Solving Work and Time Problems: A & B Collaboration

This problem involves calculating the time it takes for two people, A and B, to complete a work together. We need to determine their combined efficiency based on the information given about A's individual work and their joint effort on the remaining part.

Step 1: Calculate A's Individual Work Rate

We are given that A completes $\frac{7}{10}$ of the work in 15 days. The work rate is the amount of work done per day.

A's Work Rate ($R_A$) = $\frac{\text{Work Done}}{\text{Time Taken}}$

Using LaTeX for the calculation:

$R_A = \frac{7/10}{15} = \frac{7}{10 \times 15} = \frac{7}{150} \text{ work per day}$

This means A completes $\frac{7}{150}$ of the total work each day.

Step 2: Determine the Remaining Work

After A completes $\frac{7}{10}$ of the work, we need to find out how much work is left.

Remaining Work = Total Work - Work Done by A

Remaining Work = $1 - \frac{7}{10}$

Using LaTeX:

$ \text{Remaining Work} = 1 - \frac{7}{10} = \frac{10}{10} - \frac{7}{10} = \frac{3}{10} $

So, $\frac{3}{10}$ of the work is remaining.

Step 3: Calculate the Combined Work Rate of A and B

The problem states that A and B together complete this remaining work ($\frac{3}{10}$) in 5 days.

Combined Work Rate ($R_A + R_B$) = $\frac{\text{Remaining Work}}{\text{Time Taken}}$

Using LaTeX:

$ R_A + R_B = \frac{3/10}{5} = \frac{3}{10 \times 5} = \frac{3}{50} \text{ work per day} $

This is the rate at which A and B work together.

Step 4: Calculate Time Taken by A and B Together for Entire Work

We want to find the total time required for A and B, working together, to complete the entire work (1 unit). We use their combined work rate calculated in the previous step.

Time = $\frac{\text{Total Work}}{\text{Combined Work Rate}}$

Using LaTeX:

$ \text{Time}_{A+B} = \frac{1}{R_A + R_B} = \frac{1}{3/50} $ $ \text{Time}_{A+B} = \frac{50}{3} \text{ days} $

Step 5: Express the Result as a Mixed Fraction

To express the answer in a more common format, we convert the improper fraction $\frac{50}{3}$ into a mixed number.

Divide 50 by 3:

$ 50 \div 3 = 16 \text{ with a remainder of } 2 $

Therefore, the time taken is $16\frac{2}{3}$ days.