To determine how many days A and B worked together, let's break this problem into steps:

Step 1: Define Work and Efficiency

• Let the total work be W (we'll represent the work in units for simplicity).
• A can complete the work in 15 days, so A's efficiency is:

$\text{Efficiency of A} = \frac{W}{15} = \frac{1}{15} \text{ units of work per day}.$

• B is twice as efficient as A, so B's efficiency is:

$\text{Efficiency of B} = 2 \times \frac{1}{15} = \frac{2}{15} \text{ units of work per day}.$

Step 2: Set Up the Timeline

• A started the work and worked alone for some days, say $x$.
• After $x$ days, B joined A, and they worked together for $11 - x$ days.
• The total work completed in 11 days is $W$.

Step 3: Write the Equation for Total Work

The total work $W$ can be expressed as:

$\text{Work done by A alone} + \text{Work done by A and B together} = W.$

Substitute their efficiencies:

$x \cdot \frac{1}{15} + (11 - x) \cdot \left( \frac{1}{15} + \frac{2}{15} \right) = W.$

Simplify:

$x \cdot \frac{1}{15} + (11 - x) \cdot \frac{3}{15} = 1.$

Step 4: Solve for $x$

Eliminate fractions by multiplying through by 15:

$x + 3(11 - x) = 15.$

Expand and simplify:

$x + 33 - 3x = 15.$

$-2x + 33 = 15.$

$-2x = 15 - 33.$

$-2x = -18.$

$x = 9.$

So, A worked alone for $x = 9$ days.

Step 5: Determine Days Worked Together

Since the total duration is 11 days, the days A and B worked together is:

$11 - x = 11 - 9 = 2 \text{ days}.$

Final Answer:

The correct option is B. 2 days.

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Would you like a detailed explanation of any step? 😊

Related Questions:

1. How is efficiency calculated in problems involving work?
2. If B were three times as efficient as A, how would the solution change?
3. How does changing the total time (e.g., 12 days) affect the result?
4. Can this problem be solved using ratios instead of equations?
5. What if A and B started working together from the beginning? How long would they take to complete the work?

Tip:

Always start such problems by defining the total work as a unit (e.g., $W = 1$) to make calculations easier.