Solving Work and Time Problems: Finding Individual Work Days

This problem involves understanding how individuals and groups complete a piece of work over time. We are given information about how long A takes to complete a fraction of the work and how long A and B together take to complete the whole work. Our goal is to find out how many days B alone would take to complete the same work.

Let's break down the information provided:

• A completes \(\frac{5}{7}\) of the work in 15 days.
• A and B together complete the whole work (which is 1 or \(\frac{7}{7}\) of the work) in 12 days.

We need to find the number of days B alone will take to complete the whole work.

Step-by-Step Solution to Find B's Work Time

Step 1: Find the time A takes to complete the whole work.

If A completes \(\frac{5}{7}\) of the work in 15 days, the time taken to complete the entire work (which is \(\frac{7}{7}\) or 1 unit) can be found using a proportion or by finding the time per fraction of work.

Time for 1 unit of work = \(\text{Time for } \frac{5}{7} \text{ work} \div \frac{5}{7}\)

Time taken by A to complete the whole work = \(15 \text{ days} \div \frac{5}{7} = 15 \times \frac{7}{5} \text{ days}\)

Calculating this:

\(15 \times \frac{7}{5} = \frac{15 \times 7}{5} = \frac{105}{5} = 21\) days.

So, A alone takes 21 days to complete the entire work.

Step 2: Determine the work rate of A and the combined work rate of A and B.

The work rate is the amount of work done per day. If a person takes \(D\) days to complete a work, their work rate is \(\frac{1}{D}\) of the work per day.

• Work rate of A = \(\frac{1}{\text{Time taken by A}} = \frac{1}{21}\) of the work per day.
• Work rate of A and B together = \(\frac{1}{\text{Time taken by A and B together}} = \frac{1}{12}\) of the work per day.

Step 3: Find the work rate of B.

The combined work rate of A and B is the sum of their individual work rates. That is, Work Rate (A + B) = Work Rate (A) + Work Rate (B).

We know the combined rate (\(\frac{1}{12}\)) and A's rate (\(\frac{1}{21}\)). We can find B's rate:

Work Rate (B) = Work Rate (A + B) - Work Rate (A)

Work Rate (B) = \(\frac{1}{12} - \frac{1}{21}\)

To subtract these fractions, we find a common denominator for 12 and 21. The least common multiple (LCM) of 12 and 21 is 84.

\(\frac{1}{12} = \frac{1 \times 7}{12 \times 7} = \frac{7}{84}\)

\(\frac{1}{21} = \frac{1 \times 4}{21 \times 4} = \frac{4}{84}\)

Work Rate (B) = \(\frac{7}{84} - \frac{4}{84} = \frac{7-4}{84} = \frac{3}{84}\)

Simplify the fraction \(\frac{3}{84}\) by dividing the numerator and denominator by their greatest common divisor, which is 3.

Work Rate (B) = \(\frac{3 \div 3}{84 \div 3} = \frac{1}{28}\) of the work per day.

Step 4: Calculate the time B takes to complete the whole work.

Since B completes \(\frac{1}{28}\) of the work per day, the time taken for B to complete the whole work is the reciprocal of B's work rate.

Time taken by B = \(\frac{1}{\text{Work Rate (B)}} = \frac{1}{\frac{1}{28}}\) days.

Time taken by B = \(28\) days.

Thus, B alone will complete the same work in 28 days.

Let's summarize the key times calculated:

• A takes 21 days to complete the whole work.
• A and B together take 12 days to complete the whole work.
• B takes 28 days to complete the whole work.

Detailed Calculations Overview

The calculation process involved converting the fraction of work done by A into the time taken for the whole work, determining individual and combined work rates, and then using these rates to find the required time for B.

• A's total time = \(15 \text{ days} \times \frac{7}{5} = 21 \text{ days}\)
• A's rate = \(\frac{1}{21}\) work/day
• (A+B)'s rate = \(\frac{1}{12}\) work/day
• B's rate = (A+B)'s rate - A's rate = \(\frac{1}{12} - \frac{1}{21} = \frac{7}{84} - \frac{4}{84} = \frac{3}{84} = \frac{1}{28}\) work/day
• B's total time = \(\frac{1}{\text{B's rate}} = \frac{1}{\frac{1}{28}} = 28 \text{ days}\)

The final answer is 28 days.

Revision Table: Work and Time Concepts

Additional Information: Efficiency and Combined Work

Work and time problems often rely on the concept of efficiency, which is directly related to the work rate. A more efficient person has a higher work rate and takes less time to complete the same amount of work.

When multiple people work together, their efficiencies (or work rates) add up, allowing them to complete the work faster than any individual could alone (assuming they don't hinder each other). This is why A and B together take 12 days, which is less than the 21 days A alone takes.

The fraction of work concept is also key. If a person completes a fraction \(f\) of the work in \(d\) days, the time they would take for the whole work is \(d/f\). In our problem, A completes \(\frac{5}{7}\) of the work in 15 days, so the time for the whole work is \(15 / (\frac{5}{7}) = 15 \times \frac{7}{5} = 21\) days.

Understanding these fundamental principles helps in solving various work and time problems efficiently.