A can do a job in 16 days and B can do the same job in 12 days. With the help of C, they can finish the job in 6 days only. Then, C alone can finish it in

Options

A. 34 days
B. 22 days
C. 36 days
D. 48 days

Answer

To solve the problem step by step, we will find out how long C alone can finish the job based on the work rates of A, B, and C.

### Step 1: Determine the work rates of A and B
- A can complete the job in 16 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{16} \text{ (job per day)} \]
- B can complete the job in 12 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{12} \text{ (job per day)} \]

### Step 2: Determine the combined work rate of A and B
- To find the combined work rate of A and B, we add their individual work rates:
\[ \text{Combined work rate of A and B} = \frac{1}{16} + \frac{1}{12} \]
- To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 16 and 12 is 48.
- Convert \(\frac{1}{16}\) to have a denominator of 48:
\[ \frac{1}{16} = \frac{3}{48} \]
- Convert \(\frac{1}{12}\) to have a denominator of 48:
\[ \frac{1}{12} = \frac{4}{48} \]
- Now, add the two fractions:
\[ \text{Combined work rate of A and B} = \frac{3}{48} + \frac{4}{48} = \frac{7}{48} \]

### Step 3: Determine the combined work rate of A, B, and C
- A, B, and C together can finish the job in 6 days. Therefore, their combined work rate is:
\[ \text{Combined work rate of A, B, and C} = \frac{1}{6} \]

### Step 4: Find C's work rate
- We know the combined work rate of A, B, and C is equal to the combined work rate of A and B plus C's work rate:
\[ \frac{1}{6} = \frac{7}{48} + \text{Work rate of C} \]
- To find C's work rate, we can rearrange the equation:
\[ \text{Work rate of C} = \frac{1}{6} - \frac{7}{48} \]
- To perform this subtraction, we need a common denominator. The LCM of 6 and 48 is 48.
- Convert \(\frac{1}{6}\) to have a denominator of 48:
\[ \frac{1}{6} = \frac{8}{48} \]
- Now, subtract:
\[ \text{Work rate of C} = \frac{8}{48} - \frac{7}{48} = \frac{1}{48} \]

### Step 5: Determine how long C takes to finish the job alone
- If C's work rate is \(\frac{1}{48}\) (job per day), then the time taken by C to finish the job alone is the reciprocal of the work rate:
\[ \text{Time taken by C} = \frac{1}{\frac{1}{48}} = 48 \text{ days} \]

### Final Answer
C alone can finish the job in **48 days**.

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