Work Rate Calculation
The first step is to determine the individual work rates for each worker. The rate is the amount of work done per hour.
- Worker A finishes the job in 10 hours, so A's rate is $ \frac{1}{10} $ job/hour.
- Worker B finishes the job in 15 hours, so B's rate is $ \frac{1}{15} $ job/hour.
- Worker C finishes the job in 20 hours, so C's rate is $ \frac{1}{20} $ job/hour.
Work Done by A and B Initially
Workers A and B start the job together and work for 2 hours. We calculate their combined rate first.
Combined Rate of A and B = $ \text{Rate}_A + \text{Rate}_B = \frac{1}{10} + \frac{1}{15} $
Using the least common multiple (LCM) of 10 and 15, which is 30:
Combined Rate of A and B = $ \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} $ job/hour.
The work done by A and B in these 2 hours is:
Work Done = Rate × Time = $ \frac{1}{6} \times 2 = \frac{2}{6} = \frac{1}{3} $ of the job.
Remaining Work Calculation
After A and B work for 2 hours, the remaining portion of the job is:
Remaining Work = Total Work - Work Done = $ 1 - \frac{1}{3} = \frac{2}{3} $ of the job.
Combined Rate of A, B, and C
After 2 hours, worker C joins A and B. They now work together to complete the remaining work. We need their new combined rate.
Combined Rate of A, B, and C = $ \text{Rate}_A + \text{Rate}_B + \text{Rate}_C = \frac{1}{10} + \frac{1}{15} + \frac{1}{20} $
Using the LCM of 10, 15, and 20, which is 60:
Combined Rate of A, B, and C = $ \frac{6}{60} + \frac{4}{60} + \frac{3}{60} = \frac{13}{60} $ job/hour.
Time 't' Calculation
The time 't' (in hours) required to finish the remaining work ($ \frac{2}{3} $) is calculated using the formula: Time = Work / Rate.
$ t = \frac{\text{Remaining Work}}{\text{Combined Rate of A, B, and C}} = \frac{2/3}{13/60} $
$ t = \frac{2}{3} \times \frac{60}{13} = \frac{120}{39} $
Simplifying the fraction gives:
$ t = \frac{40}{13} $ hours.
Rounding Off the Result
The question asks for the value of 't' rounded off to the nearest integer.
$ t = \frac{40}{13} \approx 3.0769 $
Rounding $3.0769$ to the nearest integer results in 3.
Therefore, the value of t is 3 hours.
Important Questions from Time & Work (Notes)
- A fresh water tap fills a fish tank in 40 minutes. The same tank is filled by a salt water tap in 120 minutes. If both the taps are open, how many minutes will it take to fill the tank?
- A completes $\frac{7}{10}$ of a work in 15 days and then he completes the remaining work with the help of B in 5 days. In how many days can A and B together complete the entire work?
Aman can do 50% of the job in 16 days, and Bhanu can do 25% of the job in 24 days. In how many days can they do $\frac{1}{4}^{th}$ of the job working together ?
- X can finish a job in $141$ days. He worked for $57$ days alone and the remaining work was completed by Y, in $84$ days. How many days would both together take to complete the entire job?
Ravina, Sujata, and Saroj can complete a work of painting separately in 32, 48, and 64 hours, respectively. They started working together, but Saroj left after 5 hours. From the 6th hour, Ravina and Sujata decided to work on alternate hours starting with Ravina. In how much time will the entire work of painting be completed?
