The capacities of A,B and C to complete a piece of work is `1:2:3`. By working together, the thre of them can complete the work in 24 days. In how many days can C alone complete it ?

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Ratios of Work Capacities
The work capacities of A, B, and C are given in the ratio of 1:2:3. This means:
- A's work capacity = 1 part
- B's work capacity = 2 parts
- C's work capacity = 3 parts

### Step 2: Calculate Total Work Capacity
To find the total work capacity, we add the parts together:
- Total capacity = A + B + C = 1 + 2 + 3 = 6 parts

### Step 3: Determine Work Done Per Day
If A, B, and C can complete the work together in 24 days, then the total work done in terms of capacity can be calculated as follows:
- Work done in 1 day = Total work / Total days = 1 (complete work) / 24 days = 1/24 of the work per day.

### Step 4: Relate Daily Work to Capacity
Since the total capacity is 6 parts, the work done by A, B, and C together in one day can also be expressed as:
- Work done in 1 day = Total capacity = 6 parts.

### Step 5: Calculate Work Done by C Alone
Now, we need to find out how many days C alone would take to complete the work. C's capacity is 3 parts.

To find the number of days C would take to complete the entire work, we can set up the following equation:
- If C does 3 parts of work in one day, then the number of days C would take to complete the entire work (6 parts) is given by:

\[ \text{Days taken by C} = \frac{\text{Total work}}{\text{C's capacity}} = \frac{6 \text{ parts}}{3 \text{ parts/day}} = 2 \text{ days} \]

### Step 6: Calculate Total Days for C Alone
Since we know that A, B, and C together complete the work in 24 days, we can use the relationship between their capacities and the total days:
- Total days for C alone = 24 days (for 6 parts) multiplied by (Total capacity / C's capacity)

\[ \text{Days taken by C} = 24 \times \frac{6}{3} = 24 \times 2 = 48 \text{ days} \]

### Final Answer
C alone can complete the work in **48 days**.

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