Efficiency Comparison Between Arun and Bobby

The problem states that Arun is 25% more efficient than Bobby. This means Arun completes more work in the same amount of time compared to Bobby.

Let's represent Bobby's efficiency (work done per day) as $E_B$. Since Arun is 25% more efficient, Arun's efficiency ($E_A$) can be written as:

$E_A = E_B + 0.25 \times E_B$
$E_A = (1 + 0.25) E_B$
$E_A = 1.25 E_B$

We can express this relationship as a ratio:

$\frac{E_A}{E_B} = 1.25 = \frac{125}{100} = \frac{5}{4}$

This ratio indicates that for every 4 units of work Bobby does, Arun does 5 units.

Calculating Individual Work Rates

We know that the time taken to complete a job is inversely proportional to the efficiency or work rate.

Arun alone can complete the job in 15 days. Let the total amount of work be $W$.

Arun's work rate ($R_A$) is:

$R_A = \frac{\text{Total Work}}{\text{Time taken by Arun}} = \frac{W}{15}$ (units of work per day)

Since the ratio of efficiencies is $\frac{E_A}{E_B} = \frac{5}{4}$, the ratio of their work rates is the same: $\frac{R_A}{R_B} = \frac{5}{4}$.

We can find Bobby's work rate ($R_B$) using Arun's rate:

$R_B = \frac{4}{5} \times R_A$
$R_B = \frac{4}{5} \times \frac{W}{15}$
$R_B = \frac{4W}{75}$ (units of work per day)

Combined Work Rate Calculation

When Arun and Bobby work together, their work rates add up. The combined work rate ($R_{A+B}$) is:

$R_{A+B} = R_A + R_B$
$R_{A+B} = \frac{W}{15} + \frac{4W}{75}$

To add these fractions, we find a common denominator, which is 75:

$R_{A+B} = \frac{5 \times W}{5 \times 15} + \frac{4W}{75}$
$R_{A+B} = \frac{5W}{75} + \frac{4W}{75}$
$R_{A+B} = \frac{5W + 4W}{75}$
$R_{A+B} = \frac{9W}{75}$

Simplifying the fraction:

$R_{A+B} = \frac{3W}{25}$ (units of work per day)

Calculating Time Taken Together

The time it takes for them to complete the job working together is the total work divided by their combined rate:

Time Together = $\frac{\text{Total Work}}{\text{Combined Work Rate}}$
Time Together = $\frac{W}{R_{A+B}}$
Time Together = $\frac{W}{\frac{3W}{25}}$

To divide by a fraction, we multiply by its reciprocal:

Time Together = $W \times \frac{25}{3W}$
Time Together = $\frac{25}{3}$ days

To express this as a mixed number:

$\frac{25}{3} = 8 \text{ remainder } 1$
So, Time Together = $8\frac{1}{3}$ days.

Therefore, Arun and Bobby working together will take $8\frac{1}{3}$ days to complete the job.