Solve Work and Time Problems: Finding Individual Time

This problem involves calculating the time taken by an individual to complete a task based on the time taken by others, both individually and in combination. The key concept here is the work rate, which is the amount of work done per unit of time.

Understanding Work Rate

If a person can complete a piece of work in \(T\) hours, their work rate is \(1/T\) of the work per hour. When multiple people work together, their work rates add up to find their combined work rate.

Given Information

• Ram can do the work in 18 hours.
• Shyam and Naresh can do the work together in 9 hours.
• Ram and Naresh can do the work together in 12 hours.

Calculating Individual and Combined Work Rates

From the given information, we can find the work rate of each person or combination:

• Ram's work rate = \(\frac{1}{18}\) work/hour.
• Shyam and Naresh's combined work rate = \(\frac{1}{9}\) work/hour.
• Ram and Naresh's combined work rate = \(\frac{1}{12}\) work/hour.

Finding Naresh's Work Rate

We know the combined work rate of Ram and Naresh, and Ram's individual work rate. We can subtract Ram's rate from the combined rate to find Naresh's rate:

Naresh's work rate = (Ram and Naresh's combined rate) - (Ram's work rate)

Naresh's work rate = \(\frac{1}{12} - \frac{1}{18}\)

To subtract these fractions, we find a common denominator, which is 36.

Naresh's work rate = \(\frac{3}{36} - \frac{2}{36}\)

Naresh's work rate = \(\frac{1}{36}\) work/hour.

This means Naresh alone takes 36 hours to complete the work.

Finding Shyam's Work Rate

We know the combined work rate of Shyam and Naresh, and we just found Naresh's individual work rate. We can subtract Naresh's rate from the combined rate to find Shyam's rate:

Shyam's work rate = (Shyam and Naresh's combined rate) - (Naresh's work rate)

Shyam's work rate = \(\frac{1}{9} - \frac{1}{36}\)

To subtract these fractions, we find a common denominator, which is 36.

Shyam's work rate = \(\frac{4}{36} - \frac{1}{36}\)

Shyam's work rate = \(\frac{3}{36}\)

Shyam's work rate = \(\frac{1}{12}\) work/hour.

Calculating Shyam's Time to Complete the Work Alone

Shyam's work rate is \(\frac{1}{12}\) work/hour. The time taken by Shyam alone is the reciprocal of his work rate.

Time taken by Shyam = \(\frac{1}{\text{Shyam's work rate}}\)

Time taken by Shyam = \(\frac{1}{\frac{1}{12}}\)

Time taken by Shyam = \(1 \times 12\)

Time taken by Shyam = 12 hours.

Therefore, Shyam alone will take 12 hours to do the piece of work.

Revision Table: Work and Time Key Concepts

Additional Information: Efficiency in Work and Time

The concept of work rate is closely related to efficiency. A higher work rate means higher efficiency, as more work is done in the same amount of time. Problems might sometimes refer to people being 'twice as efficient' as others, which directly relates their work rates. If person A is twice as efficient as person B, A's work rate is double B's work rate. Consequently, A would take half the time B takes to do the same work.

Efficiency is inversely proportional to the time taken, assuming the total amount of work is constant.