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.
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.
Let Rakesh's age = \( R \), Mahesh's age = \( M \)
Given: \( R + M = 60 \)
From the statement:
Rakesh says, "I am as old as you were when I was one-third as old as you are"
That is, when Rakesh was \( \frac{1}{3}M \), Mahesh's age was \( R \)
So, time passed = \( R - \frac{1}{3}M \)
Then Mahesh's age was: \( M - (R - \frac{1}{3}M) = R \)
Now solve the equation:
\( M - (R - \frac{1}{3}M) = R \)
\( M - R + \frac{1}{3}M = R \)
\( \frac{4}{3}M = 2R \)
\( 4M = 6R \Rightarrow 2M = 3R \Rightarrow M = \frac{3}{2}R \)
Now plug into \( R + M = 60 \)
\( R + \frac{3}{2}R = 60 \Rightarrow \frac{5}{2}R = 60 \Rightarrow R = 24 \)
\( M = \frac{3}{2} \times 24 = 36 \)
We want: Arjun’s birth year.
\[ 1964 + 35 = 1999 \] So, the mother was born in 1999.
\[ 1999 - 25 = 1974 \] Therefore, Arjun was born in 1974.
✅ Final Answer: Arjun was born in 1974
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