(i) One year ago, a man was 8 time as old as his son. Now, his age is equal to the square of his son's age. Find their present ages.
(ii) A man is `3(1)/(2)` times as old as his son. If the sum of the squares of their ages is 1325, find the ages of the father and the son.

Explanation

### Solution:

**Part (i)**

1. **Define the Variables:**
Let the present age of the father be \( X \) years and the present age of the son be \( Y \) years.

2. **Set Up the Equations:**
- From the problem, we know that one year ago, the father was 8 times as old as his son:
\[ X - 1 = 8(Y - 1) \]
- The problem also states that the father's current age is equal to the square of his son's current age:
\[ X = Y^2 \]

3. **Substitute and Rearrange:**
Substitute \( X = Y^2 \) into the first equation:
\[ Y^2 - 1 = 8(Y - 1) \]
Expanding this gives:
\[ Y^2 - 1 = 8Y - 8 \]
Rearranging leads to:
\[ Y^2 - 8Y + 7 = 0 \]

4. **Factor the Quadratic:**
Now, we can factor the quadratic equation:
\[ (Y - 7)(Y - 1) = 0 \]
This gives us two possible solutions for \( Y \):
\[ Y = 7 \quad \text{or} \quad Y = 1 \]

5. **Find Corresponding Father's Age:**
- If \( Y = 7 \):
\[ X = Y^2 = 7^2 = 49 \]
- If \( Y = 1 \):
\[ X = Y^2 = 1^2 = 1 \]
However, the case \( Y = 1 \) is not valid since the father's age cannot be equal to the son's age.

6. **Conclusion for Part (i):**
Therefore, the present ages are:
- Father's age: \( 49 \) years
- Son's age: \( 7 \) years

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**Part (ii)**

1. **Define the Variables:**
Let the present age of the father be \( X \) years and the present age of the son be \( Y \) years.

2. **Set Up the Equations:**
- From the problem, we know that the father is \( 3\frac{1}{2} \) times as old as his son:
\[ X = \frac{7}{2}Y \]
- The problem also states that the sum of the squares of their ages is 1325:
\[ X^2 + Y^2 = 1325 \]

3. **Substitute and Rearrange:**
Substitute \( X = \frac{7}{2}Y \) into the second equation:
\[ \left(\frac{7}{2}Y\right)^2 + Y^2 = 1325 \]
This simplifies to:
\[ \frac{49}{4}Y^2 + Y^2 = 1325 \]
Converting \( Y^2 \) to a fraction gives:
\[ \frac{49}{4}Y^2 + \frac{4}{4}Y^2 = 1325 \]
Combining the terms:
\[ \frac{53}{4}Y^2 = 1325 \]

4. **Solve for \( Y^2 \):**
Multiply both sides by 4:
\[ 53Y^2 = 5300 \]
Divide by 53:
\[ Y^2 = \frac{5300}{53} = 100 \]
Taking the square root gives:
\[ Y = 10 \]

5. **Find Corresponding Father's Age:**
Substitute \( Y \) back to find \( X \):
\[ X = \frac{7}{2}Y = \frac{7}{2} \times 10 = 35 \]

6. **Conclusion for Part (ii):**
Therefore, the present ages are:
- Father's age: \( 35 \) years
- Son's age: \( 10 \) years

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