A, B and C can do a work in 25, 40 and 60 days respectively. All three start the work together and work for 5 days then A left and B left 10 days before the completion of work, how much time shall be taken to do whole work?

Answer

To solve the problem step by step, we will first calculate the work done by A, B, and C, and then determine how long it takes to complete the entire work.

### Step 1: Determine the work rates of A, B, and C.
- A can complete the work in 25 days, so A's work rate is \( \frac{1}{25} \) of the work per day.
- B can complete the work in 40 days, so B's work rate is \( \frac{1}{40} \) of the work per day.
- C can complete the work in 60 days, so C's work rate is \( \frac{1}{60} \) of the work per day.

### Step 2: Find the combined work rate of A, B, and C.
To find the combined work rate, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{25} + \frac{1}{40} + \frac{1}{60} \]

To add these fractions, we need to find a common denominator. The LCM of 25, 40, and 60 is 600.

Calculating each work rate:
- \( \frac{1}{25} = \frac{24}{600} \)
- \( \frac{1}{40} = \frac{15}{600} \)
- \( \frac{1}{60} = \frac{10}{600} \)

Now, adding them together:
\[ \text{Combined work rate} = \frac{24 + 15 + 10}{600} = \frac{49}{600} \]

### Step 3: Calculate the work done in the first 5 days.
In 5 days, the amount of work done by A, B, and C together is:
\[ \text{Work done in 5 days} = 5 \times \frac{49}{600} = \frac{245}{600} \]

### Step 4: Determine the remaining work.
The total work is considered as 1 (or 600 units). The remaining work after 5 days is:
\[ \text{Remaining work} = 1 - \frac{245}{600} = \frac{355}{600} \]

### Step 5: Determine when B leaves the work.
Let \( x \) be the total number of days taken to complete the work. B leaves 10 days before the work is completed, which means B works for \( x - 10 \) days.

### Step 6: Calculate the work done by B and C after A leaves.
After 5 days, A leaves, and B and C continue working. Their combined work rate is:
\[ \text{B's work rate} + \text{C's work rate} = \frac{1}{40} + \frac{1}{60} \]

Finding a common denominator (LCM of 40 and 60 is 120):
- \( \frac{1}{40} = \frac{3}{120} \)
- \( \frac{1}{60} = \frac{2}{120} \)

Adding them together:
\[ \text{B + C work rate} = \frac{3 + 2}{120} = \frac{5}{120} = \frac{1}{24} \]

### Step 7: Calculate the work done by B and C.
Let \( y \) be the number of days B and C work together after A leaves. Thus:
\[ y = x - 5 - 10 = x - 15 \]
The work done by B and C together is:
\[ \text{Work done by B and C} = (x - 15) \times \frac{1}{24} \]

### Step 8: Set up the equation for total work.
The total work done can be expressed as:
\[ \frac{245}{600} + (x - 15) \times \frac{1}{24} = 1 \]

### Step 9: Solve for \( x \).
Rearranging the equation:
\[ (x - 15) \times \frac{1}{24} = 1 - \frac{245}{600} \]
Calculating the right side:
\[ 1 - \frac{245}{600} = \frac{600 - 245}{600} = \frac{355}{600} \]
Thus:
\[ (x - 15) \times \frac{1}{24} = \frac{355}{600} \]
Multiplying both sides by 24:
\[ x - 15 = \frac{355 \times 24}{600} \]
Calculating the right side:
\[ \frac{355 \times 24}{600} = \frac{8520}{600} = 14.2 \]
Thus:
\[ x - 15 = 14.2 \implies x = 29.2 \]

### Step 10: Finalize the total time taken.
Since we are looking for the total time taken, we round \( x \) to the nearest whole number:
\[ \text{Total time taken} = 29.2 \text{ days} \approx 30 \text{ days} \]

### Final Answer:
The total time taken to complete the work is approximately **30 days**.