Teams A, B, and C consist of five, eight, and ten members, respectively, such that every member within a team is equally productive. Working separately, teams A, B, and C can complete a certain job in 40 hours, 50 hours, and 4 hours, respectively. Two members from team A, three members from team B, and one member from team C together start the job, and the member from team C leaves after 23 hours. The number of additional member(s) from team B, that would be required to replace the member from team C, to finish the job in the next one hour, is
Show Hint
Compute total work done in each phase.
Subtract from 1 (or total work) to find the remaining part.
Use the required time and remaining work to find how many extra workers are needed.
- \(1\)
- \(2\)
- \(3\)
- \(4\)
The Correct Option is B
Approach Solution - 1
To solve this problem, we need to find how the work is distributed among the teams and calculate the additional members required to complete the task.
- First, find the work rate of each team:
- Team A: The entire team takes 40 hours to complete the work. Therefore, the work rate per member of team A is: \(\frac{1}{40 \times 5} = \frac{1}{200}\) of the job per hour.
- Team B: The entire team takes 50 hours to complete the work. Therefore, the work rate per member of team B is: \(\frac{1}{50 \times 8} = \frac{1}{400}\) of the job per hour.
- Team C: The entire team takes 4 hours to complete the job. Therefore, the work rate per member of team C is: \(\frac{1}{4 \times 10} = \frac{1}{40}\) of the job per hour.
- Given that 2 members from team A, 3 members from team B, and 1 member from team C work together:
- Combined work rate: \(2 \times \frac{1}{200} + 3 \times \frac{1}{400} + 1 \times \frac{1}{40}\)
- Simplifying the above: \(\frac{2}{200} + \frac{3}{400} + \frac{1}{40} = \frac{1}{100} + \frac{3}{400} + \frac{1}{40}\)
- Converting all fractions to a common denominator (400): \(\frac{4}{400} + \frac{3}{400} + \frac{10}{400} = \frac{17}{400}\) of the job per hour.
- After 23 hours of work, the amount of work completed:
- \(23 \times \frac{17}{400} = \frac{391}{400}\)
- Remaining work: \(1 - \frac{391}{400} = \frac{9}{400}\)
- The member from team C leaves. The team’s new combined rate without the member from team C for team C becomes: \(\frac{4}{400} + \frac{3}{400} = \frac{7}{400}\) of the job per hour.
- We need to finish \(\frac{9}{400}\) of the job in 1 hour. Let \(x\) be the additional members from team B needed:
- Equation: \(\frac{7}{400} + x \times \frac{1}{400} = \frac{9}{400}\)
- Solving: \(\frac{7 + x}{400} = \frac{9}{400}\)
- Thus, \(7 + x = 9\)
- So, \(x = 2\)
- Therefore, the number of additional member(s) from team B required is 2.
Approach Solution -2
Step 1: Find individual efficiencies of team members. Let the total job be \(1\) unit. \[ \text{Team A: } 5 \text{ members finish in } 40 \text{ hours} \Rightarrow \text{rate of team A} = \frac{1}{40}. \] So, each member of team A has rate \[ a = \frac{1}{40 \times 5} = \frac{1}{200}. \] \[ \text{Team B: } 8 \text{ members finish in } 50 \text{ hours} \Rightarrow \text{rate of team B} = \frac{1}{50}. \] So, each member of team B has rate \[ b = \frac{1}{50 \times 8} = \frac{1}{400}. \] \[ \text{Team C: } 10 \text{ members finish in } 4 \text{ hours} \Rightarrow \text{rate of team C} = \frac{1}{4}. \] So, each member of team C has rate \[ c = \frac{1}{4 \times 10} = \frac{1}{40}. \]
Step 2: Work done in the first 23 hours. Workers initially: 2 from A, 3 from B, 1 from C. Their combined rate: \[ 2a + 3b + c = 2 \cdot \frac{1}{200} + 3 \cdot \frac{1}{400} + \frac{1}{40} = \frac{1}{100} + \frac{3}{400} + \frac{1}{40}. \] Convert to denominator 400: \[ \frac{1}{100} = \frac{4}{400},\quad \frac{1}{40} = \frac{10}{400}. \] So: \[ 2a + 3b + c = \frac{4}{400} + \frac{3}{400} + \frac{10}{400} = \frac{17}{400}. \] Work done in 23 hours: \[ \text{Work}_1 = 23 \times \frac{17}{400} = \frac{391}{400}. \] Remaining work: \[ \text{Work}_{\text{rem}} = 1 - \frac{391}{400} = \frac{9}{400}. \]
Step 3: Replace team C member with extra members from team B. After 23 hours, the member from C leaves. Remaining workers: 2 from A, 3 from B, plus \(x\) additional members from B. New combined rate: \[ 2a + (3 + x)b = 2 \cdot \frac{1}{200} + (3 + x)\cdot \frac{1}{400} = \frac{1}{100} + \frac{3 + x}{400} = \frac{4}{400} + \frac{3 + x}{400} = \frac{7 + x}{400}. \] We want the remaining work \(\dfrac{9}{400}\) to be finished in the next 1 hour: \[ \frac{7 + x}{400} \times 1 = \frac{9}{400} \quad \Rightarrow \quad 7 + x = 9 \quad \Rightarrow \quad x = 2. \] So, the number of additional members from team B required is: \[ \boxed{2}. \]
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