Work and Time: Calculating Combined Effort

This solution explains how to find the time taken for two individuals, A and B, to complete a work together, given their relative speeds and one individual's completion time.

Step 1: Understanding Relative Speeds

We are given that A works thrice as fast as B. This means the ratio of their work rates (speed) is:

$ \frac{\text{Speed}_A}{\text{Speed}_B} = \frac{3}{1} $

This implies that for every unit of work B does, A does 3 units of work in the same amount of time.

Step 2: Calculating Individual Work Rates

B completes the work in 256 days independently.

The amount of work B does in 1 day is:

$ \text{Work Rate}_B = \frac{1}{256} \text{ of the total work} $

Since A works 3 times as fast as B, A's work rate is 3 times B's work rate.

$ \text{Work Rate}_A = 3 \times \text{Work Rate}_B = 3 \times \frac{1}{256} = \frac{3}{256} \text{ of the total work per day} $

Step 3: Calculating Combined Work Rate

When A and B work together, their work rates add up to find their combined rate.

$ \text{Combined Rate} = \text{Work Rate}_A + \text{Work Rate}_B $

$ \text{Combined Rate} = \frac{3}{256} + \frac{1}{256} = \frac{4}{256} $

Simplifying the fraction gives the combined rate:

$ \text{Combined Rate} = \frac{1}{64} \text{ of the total work per day} $

Step 4: Calculating Combined Time

The total time taken to complete the work together is the reciprocal of the combined work rate.

$ \text{Time taken together} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{64}} $

$ \text{Time taken together} = 64 \text{ days} $

Final Answer

Therefore, A and B can together finish the work in 64 days.