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:
A$2$
B$4$
C$1$
D$3$
âš¡ AZUCATION SHORTCUT:
Rate of 1 member: $A = \frac{1}{200}$, $B = \frac{1}{400}$, $C = \frac{1}{40}$.
Using $B$ as 1 unit: $A = 2$ units, $C = 10$ units. Total Job = $400$ units.
Work done in 23 hrs: $(2\times2 + 3\times1 + 1\times10) \times 23 = 17 \times 23 = 391$ units.
Remaining = $9$ units. Current rate ($2A+3B$) = $7$ units.
Need $2$ more units $\implies$ 2 members of B.
Detailed Step-by-Step Solution
Step 1: Calculate individual productivity
Let the total work be $W$ units.
- Team A (5 members) takes 40 hrs: $1 \text{ member of A} = \frac{W}{5 \times 40} = \frac{W}{200}$ per hour.
- Team B (8 members) takes 50 hrs: $1 \text{ member of B} = \frac{W}{8 \times 50} = \frac{W}{400}$ per hour.
- Team C (10 members) takes 4 hrs: $1 \text{ member of C} = \frac{W}{10 \times 4} = \frac{W}{40}$ per hour.
Step 2: Simplify ratios (Let $W = 400$ units)
- Rate of 1 member of A ($r_A$) = $2$ units/hr.
- Rate of 1 member of B ($r_B$) = $1$ unit/hr.
- Rate of 1 member of C ($r_C$) = $10$ units/hr.
Step 3: Work done in the first 23 hours
Group: 2 from A, 3 from B, 1 from C.
Total Hourly Rate = $(2 \times 2) + (3 \times 1) + (1 \times 10) = 4 + 3 + 10 = 17$ units/hr.
Work done in 23 hrs $= 17 \times 23 = 391$ units.
Remaining Work $= 400 - 391 = 9$ units.
Step 4: Final calculation
To finish $9$ units in $1$ hour, the required rate is $9$ units/hr.
The member from C leaves. Remaining members (2A + 3B) provide $(2\times2 + 3\times1) = 7$ units/hr.
Additional rate needed $= 9 - 7 = 2$ units/hr.
Since 1 member of Team B provides 1 unit/hr, 2 additional members of Team B are required.
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