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

The correct option is 2.

To find the number of additional members needed from Team B, let us determine the individual productivity (rate of work) for a single member of each team.

Step 1: Calculate the work rate of a single member from each team
Let the total work of the job be represented by 1 unit.
• Team A: consists of 5 members and can complete the job in 40 hours.
Therefore, the rate of work of the entire Team A is:
$$R_{\mathrm{A,\; team}}=\frac{1}{40}\text{ job per hour}$$
The work rate of 1 individual member of Team A is:
$$r_A=\frac{1}{5\times 40}=\frac{1}{200}\text{ job per hour}$$
• Team B: consists of 8 members and can complete the job in 50 hours.
The work rate of 1 individual member of Team B is:
$$r_B=\frac{1}{8\times 50}=\frac{1}{400}\text{ job per hour}$$
• Team C: consists of 10 members and can complete the job in 4 hours.
The work rate of 1 individual member of Team C is:
$$r_C=\frac{1}{10\times 4}=\frac{1}{40}\text{ job per hour}$$

Step 2: Calculate the work completed in the first 23 hours
Initially, the group working together consists of 2 members from Team A, 3 members from Team B, and 1 member from Team C.
The combined hourly rate of this group is:
$$R_{\mathrm{initial}}=(2\times r_A)+(3\times r_B)+(1\times r_C)$$
Substituting the values we calculated:
$$R_{\mathrm{initial}}=\frac{2}{200}+\frac{3}{400}+\frac{1}{40}$$
Finding a common denominator of 400:
$$R_{\mathrm{initial}}=\frac{4}{400}+\frac{3}{400}+\frac{10}{400}=\frac{17}{400}\text{ job per hour}$$
The work done by this group in 23 hours is:
$$W_{\mathrm{done}}=23\times \frac{17}{400}=\frac{391}{400}$$

Step 3: Calculate the remaining work
The portion of the job remaining after 23 hours is:

Step 4: Determine the additional members required from Team B
The member from Team C leaves the job. To finish the remaining work in the next 1 hour, let $x$ be the number of additional members from Team B recruited to replace them.
The new group now consists of:
• 2 members of Team A
• $3+x$ members of Team B

The total work rate of this new group must be equal to the remaining work divided by the remaining time (1 hour):
$$(2\times r_A)+\left(3+x\right)\times r_B=\frac{{\mathrm{W<!--\; <mi-->remaining}}_{}}{1\text{ hour}}$$
Substituting the known values:
$$(2\times \frac{\mathrm{1<!--\; <mn-->200}}{})+(\mathrm{3<!--\; <mo-->+\; <mi>x</mi>\; <mo>)</mo>}\times \frac{\mathrm{1<!--\; <mn-->400}}{}=\frac{9}{400}$$
Simplify the equation:
$$\frac{4}{400}+\frac{3+x}{400}=\frac{9}{400}$$
Since the denominators are all equal to 400, we can equate the numerators:


$$x=2$$
Thus, 2 additional members from Team B are required to finish the job on time.