Joe complete `(1)/(4)` th of a task in 10 days , and Jack can complete `(2)/(5)` th of the same task in 9 days. How long will Joe and Jack take to complete the task working together ?

Options

A. `9 (3)/(8)` days
B. `6 (1)/(10)` days
C. `14 (2)/(5)` days
D. `17` days

Answer

To solve the problem of how long Joe and Jack will take to complete the task working together, we can follow these steps:

### Step 1: Determine Joe's total time to complete the task
Joe completes \( \frac{1}{4} \) of the task in 10 days. To find out how long it takes him to complete the entire task, we can set up the equation:

\[ \text{Total time for Joe} = 10 \text{ days} \times 4 = 40 \text{ days} \]

### Step 2: Determine Jack's total time to complete the task
Jack completes \( \frac{2}{5} \) of the task in 9 days. To find out how long it takes him to complete the entire task, we can set up the equation:

\[ \text{Total time for Jack} = 9 \text{ days} \times \frac{5}{2} = \frac{45}{2} \text{ days} = 22.5 \text{ days} \]

### Step 3: Calculate the work rates (efficiencies) of Joe and Jack
The work rate (efficiency) is calculated as the total work divided by the total time taken. Since the total work can be considered as 1 unit of work:

- Joe's efficiency:
\[ \text{Efficiency of Joe} = \frac{1 \text{ work}}{40 \text{ days}} = \frac{1}{40} \text{ work/day} \]

- Jack's efficiency:
\[ \text{Efficiency of Jack} = \frac{1 \text{ work}}{22.5 \text{ days}} = \frac{1}{22.5} \text{ work/day} \]

### Step 4: Find a common denominator to add efficiencies
To add the efficiencies, we need a common denominator. The least common multiple (LCM) of 40 and 22.5 can be calculated. However, we can also convert Jack's efficiency to a fraction with a denominator of 90 for easier addition:

\[ \text{Efficiency of Jack} = \frac{1}{22.5} = \frac{4}{90} \]
\[ \text{Efficiency of Joe} = \frac{1}{40} = \frac{2.25}{90} = \frac{9}{360} \]

### Step 5: Combine the efficiencies
Now we can add the efficiencies:

\[ \text{Combined efficiency} = \frac{1}{40} + \frac{1}{22.5} = \frac{9}{360} + \frac{16}{360} = \frac{25}{360} \]

### Step 6: Calculate the time taken to complete the task together
Now that we have the combined efficiency, we can find the total time taken to complete the task together:

\[ \text{Total time} = \frac{1 \text{ work}}{\text{Combined efficiency}} = \frac{1}{\frac{25}{360}} = \frac{360}{25} = 14.4 \text{ days} \]

### Step 7: Convert to mixed number
To express 14.4 days as a mixed number:

\[ 14.4 \text{ days} = 14 \text{ days} + 0.4 \text{ days} = 14 \text{ days} + \frac{2}{5} \text{ days} \]

Thus, the final answer is:

\[ \text{Joe and Jack will take } 14 \text{ days and } \frac{2}{5} \text{ days to complete the task together.} \]

### Final Answer:
**14 days and 2/5 days (Option C)**