A and B together can do a piece of work in 6 days. A alone can do it in 10 days. What time will B require to do it alone ?

Options

A. 20 days
B. 15 days
C. 25 days
D. 30 days

Answer

To solve the problem step by step, we will use the concept of work done by A and B together and individually.

### Step 1: Understand the Work Rate
A and B together can complete the work in 6 days. This means their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1}{6} \text{ (work per day)} \]

### Step 2: Determine A's Work Rate
A can complete the work alone in 10 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{10} \text{ (work per day)} \]

### Step 3: Set Up the Equation for B's Work Rate
Let B's work rate be \( \frac{1}{b} \), where \( b \) is the number of days B takes to complete the work alone. The combined work rate of A and B can be expressed as:
\[ \text{Work rate of A} + \text{Work rate of B} = \text{Work rate of A and B} \]
Substituting the known values:
\[ \frac{1}{10} + \frac{1}{b} = \frac{1}{6} \]

### Step 4: Solve for B's Work Rate
To solve for \( \frac{1}{b} \), we first isolate it:
\[ \frac{1}{b} = \frac{1}{6} - \frac{1}{10} \]
To subtract these fractions, we need a common denominator. The least common multiple of 6 and 10 is 30. Thus, we convert the fractions:
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{10} = \frac{3}{30} \]
Now substituting back:
\[ \frac{1}{b} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15} \]

### Step 5: Find B's Time to Complete the Work Alone
Now, we can find \( b \):
\[ b = 15 \]
This means B alone can complete the work in 15 days.

### Final Answer
B will require **15 days** to complete the work alone.

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