Ashish, Binay and Joseph can do a job in 20, 30 and 40 days respectively. The three started the job together, Ashish left the job 4 days before it was completed and Binay left the job 3 days before it was completed. In how many days was the job completed?

Options

A. 14
B. 12
C. 16
D. 15

Answer

To solve the problem step by step, we will first determine the work done by each person per day, then calculate the total work, and finally find out how many days the job was completed.

### Step 1: Determine the work done by each person per day
- Ashish can complete the job in 20 days, so his work rate is:
\[ \text{Work rate of Ashish} = \frac{1}{20} \text{ of the job per day} \]
- Binay can complete the job in 30 days, so his work rate is:
\[ \text{Work rate of Binay} = \frac{1}{30} \text{ of the job per day} \]
- Joseph can complete the job in 40 days, so his work rate is:
\[ \text{Work rate of Joseph} = \frac{1}{40} \text{ of the job per day} \]

### Step 2: Find the total work done by all three in one day
To find the combined work rate of Ashish, Binay, and Joseph, we need to find the least common multiple (LCM) of their completion times (20, 30, and 40 days). The LCM of these numbers is 120 days.

Now, we calculate the amount of work each person can do in one day based on the LCM:
- Ashish's daily work:
\[ \frac{120}{20} = 6 \text{ units of work} \]
- Binay's daily work:
\[ \frac{120}{30} = 4 \text{ units of work} \]
- Joseph's daily work:
\[ \frac{120}{40} = 3 \text{ units of work} \]

Total work done by all three in one day:
\[ \text{Total work per day} = 6 + 4 + 3 = 13 \text{ units of work} \]

### Step 3: Set up the equation based on the problem statement
Let \( x \) be the total number of days the job took to complete. According to the problem:
- Ashish left 4 days before completion, so he worked for \( x - 4 \) days.
- Binay left 3 days before completion, so he worked for \( x - 3 \) days.
- Joseph worked for the entire \( x \) days.

The total work done can be expressed as:
\[ \text{Work done by Ashish} + \text{Work done by Binay} + \text{Work done by Joseph} = \text{Total work} \]
\[ 6(x - 4) + 4(x - 3) + 3x = 120 \]

### Step 4: Solve the equation
Expanding the equation:
\[ 6x - 24 + 4x - 12 + 3x = 120 \]
Combining like terms:
\[ (6x + 4x + 3x) - 36 = 120 \]
\[ 13x - 36 = 120 \]
Adding 36 to both sides:
\[ 13x = 156 \]
Dividing by 13:
\[ x = 12 \]

### Conclusion
The job was completed in **12 days**.