The ages of two friends Ani and Biju differ by 3 years. Ani's father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.

Answer

To solve the problem, we will define the variables and set up equations based on the information given.

1. **Define Variables:**
- Let the age of Ani be \( X \) years.
- Let the age of Biju be \( Y \) years.
- Let the age of Cathy be \( C \) years.
- Let the age of Dharam (Ani's father) be \( D \) years.

2. **Set Up the Equations:**
- From the problem, we know that the ages of Ani and Biju differ by 3 years:
\[ |X - Y| = 3 \]
- We can assume \( X - Y = 3 \) (Ani is older than Biju) or \( Y - X = 3 \) (Biju is older than Ani). We'll use \( X - Y = 3 \) for our calculations.
- Ani's father, Dharam, is twice as old as Ani:
\[ D = 2X \]
- Biju is twice as old as his sister, Cathy:
\[ Y = 2C \]
- The ages of Cathy and Dharam differ by 30 years:
\[ |D - C| = 30 \]
- We can assume \( D - C = 30 \) (Dharam is older than Cathy) for our calculations.

3. **Substituting the Equations:**
- From \( D = 2X \) and \( D - C = 30 \), we can substitute \( D \):
\[ 2X - C = 30 \quad \Rightarrow \quad C = 2X - 30 \]
- From \( Y = 2C \), substituting for \( C \):
\[ Y = 2(2X - 30) = 4X - 60 \]

4. **Now we have two equations:**
- \( X - Y = 3 \)
- \( Y = 4X - 60 \)

5. **Substituting \( Y \) in the first equation:**
\[ X - (4X - 60) = 3 \]
\[ X - 4X + 60 = 3 \]
\[ -3X + 60 = 3 \]
\[ -3X = 3 - 60 \]
\[ -3X = -57 \]
\[ X = 19 \]

6. **Finding \( Y \):**
- Substitute \( X \) back into the equation for \( Y \):
\[ Y = 4(19) - 60 \]
\[ Y = 76 - 60 \]
\[ Y = 16 \]

7. **Final Ages:**
- The age of Ani is \( 19 \) years.
- The age of Biju is \( 16 \) years.

### Summary:
- The ages of Ani and Biju are \( 19 \) years and \( 16 \) years, respectively.