Five years ago, the ratio of the ages of Amit and Bhanu was 2 : 3. Fifteen years from now, their ages will be in the ratio of 5 : 6. Find the sum of their present ages. (Give your answer in years and months, if relevant.)

The correct option is 43 years, 4 months.

Let us solve the problem step-by-step by setting up algebraic equations representing Amit's and Bhanu's ages.

Step 1: Define variables for their ages 5 years ago
Five years ago, the ratio of the ages of Amit and Bhanu was 2 : 3.
Therefore, we can represent their ages five years ago as:
Amit's age 5 years ago = 2x
Bhanu's age 5 years ago = 3x
where x is a common multiplier.

Step 2: Write expressions for their present ages
Since those were their ages 5 years ago, we add 5 to each to find their present ages:
Amit's present age = 2x + 5
Bhanu's present age = 3x + 5

Step 3: Write expressions for their ages 15 years from now
To find their ages 15 years in the future, we add 15 to their present ages:
Amit's age in 15 years = (2x + 5) + 15 = 2x + 20
Bhanu's age in 15 years = (3x + 5) + 15 = 3x + 20

Step 4: Use the ratio given for their future ages to solve for x
We are given that fifteen years from now, their ages will be in the ratio of 5 : 6. We can write this relation as:

$$\frac{2x+20}{3x+20}=\frac{5}{6}$$

Now, we cross-multiply to solve the equation:
6 * (2x + 20) = 5 * (3x + 20)
12x + 120 = 15x + 100

Rearranging the terms to solve for x:
120 - 100 = 15x - 12x
20 = 3x
x = 20/3

Step 5: Calculate the sum of their present ages
The sum of their present ages is:
Sum = (2x + 5) + (3x + 5) = 5x + 10

Substitute the value of x = 20/3 into the sum expression:

$$\text{Sum}=5\left(\frac{20}{3}\right)+10$$

$$\text{Sum}=\frac{100}{3}+10$$

$$\text{Sum}=\frac{100+30}{3}=\frac{130}{3}\text{years}$$

Step 6: Convert the sum into years and months
To convert 130/3 years into years and months:
130 / 3 = 43 with a remainder of 1.
So, 130/3 years = 43 years + 1/3 year.

Since 1 year has 12 months:

$$\frac{1}{3}\times 12\text{months}=4\text{months}$$

Thus, the sum of their present ages is 43 years and 4 months.