The sum of the ages of Doris and Fred is y years. If Doris is 12 years older than Fred,...

1. Translate the problem requirements: We need to find Fred's age y years from now. Currently, Doris and Fred's ages sum to y years, and Doris is 12 years older than Fred. We need to express Fred's future age in terms of y.
2. Set up current age relationships: Use the given constraints to establish equations for Doris's and Fred's current ages in terms of y.
3. Calculate Fred's current age: Solve the system of equations to find Fred's current age expressed in terms of y.
4. Project Fred's future age: Add y years to Fred's current age to find how old he will be y years from now.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We want to know how old Fred will be y years from now.

Here's what we know right now:
• Doris and Fred's current ages add up to y years
• Doris is currently 12 years older than Fred

We need to express Fred's future age (y years from now) in terms of y.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Set up current age relationships

Let's call Fred's current age 'F' and Doris's current age 'D'.

From the problem, we can write two simple relationships in plain English:
• "The sum of their ages is y years" means: \(\mathrm{D + F = y}\)
• "Doris is 12 years older than Fred" means: \(\mathrm{D = F + 12}\)

These are our two equations that describe the current situation.

3. Calculate Fred's current age

Now we can find Fred's current age by combining our two relationships.

Since \(\mathrm{D = F + 12}\), we can substitute this into our first equation:
\(\mathrm{D + F = y}\)
\(\mathrm{(F + 12) + F = y}\)

Simplifying:
\(\mathrm{F + 12 + F = y}\)
\(\mathrm{2F + 12 = y}\)
\(\mathrm{2F = y - 12}\)
\(\mathrm{F = \frac{y - 12}{2}}\)

So Fred's current age is \(\mathrm{\frac{y - 12}{2}}\) years old.

4. Project Fred's future age

The question asks for Fred's age y years from now.

In plain English: If someone is currently a certain age, and we want to know their age some number of years in the future, we simply add those years to their current age.

Fred's current age: \(\mathrm{\frac{y - 12}{2}}\)
Years to add: y

Fred's age y years from now = Current age + y years
\(\mathrm{= \frac{y - 12}{2} + y}\)

Let's simplify this expression:
\(\mathrm{= \frac{y - 12}{2} + \frac{2y}{2}}\)
\(\mathrm{= \frac{y - 12 + 2y}{2}}\)
\(\mathrm{= \frac{3y - 12}{2}}\)
\(\mathrm{= \frac{3y}{2} - 6}\)

Final Answer

Fred will be \(\mathrm{\frac{3y}{2} - 6}\) years old y years from now.

This matches answer choice (D) \(\mathrm{\frac{3y}{2} - 6}\).

Let's verify with a quick check: If \(\mathrm{y = 20}\), then currently Doris and Fred's ages sum to 20. Fred would be \(\mathrm{\frac{20-12}{2} = 4}\) years old now, and Doris would be 16. In 20 years, Fred would be 24 years old. Using our formula: \(\mathrm{\frac{3(20)}{2} - 6 = 30 - 6 = 24}\) ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what the question is asking for
Students often confuse "Fred's age y years from now" with "Fred's current age." They may solve for Fred's current age and stop there, forgetting that the question asks for his age after y additional years have passed. This is a critical reading comprehension error that leads to selecting answer choice (C) \(\mathrm{\frac{y}{2} - 6}\) instead of the correct answer.

2. Setting up the age relationship incorrectly
When told "Doris is 12 years older than Fred," some students may write \(\mathrm{F = D + 12}\) instead of \(\mathrm{D = F + 12}\). This reversal of the relationship will lead to incorrect equations and ultimately the wrong answer. Students need to carefully translate "A is older than B" as A = B + (difference).

3. Confusion about the variable y
Students may struggle with y representing both "the sum of current ages" and "the number of years into the future." This dual usage can cause confusion when setting up equations, leading some students to think these represent different values or to misapply y in their calculations.

