Let's solve this problem step by step.

Step 1: Determine the work rates of A and B together.

• A and B together can complete the work in 4 days.
• So, their combined work rate is $\frac{1}{4}$ of the work per day.

Step 2: Determine the work rate of A alone.

• A alone can complete the work in 12 days.
• So, A's work rate is $\frac{1}{12}$ of the work per day.

Step 3: Determine the work rate of B alone.

• Let B's work rate be $\frac{1}{x}$ of the work per day, where $x$ is the number of days B takes to complete the work alone.

Since A and B together complete $\frac{1}{4}$ of the work per day: $\text{A's rate} + \text{B's rate} = \frac{1}{4}$ $\frac{1}{12} + \frac{1}{x} = \frac{1}{4}$

Step 4: Solve for $x$.

To solve for $x$, first subtract $\frac{1}{12}$ from both sides of the equation: $\frac{1}{x} = \frac{1}{4} - \frac{1}{12}$

To subtract these fractions, find a common denominator (which is 12): $\frac{1}{4} = \frac{3}{12}$ $\frac{1}{x} = \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6}$

Thus, $x = 6$.

Final Answer:

B alone can complete the work in 6 days.

Would you like more details or have any questions?

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Here are 5 related questions for further practice:

1. If A and B together can complete a task in 5 days, and B alone can do it in 15 days, how long would it take for A alone to complete it?
2. If A, B, and C together can finish a job in 3 days, and A alone takes 6 days, while B takes 9 days, how long will C alone take to finish the job?
3. If A takes twice as long as B to finish a job, and together they finish it in 8 days, how long will it take each to finish it individually?
4. A can finish a job in 10 days, B can do the same job in 15 days, and C can do it in 20 days. How long will it take for all three to finish the job together?
5. If A alone can finish a job in 7 days, and B alone can finish it in 5 days, but A starts the job and works for 2 days before B joins, how long will it take them to complete the job together?

Tip: When working with rates, it's often easier to express the work done as a fraction of the whole job, as this helps in understanding combined efforts.