Four years ago, the ratio of the ages of Aarav and Kabir was 5:2. Eight years from now, Aarav's age will be 1.75 times that of Kabir. Find the ratio of the ages of Aarav to Kabir two years from now.

The correct option is 2:1.

Let the present age of Aarav be
$A$
years and the present age of Kabir be
$K$
years.

Step 1: Form an equation from the ratio of their ages 4 years ago.
Four years ago, Aarav's age was
$A-4$
and Kabir's age was
$K-4$
.
According to the question, the ratio of their ages 4 years ago was 5:2:

$\frac{A-4}{K-4}=\frac{5}{2}$

Cross-multiplying to simplify:
$2(A-4)=5(K-4)$
$2A-8=5K-20$
$2A=5K-12$     (Equation 1)

Step 2: Form an equation from their ages 8 years from now.
Eight years from now, Aarav's age will be
$A+8$
and Kabir's age will be
$K+8$
.
We are given that Aarav's age will be 1.75 times Kabir's age. Note that
$1.75=\frac{7}{4}$
:

$A+8=1.75\times (K+8)$

$A+8=\frac{7}{4}(K+8)$

$4(A+8)=7(K+8)$

$4A+32=7K+56$

$4A=7K+24$     (Equation 2)

Step 3: Solve the equations to find present ages.
From Equation 1, we have
$2A=5K-12$
.
Rewriting
$4A$
as
$2\left(2A\right)$
and substituting Equation 1 into Equation 2:

$2(5K-12)=7K+24$

$10K-24=7K+24$

$10K-7K=24+24$

$3K=48$

$K=16$

Now substitute
$K=16$
into Equation 1:

$2A=5\left(16\right)-12$

$2A=80-12=68$

$A=34$

Thus, Aarav's present age is 34 years and Kabir's present age is 16 years.

Step 4: Calculate the ratio of their ages 2 years from now.
Aarav's age in 2 years =
$34+2=36$
years.
Kabir's age in 2 years =
$16+2=18$
years.
Ratio of Aarav's age to Kabir's age =
$36:18=2:1$
.

Hence, the ratio of the ages of Aarav to Kabir two years from now is 2:1.