CAT 2025

Slot 3

QA

Question & SolutionQues & Sol

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

1. Concept Used

• Topic: Time and Work — Individual Member Productivity & Partial Work Completion
• Formula: $$ \text{Total Work} = \text{Number of Members} \times \text{Time Taken} \times \text{Per Member Per Hour Output} $$

2. Calculation

Let the total work be $W$ units, and let the per-hour productivity of each member from teams A, B, and C be $a$, $b$, and $c$ respectively.

Since each team independently completes the same job $W$:

$$W = 5 \times 40 \times a \implies a = \frac{W}{200}$$

$$W = 8 \times 50 \times b \implies b = \frac{W}{400}$$

$$W = 10 \times 4 \times c \implies c = \frac{W}{40}$$

Notice that one member of team C is 10 times more productive per hour than one member of team A, and 10 times more productive than one member of team B (scaled). This is the key insight — team C is extremely efficient per member.

Phase 1 (First 23 hours): 2 members from A + 3 members from B + 1 member from C work together.

Combined per-hour output:

$$2a + 3b + c = \frac{2W}{200} + \frac{3W}{400} + \frac{W}{40}$$

$$= \frac{4W}{400} + \frac{3W}{400} + \frac{10W}{400} = \frac{17W}{400}$$

Work completed in 23 hours:

$$23 \times \frac{17W}{400} = \frac{391W}{400}$$

Remaining work after 23 hours:

$$W - \frac{391W}{400} = \frac{9W}{400}$$

Phase 2 (Next 1 hour): The C member leaves. Only 2 from A and 3 from B remain, plus $$n$$ additional members from B are added.

Current output of remaining members (A and B only) per hour:

$$2a + 3b = \frac{2W}{200} + \frac{3W}{400} = \frac{4W}{400} + \frac{3W}{400} = \frac{7W}{400}$$

Work that can be done by existing members in 1 hour: $\frac{7W}{400}$

Shortfall (work still left after existing members contribute for 1 hour):

$$\frac{9W}{400} - \frac{7W}{400} = \frac{2W}{400}$$

Each additional member from B contributes $b = \frac{W}{400}$ per hour.

Number of additional B members required:

$$n = \frac{2W/400}{W/400} = 2$$

3. Solution

Answer = Option B ✅

The number of additional members required from team B to finish the remaining job in exactly one hour is 2.