If 24 people can build 90 identical walls in 25 days, then how many more days will 27 people require to build 162 such walls?

The correct option is 15.

To solve this problem, we can use the concept of work equivalence, which relates the number of people, the number of days, and the work done.
Let:
- $M_1$ be the initial number of people = 24 people
- $W_1$ be the initial work done (number of walls) = 90 walls
- $D_1$ be the initial number of days = 25 days
- $M_2$ be the new number of people = 27 people
- $W_2$ be the new work done (number of walls) = 162 walls
- $D_2$ be the total number of days required by 27 people to build 162 walls

The relationship between men, days, and work is given by the formula:

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

Substitute the given values into the formula:

$\frac{24\times 25}{90}=\frac{27\times D_2}{162}$

Simplify the fractions:
First, simplify the left-hand side:
$\frac{24\times 25}{90}=\frac{600}{90}=\frac{20}{3}$

Next, simplify the right-hand side. Notice that 162 is divisible by 27:
$\frac{162}{27}=6$
So, the right-hand side becomes:
$\frac{27\times D_2}{162}=\frac{D_2}{6}$

Now, set the simplified expressions equal to each other to solve for $D_2$:

$\frac{20}{3}=\frac{D_2}{6}$

Multiply both sides by 6:
$D_2=\frac{20}{3}\times 6$
$D_2=20\times 2$
$D_2=40\text{days}$

Thus, 27 people will require 40 days to build 162 walls.

The question asks for how many more days 27 people will require compared to the original 25 days:
$\text{More days required}=D_2-D_1$
$\text{More days required}=40-25=15\text{days}$

Therefore, 27 people will require 15 more days.