A, B and C can do a piece of work in 12, 15 and 20 days respectively. In how many days will all three do this work together ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the individual work rates of A, B, and C.
- A can complete the work in 12 days, so A's work rate is \( \frac{1}{12} \) of the work per day.
- B can complete the work in 15 days, so B's work rate is \( \frac{1}{15} \) of the work per day.
- C can complete the work in 20 days, so C's work rate is \( \frac{1}{20} \) of the work per day.

### Step 2: Find a common work unit.
To make calculations easier, we can assume the total work is the least common multiple (LCM) of the days taken by A, B, and C.
- The LCM of 12, 15, and 20 is 60 units.

### Step 3: Calculate the individual work done per day.
- A's work per day:
\[ \text{Work by A} = \frac{60 \text{ units}}{12 \text{ days}} = 5 \text{ units/day} \]
- B's work per day:
\[ \text{Work by B} = \frac{60 \text{ units}}{15 \text{ days}} = 4 \text{ units/day} \]
- C's work per day:
\[ \text{Work by C} = \frac{60 \text{ units}}{20 \text{ days}} = 3 \text{ units/day} \]

### Step 4: Calculate the total work done per day when A, B, and C work together.
- Total work done per day by A, B, and C:
\[ \text{Total work per day} = 5 + 4 + 3 = 12 \text{ units/day} \]

### Step 5: Calculate the total time taken to complete the work together.
- Using the formula:
\[ \text{Number of days} = \frac{\text{Total work}}{\text{Total work per day}} = \frac{60 \text{ units}}{12 \text{ units/day}} = 5 \text{ days} \]

### Final Answer:
A, B, and C together will complete the work in **5 days**.

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