Ram can do a piece of work in 18 hours. Shyam and Naresh can do it in 9 hours. Ram and Naresh can do it in 12 hours. How long will Shyam alone take to do it?
12 hours
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
| Concept | Formula/Relation | Explanation |
|---|---|---|
| Work Rate | Work Rate = \( \frac{1}{\text{Time taken}} \) | The amount of work done per unit of time. |
| Total Work | Total Work = Rate \(\times\) Time | Often considered as 1 unit for a single task. |
| Combined Rate | Rate\(_1\) + Rate\(_2\) + ... | Sum of individual work rates when people work together. |
| Time from Rate | Time = \( \frac{1}{\text{Work Rate}} \) | Inverse relationship between rate and time for a fixed amount of work. |
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.
Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
