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? Read more

Steps to Solve

1. Calculate A's Daily Work Rate

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

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

2. Calculate Work Done by A in 7 Days

In 7 days, A will complete:

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

3. Determine Remaining Work After A's Contribution

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

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

4. 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:

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

5. 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:

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

6. 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:

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

7. 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:

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