Present Age of the Mother

Let mother’s present age = $M$, person’s present age = $P$.
Given:
$P = \frac{2}{5} M$ ...(1)
After 8 years:
$P + 8 = \frac{1}{2} (M + 8)$ ...(2)

Substitute $P$ from (1) into (2):
$$\frac{2}{5}M + 8 = \frac{1}{2}(M + 8)$$ Multiply both sides by 10 to clear denominators:
$$4M + 80 = 5M + 40$$ $$80 - 40 = 5M - 4M$$ $$40 = M$$

Answer: The mother’s present age is 40 years.

Average Age of Anu and Dhanu

Let the ages be:
Anu = $A$, Bhanu = $B$, Chanu = $C$, Dhanu = $D$

Given:
1) $A + B = 10 + B + C + D \Rightarrow A = 10 + C + D$
2) Average age of Chanu and Dhanu = 19
$\Rightarrow \frac{C + D}{2} = 19 \Rightarrow C + D = 38$
3) Dhanu is 10 years elder than Chanu:
$D = C + 10$

Substitute $D$ in $C + D = 38$:
$$C + (C + 10) = 38 \Rightarrow 2C = 28 \Rightarrow C = 14$$ $$D = 14 + 10 = 24$$

Now, $A = 10 + C + D = 10 + 14 + 24 = 48$

Average age of Anu and Dhanu:
$$\frac{A + D}{2} = \frac{48 + 24}{2} = \frac{72}{2} = 36$$

Answer: The average age of Anu and Dhanu is 36 years.

Given:

• Arjun is 25 years younger than his mother.
• Arjun’s brother was born in 1964 and is 35 years younger than their mother.

We want: Arjun’s birth year.

Step 1: Calculate the mother's birth year:

$$1964 + 35 = 1999$$ So, the mother was born in 1999.

Step 2: Arjun is 25 years younger than his mother:

$$1999 - 25 = 1974$$ Therefore, Arjun was born in 1974.

✅ Final Answer: Arjun was born in 1974