Today Anselm is three times as old as his brother Bartholomew, and Bartholomew is 4 years younger than his sister Catherine. If Anselm, Bartholomew, and Catherine are all alive 5 years from today, which of the following must be true on that day ?
I. Anselm is three times as old as Bartholomew.
II. Catherine is 4 years older than Bartholomew.
III. Anselm is older than Catherine.

Options

A. I only
B. II only
C. III only
D. I and II

Explanation

To solve the problem, we will first define the ages of Anselm, Bartholomew, and Catherine based on the information provided.

1. **Define the Variables:**
- Let Bartholomew's current age be \( X \).
- Since Anselm is three times as old as Bartholomew, Anselm's current age will be \( 3X \).
- Bartholomew is 4 years younger than his sister Catherine, so Catherine's current age will be \( X + 4 \).

2. **Calculate Their Ages After 5 Years:**
- After 5 years, Bartholomew's age will be \( X + 5 \).
- Anselm's age will be \( 3X + 5 \).
- Catherine's age will be \( (X + 4) + 5 = X + 9 \).

3. **Evaluate Each Statement:**
- **Statement I:** Anselm is three times as old as Bartholomew.
- After 5 years, Anselm's age is \( 3X + 5 \) and Bartholomew's age is \( X + 5 \).
- For Anselm to be three times as old as Bartholomew, we need \( 3X + 5 = 3(X + 5) \).
- Simplifying gives \( 3X + 5 = 3X + 15 \), which is not true. Therefore, Statement I is false.

- **Statement II:** Catherine is 4 years older than Bartholomew.
- After 5 years, Catherine's age is \( X + 9 \) and Bartholomew's age is \( X + 5 \).
- We check if \( X + 9 = (X + 5) + 4 \).
- Simplifying gives \( X + 9 = X + 9 \), which is true. Therefore, Statement II is true.

- **Statement III:** Anselm is older than Catherine.
- After 5 years, Anselm's age is \( 3X + 5 \) and Catherine's age is \( X + 9 \).
- We need to check if \( 3X + 5 > X + 9 \).
- Simplifying gives \( 3X + 5 - X > 9 \) or \( 2X + 5 > 9 \).
- This simplifies to \( 2X > 4 \) or \( X > 2 \).
- This means Anselm will be older than Catherine only if Bartholomew is older than 2 years. Thus, Statement III is not necessarily true for all values of \( X \).

4. **Conclusion:**
- Only Statement II is true. Therefore, the answer is that only Statement II must be true.

### Summary of the Steps:
1. Define the ages of Bartholomew, Anselm, and Catherine.
2. Calculate their ages after 5 years.
3. Evaluate each statement based on the calculated ages.
4. Conclude which statements are true.