A father can do a job as fast as his two sons working together. If one son does the job in 3 hours and the other in 6 hours, the number of hours taken by the father, to do the job alone is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we need to determine how long it takes for the father to complete the job alone, given the work rates of his two sons.

### Step 1: Determine the work rates of the sons.
- **Son 1** can complete the job in 3 hours. Therefore, his work rate (efficiency) is:
\[ \text{Efficiency of Son 1} = \frac{1 \text{ job}}{3 \text{ hours}} = \frac{1}{3} \text{ jobs per hour} \]

- **Son 2** can complete the job in 6 hours. Therefore, his work rate (efficiency) is:
\[ \text{Efficiency of Son 2} = \frac{1 \text{ job}}{6 \text{ hours}} = \frac{1}{6} \text{ jobs per hour} \]

### Step 2: Calculate the combined work rate of both sons.
To find the combined efficiency of both sons working together, we add their individual efficiencies:
\[ \text{Combined Efficiency} = \text{Efficiency of Son 1} + \text{Efficiency of Son 2} = \frac{1}{3} + \frac{1}{6} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 6 is 6:
\[ \frac{1}{3} = \frac{2}{6} \]
Now we can add:
\[ \text{Combined Efficiency} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \text{ jobs per hour} \]

### Step 3: Determine the time taken by the father to complete the job.
Since the father can do the job as fast as his two sons working together, his efficiency is also \(\frac{1}{2}\) jobs per hour. To find out how long it takes for the father to complete 1 job, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} \]
Here, the total work is 1 job, so:
\[ \text{Time taken by the father} = \frac{1 \text{ job}}{\frac{1}{2} \text{ jobs per hour}} = 2 \text{ hours} \]

### Final Answer:
The father takes **2 hours** to complete the job alone.

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