Hari and Mohan can do a work in 20 and 25 days respectively. After doing collectively 10 days of work, they leave the work due to illness and Shyam completes rest of the work in 3 days. How many days Shyam alone can take to complete the whole work?
30 days
Work and Time Problem Solution
This solution will explain step-by-step how to determine the total number of days Shyam would take to complete the entire work alone, based on the contributions of Hari and Mohan. We will calculate individual work rates, combined work rates, and then use the remaining work to find Shyam's efficiency.
Hari's Work Rate
First, let's understand how much work Hari can do in one day. Hari can complete the entire work in 20 days. This means his daily work rate is the reciprocal of the total days he takes.
- Hari's 1-day work = $\frac{1}{\text{Total days Hari takes}}$
- Hari's 1-day work = $\frac{1}{20}$ of the total work
Mohan's Work Rate
Similarly, we calculate Mohan's daily work rate. Mohan can complete the entire work in 25 days.
- Mohan's 1-day work = $\frac{1}{\text{Total days Mohan takes}}$
- Mohan's 1-day work = $\frac{1}{25}$ of the total work
Combined Work Rate of Hari and Mohan
Next, we find their combined efficiency when they work together. To do this, we add their individual daily work rates.
- Combined 1-day work of Hari and Mohan = Hari's 1-day work + Mohan's 1-day work
- Combined 1-day work = $\frac{1}{20} + \frac{1}{25}$
- To add these fractions, we find a common denominator, which is 100.
- Combined 1-day work = $\frac{1 \times 5}{20 \times 5} + \frac{1 \times 4}{25 \times 4} = \frac{5}{100} + \frac{4}{100} = \frac{5+4}{100} = \frac{9}{100}$ of the total work
Work Completed by Hari and Mohan Together
Hari and Mohan work collectively for 10 days. We multiply their combined daily work rate by the number of days they worked.
- Work done by Hari and Mohan in 10 days = Combined 1-day work $\times$ Number of days worked
- Work done in 10 days = $\frac{9}{100} \times 10 = \frac{90}{100} = \frac{9}{10}$ of the total work
Remaining Work Calculation
After Hari and Mohan leave, Shyam completes the rest of the work. To find out how much work is remaining, we subtract the work already done from the total work (which is considered as 1 unit).
- Remaining work = Total work - Work done by Hari and Mohan
- Remaining work = $1 - \frac{9}{10} = \frac{10-9}{10} = \frac{1}{10}$ of the total work
Shyam's Daily Work Rate
Shyam completes this remaining work in 3 days. We can use this information to find Shyam's daily work rate.
- Shyam's 1-day work = $\frac{\text{Remaining work}}{\text{Days Shyam took}}$
- Shyam's 1-day work = $\frac{1/10}{3} = \frac{1}{10} \times \frac{1}{3} = \frac{1}{30}$ of the total work
Total Days for Shyam to Complete Work Alone
Finally, to find out how many days Shyam alone would take to complete the whole work, we take the reciprocal of Shyam's 1-day work rate.
- Days Shyam alone takes = $\frac{1}{\text{Shyam's 1-day work}}$
- Days Shyam alone takes = $\frac{1}{1/30} = 30$ days
Therefore, Shyam alone can complete the entire work in 30 days.
| Person | Days to complete work | 1-day work rate |
|---|---|---|
| Hari | 20 days | $\frac{1}{20}$ |
| Mohan | 25 days | $\frac{1}{25}$ |
| Hari & Mohan (combined) | N/A | $\frac{1}{20} + \frac{1}{25} = \frac{9}{100}$ |
| Step | Description | Calculation |
|---|---|---|
| 1 | Work done by Hari & Mohan in 10 days | $10 \times \frac{9}{100} = \frac{9}{10}$ |
| 2 | Remaining work | $1 - \frac{9}{10} = \frac{1}{10}$ |
| 3 | Shyam's 1-day work (for remaining work) | $\frac{1/10}{3} = \frac{1}{30}$ |
| 4 | Days Shyam alone takes to complete whole work | $\frac{1}{1/30} = 30$ days |
Important Questions from Work Efficiency
A and B together can complete a certain work in 20 days whereas B and C together can complete it in 24 days. If A is twice as good a workman as C, then in what time will B alone do 40% of the same work?
14 men can complete a work in 15 days. If 21 men are employed, then in how many days will they complete the same work?
A can do a certain work in 15 days, while B can do the same work in 21 days. If they work together, then in how many days will the same work be completed?
To do a certain work, A and B work on alternate days with B beginning the work on the first day. A alone can complete the same work in 24 days. If the work gets completed in \(11 \frac{1}{3}\) days, then B alone can complete \(\rm \frac{7}{9}^{th}\) part of the original work in:
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 can 7 men complete the same work?
