A and B completed a work together in 8 days. Had A worked at twice the speed and B at one-third of his speed it would have taken them 6 days to complete the job. How much time would it take for B alone to do the work?

Options

A. 10
B. 20
C. 15
D. 25

Answer

To solve the problem, we need to determine how much time it would take for B alone to complete the work. Let's break down the solution step by step.

### Step 1: Define the Work Done
Let the work done by A in one day be A's efficiency and the work done by B in one day be B's efficiency. Together, A and B complete the work in 8 days.

Thus, the total work can be expressed as:
\[ \text{Total Work} = (A + B) \times 8 \]

### Step 2: Express Total Work in Terms of A and B
From the above, we can write:
\[ \text{Total Work} = 8(A + B) \]

### Step 3: Consider the Second Scenario
In the second scenario, A works at twice his speed (2A) and B works at one-third of his speed (B/3). Together, they complete the work in 6 days.

Thus, the total work can also be expressed as:
\[ \text{Total Work} = (2A + \frac{B}{3}) \times 6 \]

### Step 4: Set the Two Expressions for Total Work Equal
Since both expressions represent the same total work, we can set them equal to each other:
\[ 8(A + B) = 6(2A + \frac{B}{3}) \]

### Step 5: Simplify the Equation
Expanding both sides gives:
\[ 8A + 8B = 12A + 2B \]

Now, rearranging the equation:
\[ 8A + 8B - 12A - 2B = 0 \]
\[ -4A + 6B = 0 \]
\[ 6B = 4A \]
\[ \frac{A}{B} = \frac{6}{4} = \frac{3}{2} \]

### Step 6: Determine the Efficiency of A and B
From the ratio \( A:B = 3:2 \), we can express A and B in terms of a common variable, say \( k \):
\[ A = 3k \quad \text{and} \quad B = 2k \]

### Step 7: Calculate Total Work
Substituting A and B back into the total work equation:
\[ \text{Total Work} = 8(A + B) = 8(3k + 2k) = 8 \times 5k = 40k \]

### Step 8: Find Time Taken by B Alone
Now, to find the time taken by B alone to complete the work:
\[ \text{Time taken by B} = \frac{\text{Total Work}}{\text{Efficiency of B}} = \frac{40k}{2k} = 20 \text{ days} \]

Thus, B alone would take **20 days** to complete the work.

### Final Answer
B alone would take **20 days** to do the work.

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