Question Details

A certain number of men can complete a piece of work in 6k days, where k is a natural number. By what percent should the number of men be increased so that the work can be completed in 5k days?

Options

10%

(50/3)%

20%

25%

Show Answer

Correct Answer :

Option C

20%

Solution :

The correct option is 20%.

Here is the step-by-step derivation of the solution:

Step 1: Understand the Relationship between Men and Days
The amount of work done is directly proportional to the number of men working and the number of days they work. When the total work remains constant, the number of men ( $M$ ) and the number of days ( $D$ ) are inversely proportional to each other. This relationship can be expressed as:
$M_{1} \times D_{1} = M_{2} \times D_{2}$
where:
- $M_{1}$ is the initial number of men,
- $D_{1}$ is the initial number of days,
- $M_{2}$ is the final number of men, and
- $D_{2}$ is the final number of days.

Step 2: Substitute the Given Values
We are given that:
- The initial number of days, $D_{1} = 6 k$
- The final number of days, $D_{2} = 5 k$
Substituting these values into our relation:
$M_{1} \times 6 k = M_{2} \times 5 k$
We can divide both sides by the natural number $k$ to simplify the equation:
$6 M_{1} = 5 M_{2}$
Solving for $M_{2}$ :
$M_{2} = \frac{6}{5} M_{1}$

Step 3: Calculate the Percentage Increase in the Number of Men
The increase in the number of men is given by:
$Increase = M_{2} - M_{1}$
Substituting $M_{2}$ in terms of $M_{1}$ :
$Increase = \frac{6}{5} M_{1} - M_{1} = \frac{1}{5} M_{1}$
Now, we find the percentage increase with respect to the initial number of men ( $M_{1}$ ):
$Percentage Increase = (\frac{Increase}{M_{1}}) \times 100 %$
$Percentage Increase = (\frac{\frac{1}{5} M_{1}}{M_{1}}) \times 100 %$
$Percentage Increase = \frac{1}{5} \times 100 % = 20 %$
Thus, the number of men should be increased by 20% to complete the work in 5k days.