A can complete a work alone in 8 days. B can complete the same work alone in 12 days. C alone complete the same work in 16 days. They complete the work in 3 days with the help of D. If they get Rs.12000 for the work, then how much money does the D get?
Rs. 2250
Understanding the Problem: Work, Time, and Money Distribution
This problem involves multiple individuals working together to complete a task and then distributing the earnings based on the amount of work each person contributed. We are given the time each person (A, B, C) takes to complete the work alone, the time they take to complete the work together with an additional person (D), and the total earnings for the work. Our goal is to find out how much money D gets.
The key principle here is that the amount of money a person earns is proportional to the amount of work they do. First, we need to figure out what fraction of the total work D completed.
Calculating Individual Work Rates
The work rate of a person is the amount of work they can complete in one day. If a person can complete the entire work in 'n' days, their daily work rate is \( \frac{1}{n} \) of the total work.
- A can complete the work in 8 days.
- A's daily work rate = \( \frac{1}{8} \) of the work.
- B can complete the work in 12 days.
- B's daily work rate = \( \frac{1}{12} \) of the work.
- C can complete the work in 16 days.
- C's daily work rate = \( \frac{1}{16} \) of the work.
Combined Work Rate and D's Contribution
A, B, C, and D together complete the work in 3 days. This means their combined daily work rate is \( \frac{1}{3} \) of the work.
The combined daily work rate of A, B, C, and D is the sum of their individual daily work rates:
Combined rate (A+B+C+D) = Rate (A) + Rate (B) + Rate (C) + Rate (D)
\( \frac{1}{3} = \frac{1}{8} + \frac{1}{12} + \frac{1}{16} + \text{Rate (D)} \)
To find D's daily work rate, we subtract the combined rate of A, B, and C from the combined rate of A, B, C, and D.
First, let's find the combined daily rate of A, B, and C:
Combined rate (A+B+C) = \( \frac{1}{8} + \frac{1}{12} + \frac{1}{16} \)
To add these fractions, we find a common denominator. The least common multiple (LCM) of 8, 12, and 16 is 48.
Combined rate (A+B+C) = \( \frac{6}{48} + \frac{4}{48} + \frac{3}{48} = \frac{6+4+3}{48} = \frac{13}{48} \)
Now, we find D's daily rate:
Rate (D) = Combined rate (A+B+C+D) - Combined rate (A+B+C)
Rate (D) = \( \frac{1}{3} - \frac{13}{48} \)
Find a common denominator for 3 and 48. The LCM is 48.
Rate (D) = \( \frac{16}{48} - \frac{13}{48} = \frac{16-13}{48} = \frac{3}{48} = \frac{1}{16} \)
So, D's daily work rate is \( \frac{1}{16} \) of the work.
Calculating Work Done by Each Person in 3 Days
They completed the work in 3 days. We need to find the total work done by each person over these 3 days.
- Work done by A in 3 days = Rate (A) \(\times\) 3 days = \( \frac{1}{8} \times 3 = \frac{3}{8} \)
- Work done by B in 3 days = Rate (B) \(\times\) 3 days = \( \frac{1}{12} \times 3 = \frac{3}{12} = \frac{1}{4} \)
- Work done by C in 3 days = Rate (C) \(\times\) 3 days = \( \frac{1}{16} \times 3 = \frac{3}{16} \)
- Work done by D in 3 days = Rate (D) \(\times\) 3 days = \( \frac{1}{16} \times 3 = \frac{3}{16} \)
Let's check if the sum of work done by A, B, C, and D equals the total work (which is 1 unit):
\( \frac{3}{8} + \frac{1}{4} + \frac{3}{16} + \frac{3}{16} \)
Find a common denominator, which is 16.
\( \frac{6}{16} + \frac{4}{16} + \frac{3}{16} + \frac{3}{16} = \frac{6+4+3+3}{16} = \frac{16}{16} = 1 \)
The fractions of work done by each person are:
- A: \( \frac{3}{8} \)
- B: \( \frac{1}{4} \)
- C: \( \frac{3}{16} \)
- D: \( \frac{3}{16} \)
Distributing the Total Money
The total amount of money earned is Rs. 12000. This money is distributed among A, B, C, and D in proportion to the work they completed.
