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