Understanding the Time and Work Problem

This question asks us to find how long it takes person B to complete a certain piece of work alone. We are given the time it takes for A alone, for B and C together, and for A and C together to complete the same work.

In time and work problems, it's often helpful to think in terms of the 'rate' at which each person or group works. The rate is typically expressed as the fraction of the work done per unit of time (in this case, per hour). If someone takes $T$ hours to complete a work, their work rate is $1/T$ work per hour.

Given Information and Work Rates

Let's list the given information and convert the time taken into work rates:

• A can do the work in 16 hours.
• B and C together can do the work in 8 hours.
• A and C together can do the work in 12 hours.

Based on this, we can determine their work rates:

• Work rate of A = $\frac{1}{16}$ work per hour.
• Work rate of (B + C) = $\frac{1}{8}$ work per hour.
• Work rate of (A + C) = $\frac{1}{12}$ work per hour.

Calculating Individual Work Rates

We need to find the work rate of B alone to determine the time B takes. We can use the given combined rates and A's rate to find the individual rates of C and then B.

Step 1: Find the Work Rate of C

We know the work rate of (A + C) and the work rate of A. The work rate of (A + C) is the sum of A's work rate and C's work rate:

Work rate of (A + C) = Work rate of A + Work rate of C

Substituting the known values:

$\frac{1}{12} = \frac{1}{16} + \text{Work rate of C}$

Now, we solve for the Work rate of C:

Work rate of C = $\frac{1}{12} - \frac{1}{16}$

To subtract these fractions, find a common denominator. The least common multiple (LCM) of 12 and 16 is 48.

$\frac{1}{12} = \frac{1 \times 4}{12 \times 4} = \frac{4}{48}$

$\frac{1}{16} = \frac{1 \times 3}{16 \times 3} = \frac{3}{48}$

So, Work rate of C = $\frac{4}{48} - \frac{3}{48} = \frac{4-3}{48} = \frac{1}{48}$ work per hour.

Step 2: Find the Work Rate of B

We know the work rate of (B + C) and we just found the work rate of C. The work rate of (B + C) is the sum of B's work rate and C's work rate:

Work rate of (B + C) = Work rate of B + Work rate of C

Substituting the known values:

$\frac{1}{8} = \text{Work rate of B} + \frac{1}{48}$

Now, we solve for the Work rate of B:

Work rate of B = $\frac{1}{8} - \frac{1}{48}$

To subtract these fractions, find a common denominator. The LCM of 8 and 48 is 48.

$\frac{1}{8} = \frac{1 \times 6}{8 \times 6} = \frac{6}{48}$

So, Work rate of B = $\frac{6}{48} - \frac{1}{48} = \frac{6-1}{48} = \frac{5}{48}$ work per hour.

Calculating Time Taken by B Alone

The work rate of B is $\frac{5}{48}$ work per hour. This means B completes $\frac{5}{48}$ of the total work in one hour. To find the time B takes to complete the entire work (which is 1 unit of work), we take the reciprocal of B's work rate.

Time taken by B alone = $\frac{1}{\text{Work rate of B}} = \frac{1}{\frac{5}{48}}$ hours

Time taken by B alone = $\frac{48}{5}$ hours

Converting to Decimal

To express the time in decimal form, divide 48 by 5:

$\frac{48}{5} = 9.6$ hours.

Therefore, B alone will take 9.6 hours to complete the work.

Summary of Solution Steps

1. Determine the work rate for A, (B+C), and (A+C) from the given times.
2. Use the rates of A and (A+C) to calculate the work rate of C.
3. Use the rates of C and (B+C) to calculate the work rate of B.
4. Find the time taken by B alone by taking the reciprocal of B's work rate.
5. Convert the result to a decimal number.

Revision Table: Time and Work Formulas

Additional Information: Time and Work Concepts

Time and work problems are a common type in quantitative aptitude. They usually involve understanding the relationship between the time taken, the amount of work done, and the rate of working. The fundamental principle is that the total amount of work is constant (often taken as 1 unit). Faster workers have higher rates, meaning they complete a larger fraction of the work in the same amount of time.

When multiple people work together, their rates are typically additive, assuming they are working independently and not hindering each other. If a person's efficiency changes, their rate changes accordingly. For example, if someone becomes twice as efficient, their rate doubles, and the time taken is halved.

LCM method can also be used for these problems by considering the total work as the LCM of the individual times. For this problem, the LCM of 16, 8, and 12 is 48. We can assume the total work is 48 units. Then, the work rate of A is 48/16 = 3 units/hour, (B+C) is 48/8 = 6 units/hour, and (A+C) is 48/12 = 4 units/hour. From these rates, we can find C's rate (4 - 3 = 1 unit/hour) and then B's rate (6 - 1 = 5 units/hour). Finally, time taken by B is Total Work / B's Rate = 48 / 5 = 9.6 hours. This method often helps avoid fractions until the final step.