D's share of the money = Total money \(\times\) (Fraction of work done by D)
D's share = \( \text{Rs. } 12000 \times \frac{3}{16} \)
Calculate the value:
\( 12000 \times \frac{3}{16} = \frac{12000 \times 3}{16} = \frac{36000}{16} \)
Dividing 36000 by 16:
\( \frac{36000}{16} = \frac{18000}{8} = \frac{9000}{4} = \frac{4500}{2} = 2250 \)
So, D gets Rs. 2250.
Summary of Calculations
| Person | Time alone (days) | Daily Rate (fraction/day) | Work in 3 days (fraction) |
|---|---|---|---|
| A | 8 | \( \frac{1}{8} \) | \( \frac{1}{8} \times 3 = \frac{3}{8} \) |
| B | 12 | \( \frac{1}{12} \) | \( \frac{1}{12} \times 3 = \frac{3}{12} = \frac{1}{4} \) |
| C | 16 | \( \frac{1}{16} \) | \( \frac{1}{16} \times 3 = \frac{3}{16} \) |
| A+B+C+D | 3 | \( \frac{1}{3} \) | 1 (whole work) |
| D | Calculated from combined rate | \( \frac{1}{3} - (\frac{1}{8} + \frac{1}{12} + \frac{1}{16}) = \frac{16}{48} - \frac{13}{48} = \frac{3}{48} = \frac{1}{16} \) | \( \frac{1}{16} \times 3 = \frac{3}{16} \) |
Total work done = 1. Total money = Rs. 12000.
D's share = Total Money \( \times \) Work done by D = \( 12000 \times \frac{3}{16} = 2250 \)
D receives Rs. 2250.
Revision Table: Key Concepts in Work and Time Problems
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | Rate = \( \frac{1}{\text{Time taken to complete work}} \) |
| Total Work | Often considered as 1 unit or a value equal to the LCM of individual times. | Total Work = Rate \( \times \) Time |
| Combined Rate | Sum of individual work rates when people work together. | Rate (A+B) = Rate (A) + Rate (B) |
| Distribution of Earnings | Money is shared in proportion to the fraction of total work done by each person. | Person's Share = Total Earnings \( \times \) \( \frac{\text{Work done by person}}{\text{Total work}} \) |
Additional Information: Proportionality in Work and Money
In work and time problems where earnings are mentioned, the distribution of money is directly proportional to the work done by each individual or group. If everyone works for the same amount of time (like 3 days in this problem), the ratio of money received is the same as the ratio of their daily work rates, multiplied by the number of days they worked.
In this case, the work done ratio for A:B:C:D in 3 days is:
\( \frac{3}{8} : \frac{1}{4} : \frac{3}{16} : \frac{3}{16} \)
Multiplying by the LCM (16) to get integer ratios:
\( (\frac{3}{8} \times 16) : (\frac{1}{4} \times 16) : (\frac{3}{16} \times 16) : (\frac{3}{16} \times 16) \)
\( 6 : 4 : 3 : 3 \)
The total ratio units are \( 6+4+3+3 = 16 \). The total money is Rs. 12000.
One ratio unit corresponds to \( \frac{12000}{16} = 750 \) Rupees.
D's share corresponds to 3 ratio units.
D's share = \( 3 \times 750 = 2250 \).
This confirms the earlier calculation based directly on D's fraction of work.
Important Questions from Work and Wages
If 15 men can complete a work in 16 days by working 8 hours daily, then in how many days will 10 men complete the work by working 12 hours daily?
10 men working 8 hours a day can finish a work in 28 days. In how many days, 8 men working 5 hours a day with complete 50% of that work?
A can do a work in 12 days and B in 16 days. They undertook to do it for Rs. 6000 with the help of ‘C’ they completed the work in 6 days. What is the share of A?
Twenty lamps can be lighted for 6 hr a day for 20 days at a cost of Rs. 100. How much would be the cost of lighting 40 lamps, 8 hr for 12 days?
Six men can complete a job in two days. Four boys can complete the same job in eight days. In how many days would three men and six boys, working together, be able to complete the job?
