X can complete a task in 8 days, whereas Y can complete the same task in 10 days. If they work together, in how many days will the task be completed?

Options

A. `4(4)/(9)`
B. `(9)/(40)`
C. `4(8)/(9)`
D. 9

Answer

To solve the problem of how many days X and Y will take to complete the task together, we can follow these steps:

### Step 1: Determine the work done by each person in one day.
- X can complete the task in 8 days. Therefore, the work done by X in one day is:
\[ \text{Work done by X in one day} = \frac{1}{8} \]
- Y can complete the task in 10 days. Therefore, the work done by Y in one day is:
\[ \text{Work done by Y in one day} = \frac{1}{10} \]

### Step 2: Add the work done by both X and Y in one day.
- When X and Y work together, their combined work done in one day is:
\[ \text{Combined work} = \frac{1}{8} + \frac{1}{10} \]

### Step 3: Find a common denominator to add the fractions.
- The least common multiple (LCM) of 8 and 10 is 40. Thus, we can rewrite the fractions:
\[ \frac{1}{8} = \frac{5}{40} \quad \text{and} \quad \frac{1}{10} = \frac{4}{40} \]
- Now, we can add them:
\[ \text{Combined work} = \frac{5}{40} + \frac{4}{40} = \frac{9}{40} \]

### Step 4: Calculate the total time taken to complete the task together.
- If X and Y together complete \(\frac{9}{40}\) of the task in one day, then the total number of days \(D\) to complete the entire task (1 unit of work) is given by:
\[ D = \frac{1}{\text{Combined work}} = \frac{1}{\frac{9}{40}} = \frac{40}{9} \]

### Step 5: Convert the result into a more understandable format.
- The result \(\frac{40}{9}\) can be expressed as:
\[ \frac{40}{9} \approx 4.44 \text{ days} \quad \text{or} \quad 4 \frac{4}{9} \text{ days} \]

### Final Answer:
- Therefore, X and Y together will complete the task in \(\frac{40}{9}\) days or approximately 4 days and 4/9 of a day.

---