I can do a piece of work in 8 days, which can be done by you in 10 days How long would it take to do it if we work together ?

Options

A. `4(4)/(9)` days
B. `5(3)/(9)` days
C. `5(1)/(2)` days
D. `4(7)/(9)` days

Answer

To solve the problem step by step, we will find out how long it would take for both of us to complete the work together.

### Step 1: Determine individual work rates
- I can complete the work in 8 days. Therefore, my work rate is:
\[ \text{My work rate} = \frac{1}{8} \text{ (work per day)} \]
- You can complete the work in 10 days. Therefore, your work rate is:
\[ \text{Your work rate} = \frac{1}{10} \text{ (work per day)} \]

### Step 2: Add the work rates to find the combined work rate
To find out how much work we can do together in one day, we add our individual work rates:
\[ \text{Combined work rate} = \text{My work rate} + \text{Your work rate} = \frac{1}{8} + \frac{1}{10} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 8 and 10 is 40. We will convert both fractions to have a common denominator of 40:
\[ \frac{1}{8} = \frac{5}{40} \quad \text{(since } 1 \times 5 = 5 \text{ and } 8 \times 5 = 40\text{)} \]
\[ \frac{1}{10} = \frac{4}{40} \quad \text{(since } 1 \times 4 = 4 \text{ and } 10 \times 4 = 40\text{)} \]

### Step 4: Add the fractions
Now we can add the two fractions:
\[ \text{Combined work rate} = \frac{5}{40} + \frac{4}{40} = \frac{9}{40} \]

### Step 5: Calculate the time to complete the work together
The combined work rate of \(\frac{9}{40}\) means together we complete \(\frac{9}{40}\) of the work in one day. To find out how many days it will take to complete the entire work, we take the reciprocal of the combined work rate:
\[ \text{Time to complete work together} = \frac{1}{\frac{9}{40}} = \frac{40}{9} \text{ days} \]

### Final Answer
Thus, if we work together, it will take us \(\frac{40}{9}\) days to complete the work.

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