Understanding Work Rates
This problem involves calculating the time taken by an individual (C) to complete a job, given the times taken by others (A and B) individually and the time taken by all three together.
- The rate of work is the reciprocal of the time taken to complete the job.
- Rate = $\frac{1}{\text{Time}}$
Calculating Individual Work Rates
First, determine the work rate of A and B:
- A's time = 22 days. A's rate = $\frac{1}{22}$ of the job per day.
- B's time = 10 days. B's rate = $\frac{1}{10}$ of the job per day.
Calculating Combined Work Rate (A, B, and C)
The combined time for A, B, and C to complete the job is 2 days. Their combined work rate is:
- Combined rate (A+B+C) = $\frac{1}{2}$ of the job per day.
Finding C's Work Rate
The combined rate is the sum of individual rates: Rate(A+B+C) = Rate(A) + Rate(B) + Rate(C). We can find C's rate by subtracting A's and B's rates from the combined rate.
- Calculate the combined rate of A and B: Rate(A) + Rate(B) = $\frac{1}{22} + \frac{1}{10}$
- Find a common denominator (LCM of 22 and 10 is 110): $\frac{5}{110} + \frac{11}{110} = \frac{16}{110} = \frac{8}{55}$ job per day.
- Calculate C's rate: Rate(C) = Rate(A+B+C) - (Rate(A) + Rate(B)) Rate(C) = $\frac{1}{2} - \frac{8}{55}$
- Find a common denominator (LCM of 2 and 55 is 110): Rate(C) = $\frac{55}{110} - \frac{16}{110} = \frac{39}{110}$ job per day.
Calculating C's Time to Complete the Job
C's time is the reciprocal of C's work rate.
- C's time = $\frac{1}{\text{Rate(C)}} = \frac{1}{\frac{39}{110}}$ days
- C's time = $\frac{110}{39}$ days
Converting to Mixed Fraction
To express $\frac{110}{39}$ as a mixed fraction:
- Divide 110 by 39: $110 \div 39 = 2$ with a remainder of $32$ ($110 = 2 \times 39 + 32$).
- Therefore, C's time = $2 \frac{32}{39}$ days.
Similar Questions
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Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
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 will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
