A can complete a work in 10 days and B can complete the same work in 15 days. In how many days they together will do the same work?

Options

A. 5 days
B. 6 days
C. 8 days
D. 9 days

Answer

To solve the problem of how many days A and B will take to complete the work together, we can follow these steps:

### Step 1: Determine the work done by A in one day.
A can complete the work in 10 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{10} \]

### Step 2: Determine the work done by B in one day.
B can complete the work in 15 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{15} \]

### Step 3: Calculate the total work done by A and B in one day.
To find out how much work A and B can do together in one day, we add their individual work:
\[ \text{Total work done in one day} = \text{Work done by A} + \text{Work done by B} = \frac{1}{10} + \frac{1}{15} \]

### Step 4: Find the LCM of the denominators.
To add the fractions, we need a common denominator. The denominators are 10 and 15. The least common multiple (LCM) of 10 and 15 is 30.

### Step 5: Convert the fractions to have the same denominator.
Now we convert both fractions:
\[ \frac{1}{10} = \frac{3}{30} \quad \text{(by multiplying numerator and denominator by 3)} \]
\[ \frac{1}{15} = \frac{2}{30} \quad \text{(by multiplying numerator and denominator by 2)} \]

### Step 6: Add the fractions.
Now we can add the two fractions:
\[ \frac{3}{30} + \frac{2}{30} = \frac{5}{30} \]

### Step 7: Simplify the result.
We can simplify \(\frac{5}{30}\) to:
\[ \frac{5}{30} = \frac{1}{6} \]

### Step 8: Calculate the total number of days to complete the work together.
If A and B together can do \(\frac{1}{6}\) of the work in one day, then the total number of days to complete the work is the reciprocal of \(\frac{1}{6}\):
\[ \text{Total days} = \frac{1}{\frac{1}{6}} = 6 \]

### Final Answer:
Therefore, A and B together will complete the work in **6 days**.

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