A and B together can finish a piece of work in 11 1/9 days. B is 25% more efficient than A. If B starts working alone on the same piece of work and A joins him after 6 days, in how... A and B together can finish a piece of work in 11 1/9 days. B is 25% more efficient than A. If B starts working alone on the same piece of work and A joins him after 6 days, in how many days will the entire work be completed? Read more

Steps to Solve

1. Calculate the combined work rate of A and B

Since A and B together can finish the work in $11 \frac{1}{9}$ days, we first convert this to an improper fraction: $$ 11 \frac{1}{9} = \frac{100}{9} \text{ days} $$ The work rate of A and B together is: $$ \text{Work rate} = \frac{1 \text{ work}}{\frac{100}{9} \text{ days}} = \frac{9}{100} \text{ work/day} $$

2. Determine the work rates of A and B

Let the work rate of A be $x$. Since B is 25% more efficient than A, B's work rate is: $$ B = 1.25x $$ Thus, the combined work rate can be expressed as: $$ x + 1.25x = 2.25x $$ Setting this equal to the combined rate we found: $$ 2.25x = \frac{9}{100} $$ Now we solve for $x$: $$ x = \frac{9}{100 \times 2.25} = \frac{9}{225} = \frac{1}{25} \text{ work/day} $$ So, A’s work rate is $\frac{1}{25}$ and B’s work rate is: $$ B = 1.25 \times \frac{1}{25} = \frac{1}{20} \text{ work/day} $$

3. Calculate the work done by B alone for 6 days

In 6 days, the work done by B alone would be: $$ \text{Work done} = \text{Rate of B} \times \text{Time} = \frac{1}{20} \times 6 = \frac{6}{20} = \frac{3}{10} $$

4. Calculate the remaining work

The remaining work after B has worked for 6 days: $$ \text{Remaining work} = 1 - \frac{3}{10} = \frac{7}{10} $$

5. Calculate the time for A and B to complete the remaining work together

The combined work rate of A and B is: $$ \text{Combined rate} = \frac{1}{25} + \frac{1}{20} $$ Finding a common denominator (LCM of 25 and 20 is 100): $$ \text{Combined rate} = \frac{4}{100} + \frac{5}{100} = \frac{9}{100} \text{ work/day} $$ Now, we calculate the time required to complete the remaining work: $$ \text{Time} = \frac{\text{Remaining work}}{\text{Combined rate}} = \frac{\frac{7}{10}}{\frac{9}{100}} = \frac{7 \times 100}{10 \times 9} = \frac{700}{90} = \frac{70}{9} \text{ days} $$

6. Calculate the total time to complete the work

The total time required to complete the work is the 6 days that B worked alone plus the time A and B worked together: $$ \text{Total Time} = 6 + \frac{70}{9} = \frac{54}{9} + \frac{70}{9} = \frac{124}{9} \text{ days} \approx 13.78 \text{ days} $$