A can finish a piece of work in 42 days. He worked alone for 7 days and then B joined him. Together they could finish the remaining work in 14 days. In how many days can B alone fi... A can finish a piece of work in 42 days. He worked alone for 7 days and then B joined him. Together they could finish the remaining work in 14 days. In how many days can B alone finish the same piece of work?

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Understand the Problem

The question is asking us to determine how many days person B alone can finish a piece of work, given that person A can finish the work in 42 days, and after A worked alone for 7 days, they worked together for 14 days to complete the remaining work.

Answer

B can finish the work in 28 days.
Answer for screen readers

B can finish the same piece of work in 28 days.

Steps to Solve

  1. Calculate A's Daily Work Rate

Since A can finish the work in 42 days, A's work rate is:

$$ \text{A's work rate} = \frac{1}{42} $$

  1. Calculate Work Done by A in 7 Days

In 7 days, A will complete:

$$ \text{Work by A} = 7 \times \text{A's work rate} = 7 \times \frac{1}{42} = \frac{7}{42} = \frac{1}{6} $$

  1. Determine Remaining Work After A's Contribution

The total work is 1 (the whole piece), so the remaining work after A's contribution is:

$$ \text{Remaining work} = 1 - \frac{1}{6} = \frac{5}{6} $$

  1. Calculate Combined Work Rate for A and B

The remaining work is completed in 14 days by A and B together. The combined work rate is:

$$ \text{Combined work rate} = \frac{\text{Remaining work}}{\text{Time taken}} = \frac{5/6}{14} = \frac{5}{84} $$

  1. Express Combined Work Rate in Terms of A and B

We know A’s work rate is $\frac{1}{42}$, so we can write:

$$ \text{A's work rate} + \text{B's work rate} = \frac{5}{84} $$

Substituting A's work rate:

$$ \frac{1}{42} + \text{B's work rate} = \frac{5}{84} $$

  1. Isolate B's Work Rate

Convert $\frac{1}{42}$ to a fraction with a denominator of 84:

$$ \frac{1}{42} = \frac{2}{84} $$

So, we have:

$$ \frac{2}{84} + \text{B's work rate} = \frac{5}{84} $$

Now isolate B's work rate:

$$ \text{B's work rate} = \frac{5}{84} - \frac{2}{84} = \frac{3}{84} = \frac{1}{28} $$

  1. Determine Total Days for B to Finish Alone

If B's work rate is $\frac{1}{28}$, then the time taken by B to finish the work alone is:

$$ \text{Days for B alone} = \frac{1}{\text{B's work rate}} = \frac{1}{\frac{1}{28}} = 28 $$

B can finish the same piece of work in 28 days.

More Information

This problem demonstrates the use of rates in collaborative work. By isolating work rates and understanding their contributions, we can solve tasks accomplished by multiple workers effectively.

Tips

  • Not converting work rates to a common denominator when adding.
  • Confusing total work and remaining work after partial contributions.

AI-generated content may contain errors. Please verify critical information

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