Today Rose is twice as old as Sam and Sam is 3 years younger than Tina. If Rose, Sam, and...

1. Translate the problem requirements: Define the current age relationships and determine which relationships will remain true after 4 years pass
2. Set up current age relationships: Express all ages in terms of one variable using the given relationships
3. Apply time progression logic: Determine how each relationship changes when 4 years are added to all ages
4. Evaluate each statement systematically: Test whether each given statement holds true in the future scenario

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know today and what we need to figure out about the future.

Today's situation:

• Rose is twice as old as Sam
• Sam is 3 years younger than Tina

Future situation (4 years from today):

• All three people will be 4 years older
• We need to check which of the three statements will be true

The key insight here is that when time passes, everyone ages by the same amount. This means some relationships between ages will change, while others will stay the same.

Process Skill: TRANSLATE - Converting the age relationships into clear mathematical understanding

2. Set up current age relationships

Let's use concrete numbers to make this crystal clear, then we'll see the general pattern.

Let's say Sam is currently 10 years old (we can pick any number - the relationships will work the same way).

If Sam is 10:

• Rose is twice as old as Sam, so Rose is 20
• Sam is 3 years younger than Tina, so Tina is \(10 + 3 = 13\)

Current ages: Sam = 10, Rose = 20, Tina = 13

In general terms (using S for Sam's current age):

• Sam's age = \(\mathrm{S}\)
• Rose's age = \(2\mathrm{S}\) (twice Sam's age)
• Tina's age = \(\mathrm{S} + 3\) (3 years older than Sam)

3. Apply time progression logic

Now let's see what happens when 4 years pass. Everyone gets 4 years older.

Using our concrete example (Sam started at 10):

• Sam will be: \(10 + 4 = 14\)
• Rose will be: \(20 + 4 = 24\)
• Tina will be: \(13 + 4 = 17\)

Future ages: Sam = 14, Rose = 24, Tina = 17

In general terms:

• Sam's future age = \(\mathrm{S} + 4\)
• Rose's future age = \(2\mathrm{S} + 4\)
• Tina's future age = \((\mathrm{S} + 3) + 4 = \mathrm{S} + 7\)

The crucial insight: Age differences between people never change, but age ratios do change as people get older.

4. Evaluate each statement systematically

Now let's check each statement using our concrete example (Sam = 14, Rose = 24, Tina = 17 in the future):

Statement I: Rose is twice as old as Sam
Is 24 twice 14? No! \(24 \neq 2 \times 14 = 28\)
Rose is less than twice Sam's age now.
This makes sense: when both people age by the same amount, the older person becomes a smaller multiple of the younger person's age.

Statement II: Sam is 3 years younger than Tina
Is Sam 3 years younger than Tina? Let's check: \(17 - 14 = 3\). Yes!
Age differences never change when time passes. If Sam was 3 years younger before, he'll always be 3 years younger.

Statement III: Rose is older than Tina
Is \(24 > 17\)? Yes, Rose is older.
But wait - will this always be true? Let's check with our general formula:
Rose's future age = \(2\mathrm{S} + 4\)
Tina's future age = \(\mathrm{S} + 7\)
Is \(2\mathrm{S} + 4 > \mathrm{S} + 7\)? This gives us \(2\mathrm{S} > \mathrm{S} + 3\), or \(\mathrm{S} > 3\).
This is only true if Sam is currently older than 3. Since the problem doesn't guarantee Sam's specific age, we can't be sure this will always be true.

Process Skill: INFER - Drawing the non-obvious conclusion that age ratios change while age differences remain constant

4. Final Answer

Only Statement II must always be true: "Sam is 3 years younger than Tina."

This is because age differences between people remain constant over time, while age ratios change.

The answer is (B) II only.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the time frame
Students often confuse what needs to be evaluated. The question asks what must be true "4 years from today," but students may incorrectly analyze the current relationships instead of the future ones. They might think they need to verify today's conditions rather than determining how these relationships change over time.

Faltering Point 2: Assuming all relationships remain unchanged
Students frequently assume that if a relationship is true today, it will automatically be true in the future. They fail to recognize the crucial distinction between age differences (which remain constant) and age ratios (which change as people age). This leads them to incorrectly conclude that "Rose is twice as old as Sam" will always be true.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in age calculations
When adding 4 years to each person's age, students may make simple arithmetic mistakes, especially when working with variables. For example, they might incorrectly calculate Tina's future age as \(\mathrm{S} + 4\) instead of \((\mathrm{S} + 3) + 4 = \mathrm{S} + 7\).

Faltering Point 2: Incorrect ratio comparisons
When checking if Rose will still be twice as old as Sam, students may set up the equation incorrectly. They might check if \(2\mathrm{S} + 4 = 2(\mathrm{S} + 4)\) instead of comparing \(2\mathrm{S} + 4\) with \(2(\mathrm{S} + 4)\), failing to recognize that \(2\mathrm{S} + 4 \neq 2\mathrm{S} + 8\).

Errors while selecting the answer

Faltering Point 1: Selecting statements that are sometimes true rather than always true
The question asks which statements "must be true," meaning they need to be true in ALL cases. Students often select Statement III (Rose is older than Tina) because it's true in their specific example, without testing whether this holds for all possible values of Sam's age (e.g., if Sam were currently 2 years old, Rose would be 4 and Tina would be 5, making Tina older).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient concrete ages that satisfy the current relationships
Let's assign specific ages that make the given relationships work:

• Let Sam's current age = 12 years
• Since Rose is twice as old as Sam: Rose's current age = 24 years
• Since Sam is 3 years younger than Tina: Tina's current age = 15 years

Step 2: Calculate everyone's age 4 years from today
Adding 4 years to each person's current age:

• Sam's future age = \(12 + 4 = 16\) years
• Rose's future age = \(24 + 4 = 28\) years
• Tina's future age = \(15 + 4 = 19\) years

Step 3: Test each statement using these concrete ages
Statement I: Rose is twice as old as Sam
Check: Is \(28 = 2 \times 16\)? Is \(28 = 32\)? No, this is false.

Statement II: Sam is 3 years younger than Tina
Check: Is \(16 = 19 - 3\)? Is \(16 = 16\)? Yes, this is true.

Statement III: Rose is older than Tina
Check: Is \(28 > 19\)? Yes, this is true.

Step 4: Verify our logic with the key insight
The age difference between any two people never changes over time. Since Sam is currently 3 years younger than Tina, this 3-year gap will remain constant regardless of how much time passes.
However, proportional relationships (like "twice as old") do change as people age.

Conclusion: Only Statement II must be true, confirming answer choice (B).