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
- M is thrice as efficient as workman as N and therefore is able to finish a job in 264 days less than N. Then in how many days can N finish the entire work?
- A can do a piece of work in $1455$ days. B is $25\%$ more efficient than A. Find the number of days taken by B to do the same piece of work.
- 45 people take 18 days to dig a pond. If the pond would have to be dug in 15 days, then the number of people to be employed will be
- A stock of food is enough for 240 men for 48 days. How long will the same stock last for 160 men?
One contractor had undertaken a construction work. To complete the given job, he sought labour comprising 24 men for 12 days for working 10 hours a day. But due to festival season, as the labour had gone to their respective hometowns, he could get only 15 men and they too were willing to work 8 hours a day. With this much labour, in how many days would he be able to complete the same job?
