Time Taken to Complete Work
The time taken to complete the total work \( K \) together is given by:
\[ \text{Time} = \frac{K}{H + M} = \frac{75M}{\frac{5}{2}M} = 75 \cdot \frac{2}{5} = 30 \text{ days} \]

### Final Answer
Thus, both Hemant and Manoj can complete the work together in **30 days**.

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