A, b and C can alone complete a work in 10, 12 and 15 days
respectively. A and C started the work and after working for
4 days, A left and B joined. In how many days the total work
was completed ?

Options

A. `6 5/9 days `
B. ` 6 2/9 days `
C. 6 days
D. ` 5 4/9 days `

Explanation

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

### Step 1: Determine the work done by A, B, and C in one day.
- A can complete the work in 10 days, so A's work in one day = \( \frac{1}{10} \) of the work.
- B can complete the work in 12 days, so B's work in one day = \( \frac{1}{12} \) of the work.
- C can complete the work in 15 days, so C's work in one day = \( \frac{1}{15} \) of the work.

### Step 2: Convert the work done in one day into a common unit.
To make calculations easier, we can find a common denominator. The least common multiple of 10, 12, and 15 is 60. Therefore:
- A's work in one day = \( \frac{60}{10} = 6 \) units.
- B's work in one day = \( \frac{60}{12} = 5 \) units.
- C's work in one day = \( \frac{60}{15} = 4 \) units.

### Step 3: Calculate the total work done by A and C in 4 days.
- A and C together can do \( 6 + 4 = 10 \) units of work in one day.
- In 4 days, they will complete \( 10 \times 4 = 40 \) units of work.

### Step 4: Determine the remaining work after A and C have worked for 4 days.
- Total work = 60 units.
- Work done by A and C in 4 days = 40 units.
- Remaining work = \( 60 - 40 = 20 \) units.

### Step 5: Calculate the work done by B and C together.
- After 4 days, A leaves and B joins C.
- B and C together can do \( 5 + 4 = 9 \) units of work in one day.

### Step 6: Calculate the number of days required to complete the remaining work.
- Remaining work = 20 units.
- Work done by B and C in one day = 9 units.
- Days required to finish the remaining work = \( \frac{20}{9} \) days.

### Step 7: Calculate the total time taken to complete the work.
- Total time = 4 days (A and C working) + \( \frac{20}{9} \) days (B and C working).
- To combine these, we convert 4 days into a fraction: \( 4 = \frac{36}{9} \).
- Total time = \( \frac{36}{9} + \frac{20}{9} = \frac{56}{9} \) days.

### Final Answer:
The total work was completed in \( \frac{56}{9} \) days, which is approximately 6.22 days.

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