If 10 persons can complete $\frac{2}{5}^th$ part of a work in 8 days, then how many persons are required to complete the remaining work in 12 days?

Question

Accepted Answer

The correct option is 3:
This problem is a classic example of a Time and Work question, often solved using the principle of inverse proportionality, specifically the formula that relates Men (M), Days (D), and Work (W):

$$ \frac{M_1 	imes D_1}{W_1} = \frac{M_2 	imes D_2}{W_2} $$

This formula is based on the concept that the amount of work done is directly proportional to the number of people working and the number of days they work (Work $\propto$ Men $	imes$ Days). When comparing two different scenarios (Scenario 1 and Scenario 2) involving the same type of work and people with uniform efficiency, the ratio of (Men $	imes$ Days) to the Work done must remain constant.

Let's analyze the given information for both scenarios:

**Scenario 1 (Initial Work):**  * $M\_1$ (Number of persons) = 10
  * $W\_1$ (Part of work completed) = $\frac{2}{5}$
  * $D\_1$ (Number of days) = 8**Scenario 2 (Remaining Work):**  * $W\_2$ (Remaining part of work) = $1 - \frac{2}{5} = \frac{5-2}{5} = \frac{3}{5}$
  * $D\_2$ (Number of days to complete remaining work) = 12
  * $M\_2$ (Number of persons required) = ? (This is what we need to find)Now we apply the combined proportionality rule:

$$ \frac{M_1 	imes D_1}{W_1} = \frac{M_2 	imes D_2}{W_2} $$

Substitute the known values into the equation:

$$ \frac{10 	imes 8}{\frac{2}{5}} = \frac{M_2 	imes 12}{\frac{3}{5}} $$

We can simplify the work fractions by multiplying the numerator and denominator of both sides by 5:

$$ \frac{10 	imes 8}{2} = \frac{M_2 	imes 12}{3} $$

Simplify the left side:
$$ \frac{80}{2} = 40 $$

Simplify the right side:
$$ \frac{12 M\_2}{3} = 4 M\_2 $$

Therefore, the equation becomes:
$$ 40 = 4 M\_2 $$

Solve for $M\_2$:
$$ M\_2 = \frac{40}{4} $$
$$ M\_2 = 10 $$

Thus, **10 persons** are required to complete the remaining $\frac{3}{5}$ of the work in 12 days.

The incorrect option is 1:
Option 1 suggests that 6 persons are required. If 6 persons were to complete $3/5$ of the work in 12 days, their output rate ($M \times D / W$) would be significantly lower than the established rate of 10 persons working for 8 days to complete $2/5$ of the work. Since $M\_2 = 10$ maintains the same overall work rate, 6 people would not be enough; they would require more than 12 days to finish the task.

The incorrect option is 2:
Option 2 suggests that 8 persons are required. Using the calculation, we found that 10 persons are necessary to maintain the required output rate and complete the remaining work in the specified 12 days. If only 8 persons were assigned, they would take longer than 12 days to finish the work, making this option incorrect as it fails to meet the time constraint given in the problem.

The incorrect option is 4:
Option 4 suggests that 12 persons are required. If 12 persons were put on the job, they would be able to complete the remaining work in less than 12 days, as 10 people is the exact requirement for a 12-day completion timeline. In scenarios where a specific time frame (12 days) must be met, assigning more than the calculated number of workers (10) would lead to premature completion or inefficient use of resources, but 10 is the precise number needed.