Mahesh and Suresh can complete a work alone in 20 days and 30 days respectively. They received Rs. 3300 for completing the work together. What is the share of Mahesh?
Rs. 1980
Understanding Work and Wages Problems
This problem involves the concept of work and wages. When multiple individuals work together to complete a task and receive a total wage, the wage is typically distributed among them in proportion to the amount of work they complete. If they work for the same duration (as implied when they complete the same total work together), their share of the wages is proportional to their efficiency or their rate of doing work.
Calculating Individual Work Rates
The rate at which a person completes work is the inverse of the time taken to complete the entire work alone. If Mahesh takes 20 days to complete the work alone, his daily work rate is $\frac{1}{20}$ of the total work.
- Mahesh's time to complete work alone = 20 days
- Mahesh's daily work rate = $\frac{1}{20}$ work per day
Similarly, if Suresh takes 30 days to complete the work alone, his daily work rate is $\frac{1}{30}$ of the total work.
- Suresh's time to complete work alone = 30 days
- Suresh's daily work rate = $\frac{1}{30}$ work per day
Determining the Ratio of Shares
Since Mahesh and Suresh work together to complete the work, and assuming they work for the same duration until the work is finished, their shares in the total wage will be directly proportional to their individual work rates (efficiency).
The ratio of their daily work rates is:
Mahesh's Rate : Suresh's Rate = $\frac{1}{20} : \frac{1}{30}$
To simplify this ratio and work with whole numbers, we can multiply both parts of the ratio by the least common multiple (LCM) of 20 and 30, which is 60.
Ratio = $\frac{1}{20} \times 60 : \frac{1}{30} \times 60$
Ratio = $3 : 2$
So, the total wage of Rs. 3300 should be divided between Mahesh and Suresh in the ratio $3:2$.
Calculating Mahesh's Share
The total ratio parts are $3 + 2 = 5$.
Mahesh's share corresponds to 3 out of these 5 parts.
Total wage received for completing the work = Rs. 3300
Mahesh's share = (Mahesh's ratio part / Total ratio parts) $\times$ Total Wage
Mahesh's share = $\frac{3}{5} \times 3300$
Mahesh's share = $3 \times \frac{3300}{5}$
Mahesh's share = $3 \times 660$
Mahesh's share = Rs. 1980
We can also calculate Suresh's share for completeness:
Suresh's share = (Suresh's ratio part / Total ratio parts) $\times$ Total Wage
Suresh's share = $\frac{2}{5} \times 3300$
Suresh's share = $2 \times 660$
Suresh's share = Rs. 1320
Check: Mahesh's share + Suresh's share = $1980 + 1320 = 3300$. This matches the total wage.
Summary of Shares
| Person | Time Alone (days) | Daily Rate | Ratio Part | Share (Rs.) |
|---|---|---|---|---|
| Mahesh | 20 | $\frac{1}{20}$ | 3 | 1980 |
| Suresh | 30 | $\frac{1}{30}$ | 2 | 1320 |
| Total | - | - | 5 | 3300 |
Thus, the share of Mahesh for completing the work together is Rs. 1980.
Revision Table: Work and Wages Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Work Rate | Amount of work done per unit of time. It's the reciprocal of the time taken to complete the whole work. | Mahesh's rate = $1/20$, Suresh's rate = $1/30$. |
| Ratio of Shares | When people work together for the same time, wages are divided in the ratio of their work rates (efficiency). | Ratio of shares = Mahesh's Rate : Suresh's Rate = $3:2$. |
| Total Ratio Parts | Sum of the parts in the ratio used for distribution. | Total parts = $3+2=5$. |
| Individual Share Calculation | (Individual's Ratio Part / Total Ratio Parts) $\times$ Total Wage. | Mahesh's share = $(3/5) \times 3300$. |
Additional Information: Alternative Approach using LCM (Total Work)
An alternative method to solve work and wages problems is using the LCM of the times taken to represent the 'total work units'.
- Time for Mahesh = 20 days
- Time for Suresh = 30 days
Let the total work be the LCM of 20 and 30, which is 60 units.
- Mahesh's daily efficiency = Total work / Mahesh's time = $60 / 20 = 3$ units/day
- Suresh's daily efficiency = Total work / Suresh's time = $60 / 30 = 2$ units/day
When they work together for the same period, their shares are proportional to their daily efficiency.
Ratio of efficiencies = Mahesh's efficiency : Suresh's efficiency = $3 : 2$.
This ratio is the same as the ratio of rates calculated earlier. The total wage (Rs. 3300) is divided in this $3:2$ ratio.
Mahesh's share = $\frac{3}{3+2} \times 3300 = \frac{3}{5} \times 3300 = 3 \times 660 = \text{Rs. } 1980$.
This LCM method often makes the calculation of the ratio simpler by avoiding fractions initially.
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?
