A person can do a job as fast as his two sons working together. If one son does the job in 6 days and the other in 12 days, how many days does it take the father to do the job ?

Answer

To solve the problem step by step, we will determine how long it takes the father to complete the job based on the work rates of his two sons.

### Step 1: Determine the work rates of the sons.
- The first son can complete the job in 6 days. Therefore, his work rate is:
\[ \text{Work rate of Son 1} = \frac{1}{6} \text{ of the job per day} \]
- The second son can complete the job in 12 days. Therefore, his work rate is:
\[ \text{Work rate of Son 2} = \frac{1}{12} \text{ of the job per day} \]

### Step 2: Combine the work rates of both sons.
To find the combined work rate of both sons working together, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{6} + \frac{1}{12} \]

### Step 3: Find a common denominator and add the fractions.
The least common multiple of 6 and 12 is 12. We can rewrite \(\frac{1}{6}\) with a denominator of 12:
\[ \frac{1}{6} = \frac{2}{12} \]
Now, we can add the two fractions:
\[ \text{Combined work rate} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \]
This means that together, both sons can complete \(\frac{1}{4}\) of the job in one day.

### Step 4: Determine the father's work rate.
According to the problem, the father can do the job as fast as his two sons working together. Therefore, the father's work rate is also \(\frac{1}{4}\) of the job per day.

### Step 5: Calculate the time taken by the father to complete the job.
If the father completes \(\frac{1}{4}\) of the job in one day, then to complete the entire job (1 whole job), it will take:
\[ \text{Time taken by father} = \frac{1}{\frac{1}{4}} = 4 \text{ days} \]

### Final Answer:
The father takes **4 days** to complete the job.

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