In an auditorium, 360 chairs are to be set up in a rectangular arrangement with x rows of exactly y...

1. Translate the problem requirements: We need 360 chairs arranged in x rows with y chairs per row, where \(\mathrm{x \times y = 360}\) and \(\mathrm{10 < x < 25}\). Find how many valid values of x exist.
2. Find all factor pairs of 360: Identify all possible combinations where one factor could represent rows and determine which factors fall within our constraint range.
3. Apply the row constraint filter: Check each factor of 360 to see which ones satisfy the condition \(\mathrm{10 < x < 25}\).
4. Count the valid arrangements: Tally up all factor values that meet our row requirement to get the total number of possible rectangular arrangements.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in everyday terms. We have an auditorium with exactly 360 chairs that need to be arranged in a rectangle. Think of it like arranging desks in a classroom - we need rows and columns that form a perfect rectangle.

The key pieces of information are:

• Total chairs = 360
• Arrangement must be rectangular with x rows and y chairs per row
• Every chair must be used, so \(\mathrm{x \times y = 360}\)
• The number of rows must satisfy: \(\mathrm{10 < x < 25}\) (so x can be 11, 12, 13... up to 24)

We need to find how many different ways we can choose the number of rows (x) such that when we multiply by the number of chairs per row (y), we get exactly 360 chairs.

Process Skill: TRANSLATE - Converting the rectangular arrangement language into the mathematical constraint \(\mathrm{x \times y = 360}\)

2. Find all factor pairs of 360

Since \(\mathrm{x \times y = 360}\), this means x must be a factor of 360. Let's find what 360 breaks down into by finding its prime factorization first.

\(\mathrm{360 = 36 \times 10 = (6^2) \times (2 \times 5) = (2 \times 3)^2 \times (2 \times 5) = 2^3 \times 3^2 \times 5}\)

Now we can find all the factors of 360. The factors are all the numbers that divide 360 evenly:

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

Each of these numbers could potentially be our number of rows (x), as long as it satisfies our constraint.

3. Apply the row constraint filter

Now we need to check which factors of 360 satisfy our constraint that \(\mathrm{10 < x < 25}\).

Looking at our list of factors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

We need factors that are greater than 10 but less than 25:

• 12 ✓ (\(\mathrm{12 \times 30 = 360}\))
• 15 ✓ (\(\mathrm{15 \times 24 = 360}\))
• 18 ✓ (\(\mathrm{18 \times 20 = 360}\))
• 20 ✓ (\(\mathrm{20 \times 18 = 360}\))
• 24 ✓ (\(\mathrm{24 \times 15 = 360}\))

Let's verify each: \(\mathrm{12 \times 30 = 360}\) ✓, \(\mathrm{15 \times 24 = 360}\) ✓, \(\mathrm{18 \times 20 = 360}\) ✓, \(\mathrm{20 \times 18 = 360}\) ✓, \(\mathrm{24 \times 15 = 360}\) ✓

Process Skill: APPLY CONSTRAINTS - Filtering the complete list of factors using the given range restriction

4. Count the valid arrangements

From our analysis, we found exactly 5 values for x (number of rows) that satisfy all our requirements:

• x = 12 (giving us 12 rows of 30 chairs each)
• x = 15 (giving us 15 rows of 24 chairs each)
• x = 18 (giving us 18 rows of 20 chairs each)
• x = 20 (giving us 20 rows of 18 chairs each)
• x = 24 (giving us 24 rows of 15 chairs each)

Therefore, there are exactly 5 different rectangular arrangements possible.

5. Final Answer

The answer is B. Five

We found exactly 5 valid values for the number of rows (12, 15, 18, 20, and 24) that satisfy both the requirement that \(\mathrm{x \times y = 360}\) and the constraint that \(\mathrm{10 < x < 25}\).

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting the constraint boundaries
Students often misread "\(\mathrm{10 < x < 25}\)" as "\(\mathrm{10 \leq x \leq 25}\)", incorrectly including x = 10 and x = 25 in their solutions. This would lead them to consider additional factors like x = 10 (\(\mathrm{10 \times 36 = 360}\)) and potentially others, resulting in an incorrect count.

Faltering Point 2: Confusing which variable represents rows vs. chairs per row
Some students may set up the equation correctly as \(\mathrm{x \times y = 360}\) but then apply the constraint "\(\mathrm{10 < x < 25}\)" to the wrong variable (y instead of x). This fundamental misunderstanding would lead them to look for factors of 360 that could represent chairs per row rather than number of rows.

Faltering Point 3: Forgetting that arrangements are determined by factor pairs, not individual factors
Students might think they need to count both (x,y) and (y,x) as separate arrangements, essentially double-counting. For example, they might count both "12 rows of 30 chairs" and "30 rows of 12 chairs" as different solutions, when the constraint only allows one of these possibilities.

Errors while executing the approach

Faltering Point 1: Incomplete factor identification
When finding all factors of 360, students may miss some factors, especially the larger ones or those that result from less obvious combinations. Missing even one factor within the constraint range (like 18 or 24) would lead to an undercount of possible arrangements.

Faltering Point 2: Arithmetic errors in verification
Students may correctly identify potential values of x but make multiplication errors when verifying that \(\mathrm{x \times y = 360}\). For example, they might incorrectly calculate \(\mathrm{18 \times 20}\) or \(\mathrm{24 \times 15}\), leading them to reject valid solutions or accept invalid ones.

Errors while selecting the answer

Faltering Point 1: Including boundary values in the final count
Even after correctly solving most of the problem, students may accidentally include x = 10 or x = 25 in their final count, forgetting that these boundary values are excluded by the strict inequality "\(\mathrm{10 < x < 25}\)". This would lead them to choose "Six" instead of "Five".