Solving the Work and Time Problem
This problem involves calculating the time taken by individuals to complete work, both individually and together. We need to find out how long Deven would take to finish the work alone.
Calculating Individual Work Rates
First, determine the rate at which Lalit and Laxman complete the work individually.
- Lalit's time to finish work = $15\text{ days}$.
- Lalit's work rate = $ \frac{1}{15} $ of the work per day.
- Laxman's time to finish work = $25\text{ days}$.
- Laxman's work rate = $ \frac{1}{25} $ of the work per day.
Work Done Together
Next, calculate the combined work rate when Lalit and Laxman work together and the amount of work they complete in $5$ days.
- Combined work rate of Lalit and Laxman = $ \frac{1}{15} + \frac{1}{25} $.
- To add these fractions, find a common denominator, which is $75$.
- Combined rate = $ \frac{5}{75} + \frac{3}{75} = \frac{8}{75} $ of the work per day.
- Work done by Lalit and Laxman in $5$ days = $ \text{Rate} \times \text{Time} = \frac{8}{75} \times 5 $.
- Work done = $ \frac{40}{75} = \frac{8}{15} $ of the total work.
Calculating Remaining Work
Determine the amount of work left after Lalit and Laxman have worked for $5$ days.
- Total work = $1$ (representing the whole job).
- Remaining work = $ 1 - \frac{8}{15} $.
- Remaining work = $ \frac{15}{15} - \frac{8}{15} = \frac{7}{15} $ of the total work.
Finding Deven's Time
The remaining work ($ \frac{7}{15} $) is completed by Lalit and Deven in $4$ days. We need to find Deven's individual time.
- Let Deven's time to finish the work alone be $D$ days.
- Deven's work rate = $ \frac{1}{D} $ of the work per day.
- Lalit's work rate is $ \frac{1}{15} $.
- Combined work rate of Lalit and Deven = $ \frac{1}{15} + \frac{1}{D} $.
- Work done by Lalit and Deven in $4$ days = $ \left(\frac{1}{15} + \frac{1}{D}\right) \times 4 $.
- This equals the remaining work: $ \left(\frac{1}{15} + \frac{1}{D}\right) \times 4 = \frac{7}{15} $.
- Distribute the $4$: $ \frac{4}{15} + \frac{4}{D} = \frac{7}{15} $.
- Isolate the term with $D$: $ \frac{4}{D} = \frac{7}{15} - \frac{4}{15} $.
- $ \frac{4}{D} = \frac{3}{15} = \frac{1}{5} $.
- Solve for $D$: $ D = 4 \times 5 = 20 $.
Therefore, Deven can finish the work alone in $20\text{ days}$.
Similar Questions
- A contractor employs 15 men to complete a work in 24 days, working 8 hours per day. After six days, he adds five more men but also reduces the working hours to six per day from the seventh day onwards. If all men work with the same efficiency throughout, how many more days will be required to finish the remaining work?
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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?