Errors while executing the approach

1. Algebraic manipulation errors when solving for Fred's current age
When substituting \(\mathrm{D = F + 12}\) into \(\mathrm{D + F = y}\) to get \(\mathrm{(F + 12) + F = y}\), students commonly make errors like: forgetting to distribute properly, writing \(\mathrm{2F + 12 = y}\) as \(\mathrm{2F = y + 12}\) instead of \(\mathrm{2F = y - 12}\), or incorrectly dividing to get \(\mathrm{F = y - 6}\) instead of \(\mathrm{F = \frac{y - 12}{2}}\).

2. Fraction arithmetic errors when adding current age and future years
When calculating \(\mathrm{\frac{y - 12}{2} + y}\), students often struggle with adding fractions and whole numbers. Common errors include: writing the result as \(\mathrm{\frac{y - 12 + y}{2}}\) instead of \(\mathrm{\frac{y - 12 + 2y}{2}}\), or making arithmetic mistakes when combining like terms, getting \(\mathrm{2y - 12}\) instead of \(\mathrm{3y - 12}\) in the numerator.

3. Simplification errors in the final expression
When converting \(\mathrm{\frac{3y - 12}{2}}\) to \(\mathrm{\frac{3y}{2} - 6}\), students may incorrectly distribute the division, writing results like \(\mathrm{\frac{3y}{2} - 12}\) or \(\mathrm{\frac{3y - 12}{2}}\) but failing to recognize this equals \(\mathrm{\frac{3y}{2} - 6}\). These algebraic manipulation errors lead to answers that don't match any of the given choices.

Errors while selecting the answer

1. Stopping at Fred's current age instead of his future age
After correctly finding Fred's current age as \(\mathrm{\frac{y - 12}{2}}\), students may mistakenly think this is the final answer. They might try to manipulate this expression to match an answer choice, potentially selecting (C) \(\mathrm{\frac{y}{2} - 6}\), not realizing they still need to add y years for his future age.

2. Selecting an answer that looks similar but is algebraically different
Students who get confused during algebraic manipulation might select answer choices that contain similar elements to their work. For example, if they correctly identify that the answer involves fractions and the number 6, they might incorrectly choose (A) \(\mathrm{y - 6}\) or (C) \(\mathrm{\frac{y}{2} - 6}\) because these look "close" to their calculations, without verifying the complete mathematical accuracy.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for y

Let's set \(\mathrm{y = 30}\). This gives us a nice number to work with since we'll be dealing with age differences of 12 years.

Step 2: Find current ages using our chosen value

With \(\mathrm{y = 30}\):

• Doris's current age + Fred's current age = 30
• Doris is 12 years older than Fred

Let Fred's current age = F
Then Doris's current age = F + 12

Setting up the equation: \(\mathrm{F + (F + 12) = 30}\)
\(\mathrm{2F + 12 = 30}\)
\(\mathrm{2F = 18}\)
\(\mathrm{F = 9}\)

So with \(\mathrm{y = 30}\): Fred is currently 9 years old, Doris is currently 21 years old.

Step 3: Calculate Fred's age y years from now

Fred's age y years from now = Current age + y years
\(\mathrm{= 9 + 30 = 39}\) years old

Step 4: Check which answer choice gives us 39 when y = 30

• (A) \(\mathrm{y - 6 = 30 - 6 = 24}\) ❌
• (B) \(\mathrm{2y - 6 = 2(30) - 6 = 54}\) ❌
• (C) \(\mathrm{\frac{y}{2} - 6 = \frac{30}{2} - 6 = 9}\) ❌
• (D) \(\mathrm{\frac{3y}{2} - 6 = \frac{3(30)}{2} - 6 = 45 - 6 = 39}\) ✓
• (E) \(\mathrm{\frac{5y}{2} - 6 = \frac{5(30)}{2} - 6 = 75 - 6 = 69}\) ❌

Step 5: Verify with another value

Let's try \(\mathrm{y = 20}\) to double-check:

Fred's current age: \(\mathrm{\frac{20 - 12}{2} = 4}\)
Fred's age 20 years from now: \(\mathrm{4 + 20 = 24}\)

Testing answer choice (D): \(\mathrm{\frac{3(20)}{2} - 6 = 30 - 6 = 24}\) ✓

The answer is (D) \(\mathrm{\frac{3y}{2} - 6}\).