If two 2-digit positive integers have their respective tens digits exchanged, the difference between the pair of integers changes by...

1. Translate the problem requirements: We have two 2-digit positive integers. When we swap their tens digits with each other, the difference between the pair changes by 4. We need to find the maximum possible original difference.
2. Set up algebraic representation: Express the two integers in terms of their tens and units digits, then write expressions for both the original difference and the new difference after exchanging tens digits.
3. Establish the constraint relationship: Use the given condition that the difference changes by 4 to create an equation relating the tens digits of the two numbers.
4. Maximize the original difference: Apply the constraint to determine what values of tens and units digits will produce the largest possible original difference between the two integers.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in everyday terms. We have two 2-digit positive integers - let's call them our "first number" and "second number."

When we exchange (swap) their tens digits with each other, something interesting happens: the difference between these two numbers changes by exactly 4.

For example, if we started with numbers 35 and 72, after swapping tens digits we'd get 75 and 32. The original difference was \(|35 - 72| = 37\), and the new difference is \(|75 - 32| = 43\). The change in difference is \(43 - 37 = 6\). But we need this change to be exactly 4.

Our goal is to find the maximum possible original difference between two such numbers that satisfy this condition.

Process Skill: TRANSLATE - Converting the problem's verbal description into a clear mathematical understanding

2. Set up algebraic representation

Now let's represent our two numbers mathematically. Any 2-digit number can be written as 10 times its tens digit plus its units digit.

Let's call our first number: \(10\mathrm{a} + \mathrm{b}\) (where 'a' is the tens digit and 'b' is the units digit)
Let's call our second number: \(10\mathrm{c} + \mathrm{d}\) (where 'c' is the tens digit and 'd' is the units digit)

After exchanging tens digits:

• First number becomes: \(10\mathrm{c} + \mathrm{b}\)
• Second number becomes: \(10\mathrm{a} + \mathrm{d}\)

The original difference is: \(|(10\mathrm{a} + \mathrm{b}) - (10\mathrm{c} + \mathrm{d})| = |10(\mathrm{a} - \mathrm{c}) + (\mathrm{b} - \mathrm{d})|\)
The new difference is: \(|(10\mathrm{c} + \mathrm{b}) - (10\mathrm{a} + \mathrm{d})| = |10(\mathrm{c} - \mathrm{a}) + (\mathrm{b} - \mathrm{d})| = |10(\mathrm{a} - \mathrm{c}) - (\mathrm{b} - \mathrm{d})|\)

3. Establish the constraint relationship

We're told that the difference changes by 4. Let's think about this carefully.

Original difference: \(|10(\mathrm{a} - \mathrm{c}) + (\mathrm{b} - \mathrm{d})|\)
New difference: \(|10(\mathrm{a} - \mathrm{c}) - (\mathrm{b} - \mathrm{d})|\)

For the change to be exactly 4, we need:
\(||10(\mathrm{a} - \mathrm{c}) - (\mathrm{b} - \mathrm{d})| - |10(\mathrm{a} - \mathrm{c}) + (\mathrm{b} - \mathrm{d})|| = 4\)

Let's say \(\mathrm{x} = 10(\mathrm{a} - \mathrm{c})\) and \(\mathrm{y} = (\mathrm{b} - \mathrm{d})\). Then we need: \(||\mathrm{x} - \mathrm{y}| - |\mathrm{x} + \mathrm{y}|| = 4\)

This equation is satisfied when either:

• x and y have the same sign and \(|\mathrm{x}| > |\mathrm{y}|\), giving us \(|\mathrm{x} - \mathrm{y}| - |\mathrm{x} + \mathrm{y}| = ±4\)
• This simplifies to either \(2\mathrm{y} = 4\) or \(2\mathrm{y} = -4\), so \(\mathrm{y} = ±2\)

Therefore: \(\mathrm{b} - \mathrm{d} = ±2\), which means \(|\mathrm{b} - \mathrm{d}| = 2\)

Process Skill: MANIPULATE - Working with absolute value expressions to derive the key constraint

4. Maximize the original difference

Now we know that the units digits must differ by exactly 2. To maximize the original difference \(|10(\mathrm{a} - \mathrm{c}) + (\mathrm{b} - \mathrm{d})|\), we want to make both terms as large as possible and have them work together.

For maximum difference:

• Make \(|\mathrm{a} - \mathrm{c}|\) as large as possible: Since tens digits range from 1 to 9, the maximum is \(|9 - 1| = 8\)
• Make \(|\mathrm{b} - \mathrm{d}| = 2\) (from our constraint)
• Ensure both terms add up rather than subtract

Let's choose \(\mathrm{a} = 9, \mathrm{c} = 1\), so \(10(\mathrm{a} - \mathrm{c}) = 80\)
Let's choose \(\mathrm{b} = 9, \mathrm{d} = 7\), so \((\mathrm{b} - \mathrm{d}) = 2\)

This gives us:

• First number: \(10(9) + 9 = 99\)
• Second number: \(10(1) + 7 = 17\)
• Original difference: \(|99 - 17| = 82\)

Let's verify: After swapping tens digits:

• First becomes: \(10(1) + 9 = 19\)
• Second becomes: \(10(9) + 7 = 97\)
• New difference: \(|19 - 97| = 78\)
• Change in difference: \(|82 - 78| = 4\) ✓

Process Skill: APPLY CONSTRAINTS - Using our derived constraint to find the maximum while satisfying all conditions

4. Final Answer

The greatest possible difference between the original pair of integers is 82.

This corresponds to answer choice C) 82.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "difference changes by 4" means

Students often confuse this constraint with "the new difference equals 4" or "the original difference equals 4." The key insight is that we need |new difference - original difference| = 4, not that either difference itself equals 4. This misinterpretation leads to setting up completely wrong equations.

2. Incorrectly setting up the algebraic representation

When exchanging tens digits, students may incorrectly think both numbers get the same tens digit, rather than understanding that the tens digits are swapped between the two numbers. For example, if we have 35 and 72, after exchange we get 75 and 32 (swapping the 3 and 7), not 75 and 75 or some other incorrect combination.

3. Missing the constraint on digit ranges

Students may forget that tens digits of 2-digit numbers must be between 1 and 9 (not 0), while units digits can be 0-9. This oversight can lead to invalid number combinations when trying to maximize the difference.

Errors while executing the approach

1. Algebraic manipulation errors with absolute values

When working with \(||10(\mathrm{a}-\mathrm{c}) - (\mathrm{b}-\mathrm{d})| - |10(\mathrm{a}-\mathrm{c}) + (\mathrm{b}-\mathrm{d})|| = 4\), students often struggle with the nested absolute values and may incorrectly conclude that any relationship between the terms works, missing that we specifically need \(|\mathrm{b}-\mathrm{d}| = 2\).

2. Sign errors when determining maximum difference

Even after correctly finding that \(|\mathrm{b}-\mathrm{d}| = 2\), students may choose digit combinations where the terms \(10(\mathrm{a}-\mathrm{c})\) and \((\mathrm{b}-\mathrm{d})\) work against each other rather than together. For maximum difference, we want both terms to have the same sign and add up, not subtract.

3. Failing to verify the constraint

Students may find a large difference but forget to check that their chosen numbers actually satisfy the "difference changes by 4" condition. This verification step is crucial because not all combinations that seem to maximize will actually satisfy the constraint.

Errors while selecting the answer

1. Selecting a difference that violates the constraint

Students might calculate a difference like 90 (from 91 and 1) that seems maximum but doesn't satisfy the "changes by 4" constraint, and select this incorrect maximum instead of the constrained maximum of 82.