If k and m are numbers such that k + m = 20 and k^2 + m^2 = 289, then...

1. Translate the problem requirements: We have two numbers k and m with a known sum (\(\mathrm{k + m = 20}\)) and known sum of squares (\(\mathrm{k^2 + m^2 = 289}\)). We need to find their product \(\mathrm{km}\) and determine which range it falls into.
2. Connect sum and sum of squares using the expansion identity: Use the algebraic relationship \(\mathrm{(k + m)^2 = k^2 + m^2 + 2km}\) to relate the given information to what we're looking for.
3. Substitute known values and solve for the product: Plug in the given values to find \(\mathrm{km}\) directly.
4. Match the calculated value to the appropriate range: Compare our result with the given answer choices to identify the correct range.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find.

We have two numbers, k and m, and we know two things about them:

• When we add them together: \(\mathrm{k + m = 20}\)
• When we square each number and add those squares: \(\mathrm{k^2 + m^2 = 289}\)

Our goal is to find the product \(\mathrm{km}\) (what we get when we multiply k and m together) and determine which range from the answer choices it falls into.

Think of this like knowing the combined age of two siblings is 20 years, and if we square each of their ages and add those squares we get 289. We want to find what we get when we multiply their ages together.

Process Skill: TRANSLATE

2. Connect sum and sum of squares using the expansion identity

Here's the key insight: there's a beautiful relationship between the sum of two numbers, their individual squares, and their product.

If you take any two numbers and add them, then square that sum, you get something very useful. Let's think about what happens when we square \(\mathrm{(k + m)}\):

When we expand \(\mathrm{(k + m)^2}\), we get:

• \(\mathrm{k^2}\) (k times itself)
• \(\mathrm{m^2}\) (m times itself)
• \(\mathrm{2km}\) (twice the product of k and m)

So: \(\mathrm{(k + m)^2 = k^2 + m^2 + 2km}\)

This is perfect! We know \(\mathrm{k + m = 20}\), and we know \(\mathrm{k^2 + m^2 = 289}\). The expansion gives us a direct way to find \(\mathrm{km}\).

3. Substitute known values and solve for the product

Now let's plug in what we know into our relationship:

\(\mathrm{(k + m)^2 = k^2 + m^2 + 2km}\)

Substituting our known values:

• \(\mathrm{k + m = 20}\), so \(\mathrm{(k + m)^2 = 20^2 = 400}\)
• \(\mathrm{k^2 + m^2 = 289}\)

So: \(\mathrm{400 = 289 + 2km}\)

To find \(\mathrm{km}\), we solve:

\(\mathrm{400 - 289 = 2km}\)
\(\mathrm{111 = 2km}\)
\(\mathrm{km = 111 ÷ 2 = 55.5}\)

So the product \(\mathrm{km}\) equals \(\mathrm{55.5}\).

4. Match the calculated value to the appropriate range

We found that \(\mathrm{km = 55.5}\).

Looking at our answer choices:

• (A) between 20 and 60: Since \(\mathrm{20 < 55.5 < 60}\), this is correct!
• (B) between 60 and 100: 55.5 is less than 60, so this is incorrect
• (C) between 100 and 150: 55.5 is much less than 100, so this is incorrect
• (D) between 150 and 200: 55.5 is much less than 150, so this is incorrect
• (E) greater than 200: 55.5 is much less than 200, so this is incorrect

Final Answer

The product \(\mathrm{km = 55.5}\), which falls between 20 and 60.

The answer is (A).

Common Faltering Points

Errors while devising the approach

1. Attempting to solve the system by substitution without recognizing the algebraic identity
Many students will try to solve \(\mathrm{k + m = 20}\) and \(\mathrm{k^2 + m^2 = 289}\) by substituting \(\mathrm{m = 20 - k}\) into the second equation, leading to a quadratic equation. While this approach works, it's more complex and time-consuming than using the identity \(\mathrm{(k + m)^2 = k^2 + m^2 + 2km}\). Students may miss this elegant shortcut and get bogged down in unnecessary calculations.

2. Not connecting the given information to find the product directly
Some students might think they need to find the individual values of k and m first before calculating their product. They don't realize that the question is specifically designed so that \(\mathrm{km}\) can be found directly without determining k and m separately. This leads to overcomplicated approaches when a direct path exists.

Errors while executing the approach

1. Arithmetic errors when expanding or calculating
Students may make calculation mistakes such as: incorrectly computing \(\mathrm{20^2 = 400}\), making errors when subtracting \(\mathrm{400 - 289 = 111}\), or incorrectly dividing \(\mathrm{111 ÷ 2 = 55.5}\). These arithmetic slips can lead to wrong values for \(\mathrm{km}\) even when the approach is correct.

2. Sign errors or incorrect algebraic manipulation
When rearranging the equation \(\mathrm{400 = 289 + 2km}\) to solve for \(\mathrm{km}\), students might make algebraic errors such as: writing \(\mathrm{2km = 400 + 289}\) instead of \(\mathrm{2km = 400 - 289}\), or forgetting to divide both sides by 2 to isolate \(\mathrm{km}\).

Errors while selecting the answer

1. Boundary confusion when checking ranges
Students who correctly calculate \(\mathrm{km = 55.5}\) might be unsure about which range it belongs to, especially if they're unclear about whether the endpoints are included. They need to verify that \(\mathrm{20 < 55.5 < 60}\) to confidently select option (A), but may second-guess themselves about boundary conditions.