The present age of Shobha is equal to one fifth of her mother sudha's age.

Let $S$ be the present age of Shobha and $M$ be the present age of her mother Sudha. We are given two pieces of information:

1. Shobha's present age is one fifth of Sudha's present age:

$$S = \frac{1}{5} M$$

2. Twenty-five years later, Shobha's age will be 4 years less than half of Sudha's age at that time:

$$S + 25 = \frac{1}{2} (M + 25) - 4$$

We need to solve these equations to find the present ages of Shobha and Sudha.

Solve for $S$ and $M$:

First, simplify the second equation:

$$S + 25 = \frac{1}{2} (M + 25) - 4$$

Multiply both sides of the equation by 2 to eliminate the fraction:

$$2(S + 25) = M + 25 - 8$$ $$2S + 50 = M + 17$$

Rearrange this to:

$$2S = M - 33$$
$$M = 2S + 33$$

Now substitute $M = 2S + 33$ into the first equation $S = \frac{1}{5} M$

$$S = \frac{1}{5} (2S + 33)$$

Multiply both sides by 5 to clear the fraction:

$$5S = 2S + 33$$

Subtract $2S$ from both sides:

$$3S = 33$$

Divide both sides by 3:

$$S = 11$$

Now find $M$ using $M = 2S + 33$:

$$M = 2(11) + 33$$
$$M = 22 + 33$$
$$M = 55$$

Verify the Solution:

• Shobha's present age is $S = 11$.
• Sudha's present age is $M = 55$.

Twenty-five years later:

• Shobha’s age will be $11 + 25 = 36$.
• Sudha’s age will be $55 + 25 = 80$.

Half of Sudha’s age twenty-five years later:

$$\frac{1}{2} \times 80 = 40$$

Shobha’s age will be 4 years less than this half-age:

$$36 = 40 - 4$$

The solution is correct. Therefore, the present ages are:

• Shobha: 11 years
• Sudha: 55 years