Two numbers differ by 2 and sum to S. Which of the following is the greater of the numbers in...

1. Translate the problem requirements: We have two numbers where one is exactly 2 more than the other, and when added together they equal \(\mathrm{S}\). We need to find an expression for the larger number in terms of \(\mathrm{S}\).
2. Set up the relationship using variables: Define the smaller number as a variable and express the larger number in terms of it, using the "differ by 2" constraint.
3. Apply the sum constraint: Use the fact that both numbers add up to \(\mathrm{S}\) to create an equation we can solve.
4. Solve for the larger number: Manipulate the equation to isolate and express the larger number directly in terms of \(\mathrm{S}\).

Execution of Strategic Approach

1. Translate the problem requirements

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

We have two numbers. Let's call them the "smaller number" and the "larger number" for clarity.

The problem gives us two key pieces of information:
• The two numbers differ by 2 (meaning one is exactly 2 more than the other)
• When we add both numbers together, we get \(\mathrm{S}\)

Our goal is to find an expression that represents the larger of these two numbers, written in terms of \(\mathrm{S}\).

Process Skill: TRANSLATE - Converting the word problem into mathematical understanding

2. Set up the relationship using variables

Now let's use algebra to represent this relationship clearly.

Let's call the smaller number "\(\mathrm{x}\)". Since the two numbers differ by 2, and we want the larger number, the larger number must be "\(\mathrm{x} + 2\)".

Think of it this way: if the smaller number is 5, then the larger number is \(5 + 2 = 7\). The difference is \(7 - 5 = 2\). ✓

So we have:
• Smaller number = \(\mathrm{x}\)
• Larger number = \(\mathrm{x} + 2\)

3. Apply the sum constraint

The problem tells us that these two numbers add up to \(\mathrm{S}\). Let's write this as an equation:

Smaller number + Larger number = \(\mathrm{S}\)

Substituting what we know:
\(\mathrm{x} + (\mathrm{x} + 2) = \mathrm{S}\)

Simplifying the left side:
\(\mathrm{x} + \mathrm{x} + 2 = \mathrm{S}\)
\(2\mathrm{x} + 2 = \mathrm{S}\)

4. Solve for the larger number

Now we need to find the larger number in terms of \(\mathrm{S}\). Remember, the larger number is \(\mathrm{x} + 2\), so we need to find what \(\mathrm{x}\) equals first.

Starting with our equation: \(2\mathrm{x} + 2 = \mathrm{S}\)

Subtract 2 from both sides: \(2\mathrm{x} = \mathrm{S} - 2\)

Divide both sides by 2: \(\mathrm{x} = \frac{\mathrm{S} - 2}{2} = \frac{\mathrm{S}}{2} - 1\)

But remember, we don't want \(\mathrm{x}\) (the smaller number). We want the larger number, which is \(\mathrm{x} + 2\):

Larger number = \(\mathrm{x} + 2\)
Larger number = \(\left(\frac{\mathrm{S}}{2} - 1\right) + 2\)
Larger number = \(\frac{\mathrm{S}}{2} - 1 + 2\)
Larger number = \(\frac{\mathrm{S}}{2} + 1\)

Final Answer

The greater of the two numbers is \(\frac{\mathrm{S}}{2} + 1\).

Looking at our answer choices, this matches choice D: \(\frac{\mathrm{S}}{2} + 1\)

Let's verify with a quick example: If \(\mathrm{S} = 10\), then our two numbers should be 4 and 6 (they differ by 2 and sum to 10). The larger number is 6. Using our formula: \(\frac{\mathrm{S}}{2} + 1 = \frac{10}{2} + 1 = 5 + 1 = 6\). ✓

Answer: D

Common Faltering Points

Errors while devising the approach

1. Misinterpreting which number is larger: Students may assume the problem is asking for the smaller number instead of the larger one, or they may set up their variables incorrectly by calling the larger number 'x' and the smaller number 'x-2', which can lead to sign errors later.

2. Confusion about the difference constraint: Students might misunderstand "differ by 2" and think it means one number is twice the other, or they may not clearly establish that if one number is x, the other must be x+2 (not x-2 if we want the larger one).

3. Forgetting the constraint requirements: Students may focus only on the sum constraint (adds to S) and forget to incorporate the difference constraint (differ by 2) into their setup, or vice versa.

Errors while executing the approach

1. Algebraic manipulation errors: When solving \(2\mathrm{x} + 2 = \mathrm{S}\), students commonly make errors like forgetting to subtract 2 from both sides first, or incorrectly dividing, leading to wrong expressions for x.

2. Substitution mistakes: After finding \(\mathrm{x} = \frac{\mathrm{S}}{2} - 1\), students may forget that they need the larger number \((\mathrm{x} + 2)\) and mistakenly think x itself is the answer, or they may make arithmetic errors when computing \(\left(\frac{\mathrm{S}}{2} - 1\right) + 2\).

3. Sign errors in simplification: When simplifying \(\left(\frac{\mathrm{S}}{2} - 1\right) + 2\) to get \(\frac{\mathrm{S}}{2} + 1\), students may incorrectly handle the signs and get \(\frac{\mathrm{S}}{2} - 1 + 2 = \frac{\mathrm{S}}{2} + 3\) or other wrong combinations.

Errors while selecting the answer

1. Selecting the smaller number instead: Students may correctly solve for both numbers but then select choice A \(\left(\frac{\mathrm{S}}{2} - 1\right)\), which represents the smaller number, instead of choice D \(\left(\frac{\mathrm{S}}{2} + 1\right)\), which represents the larger number that the question asks for.

2. Not verifying with the given constraints: Students might pick an answer that looks reasonable but fail to check whether their chosen expression actually satisfies both the sum and difference constraints when substituted back into the original problem.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for S

Let's use \(\mathrm{S} = 10\). This gives us two numbers that differ by 2 and sum to 10.

Step 2: Find the two numbers when S = 10

If the smaller number is \(\mathrm{x}\), then the larger number is \(\mathrm{x} + 2\).
Since they sum to 10: \(\mathrm{x} + (\mathrm{x} + 2) = 10\)
\(2\mathrm{x} + 2 = 10\)
\(2\mathrm{x} = 8\)
\(\mathrm{x} = 4\)

So the two numbers are 4 and 6, where 6 is the greater number.

Step 3: Test each answer choice with S = 10

We need to find which expression equals 6 when \(\mathrm{S} = 10\):

1. \(\frac{\mathrm{S}}{2} - 1 = \frac{10}{2} - 1 = 5 - 1 = 4\) ❌
2. \(\frac{\mathrm{S}}{2} = \frac{10}{2} = 5\) ❌
3. \(\frac{\mathrm{S}}{2} + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = 5 + 0.5 = 5.5\) ❌
4. \(\frac{\mathrm{S}}{2} + 1 = \frac{10}{2} + 1 = 5 + 1 = 6\) ✓
5. \(\frac{\mathrm{S}}{2} + 2 = \frac{10}{2} + 2 = 5 + 2 = 7\) ❌

Step 4: Verify with another value

Let's confirm with \(\mathrm{S} = 6\). The two numbers that differ by 2 and sum to 6 are 2 and 4.
Using choice D: \(\frac{\mathrm{S}}{2} + 1 = \frac{6}{2} + 1 = 3 + 1 = 4\) ✓

This matches our expected greater number of 4.

Answer: D