Roy is now 4 years older than Erik and half of that amount older than Iris. If in 2 years,...

1. Translate the problem requirements: Convert the verbal age relationships into mathematical expressions. Roy is 4 years older than Erik, and 2 years older than Iris (half of 4). In 2 years, Roy will be twice Erik's age. We need to find Roy's age × Iris's age in 2 years.
2. Set up current age relationships: Use Erik's current age as the base variable to express all ages in terms of one unknown, keeping calculations simple.
3. Apply the future age constraint: Use the condition that in 2 years Roy will be twice Erik's age to solve for the unknown variable.
4. Calculate future ages and final product: Determine Roy's and Iris's ages in 2 years, then multiply them together.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in everyday language first:

• Roy is 4 years older than Erik right now
• Roy is also "half of that amount" older than Iris - this means Roy is 2 years older than Iris (since half of 4 is 2)
• In 2 years from now, Roy will be twice as old as Erik will be then
• We need to find: Roy's age in 2 years multiplied by Iris's age in 2 years

Now let's convert this to mathematical expressions. We need to pick one person's age as our starting point - let's use Erik's current age and call it \(\mathrm{E}\).

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

2. Set up current age relationships

Using Erik's current age as our base variable \(\mathrm{E}\), we can express everyone's current ages:

• Erik's current age = \(\mathrm{E}\)
• Roy's current age = \(\mathrm{E} + 4\) (since Roy is 4 years older than Erik)
• Iris's current age = \((\mathrm{E} + 4) - 2 = \mathrm{E} + 2\) (since Roy is 2 years older than Iris)

Let's verify this makes sense: If Erik is 10, then Roy is 14, and Iris is 12. Roy is indeed 4 years older than Erik and 2 years older than Iris. ✓

3. Apply the future age constraint

Now we use the key constraint: "In 2 years, Roy will be twice as old as Erik."

In 2 years:

• Erik's age will be: \(\mathrm{E} + 2\)
• Roy's age will be: \((\mathrm{E} + 4) + 2 = \mathrm{E} + 6\)

The constraint tells us: Roy's age in 2 years = 2 × Erik's age in 2 years

So: \(\mathrm{E} + 6 = 2(\mathrm{E} + 2)\)
\(\mathrm{E} + 6 = 2\mathrm{E} + 4\)
\(6 - 4 = 2\mathrm{E} - \mathrm{E}\)
\(2 = \mathrm{E}\)

Therefore, Erik's current age is 2 years old.

Process Skill: APPLY CONSTRAINTS - Using the future age condition to solve for the unknown

4. Calculate future ages and final product

Now that we know Erik's current age is 2, we can find everyone's current ages:

• Erik's current age = 2
• Roy's current age = 2 + 4 = 6
• Iris's current age = 2 + 2 = 4

Let's verify our constraint: In 2 years, Erik will be 4 and Roy will be 8. Indeed, 8 = 2 × 4. ✓

In 2 years:

• Roy's age = 6 + 2 = 8
• Iris's age = 4 + 2 = 6

Therefore: Roy's age × Iris's age = 8 × 6 = 48

Final Answer

Roy's age multiplied by Iris's age in 2 years = 48

This matches answer choice (c) 48.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "half of that amount older than Iris"

Students often struggle with the phrase "half of that amount older than Iris." They may incorrectly think this means Roy is half of Erik's age older than Iris, or that Roy's age is half of some other quantity. The correct interpretation is that "that amount" refers to the 4-year difference mentioned in the first part of the sentence, so Roy is 4÷2 = 2 years older than Iris.

Faltering Point 2: Setting up inconsistent variable relationships

When establishing the age relationships, students may set up contradictory equations. For example, they might correctly state Roy = Erik + 4, but then incorrectly write Iris = Roy + 2 instead of Iris = Roy - 2. This fundamental error in translating "Roy is older than Iris" into mathematical notation will lead to incorrect results throughout the solution.

Faltering Point 3: Confusing current ages with future ages in the constraint equation

Students may set up the future age constraint incorrectly by mixing current and future ages. They might write Roy's current age = 2 × Erik's current age, rather than properly adding 2 years to both Roy's and Erik's current ages before applying the "twice as old" relationship.

Errors while executing the approach

Faltering Point 1: Algebraic manipulation errors when solving the constraint equation

When solving \(\mathrm{E} + 6 = 2(\mathrm{E} + 2)\), students commonly make distribution or simplification errors. They might incorrectly distribute to get \(\mathrm{E} + 6 = 2\mathrm{E} + 2\), or make sign errors when collecting like terms, such as writing \(6 + 4 = 2\mathrm{E} - \mathrm{E}\) instead of \(6 - 4 = 2\mathrm{E} - \mathrm{E}\).

Faltering Point 2: Failing to verify the solution against the original constraints

Students may arrive at Erik's current age = 2 but fail to check whether this satisfies all the given conditions. Without verification, they miss the opportunity to catch potential errors in their setup or calculation, and may proceed with incorrect age values.

Errors while selecting the answer

Faltering Point 1: Calculating current ages instead of future ages for the final answer

After finding that Roy's current age is 6 and Iris's current age is 4, students may immediately multiply these values to get 6 × 4 = 24, forgetting that the question asks for their ages "in 2 years." This leads them to select an incorrect answer that isn't even among the choices, causing confusion.

Faltering Point 2: Mixing up which person's ages to multiply

Students may correctly calculate all the future ages but then multiply the wrong combination, such as Roy's age × Erik's age (8 × 4 = 32) or Erik's age × Iris's age (4 × 6 = 24), instead of the required Roy's age × Iris's age (8 × 6 = 48).