If the perimeter of square region S and the perimeter of rectangular region R are equal and the sides of...

1. Translate the problem requirements: We have a square S and rectangle R with equal perimeters. Rectangle R has sides in ratio \(2:3\). We need to find the ratio of area of R to area of S.
2. Set up variables using the given ratio: Express the rectangle's dimensions in terms of a single variable using the \(2:3\) ratio to keep calculations simple.
3. Apply the equal perimeter condition: Use the fact that both shapes have the same perimeter to establish a relationship between the rectangle's dimensions and the square's side length.
4. Calculate and compare areas: Find the areas of both shapes and express their ratio in simplest form.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• We have a square called S
• We have a rectangle called R
• Both shapes have the same perimeter (distance around the edge)
• The rectangle's sides are in the ratio \(2:3\), meaning if one side has length 2 units, the other has length 3 units
• We need to find how their areas compare

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up variables using the given ratio

Since the rectangle's sides are in ratio \(2:3\), let's use a simple approach. If we call the shorter side \(2\mathrm{k}\) and the longer side \(3\mathrm{k}\) (where k is some positive number), then we automatically maintain the \(2:3\) ratio.

So our rectangle has:

• Shorter side = \(2\mathrm{k}\)
• Longer side = \(3\mathrm{k}\)

For the square, let's call each side length \(\mathrm{s}\).

3. Apply the equal perimeter condition

Now we use the fact that both shapes have equal perimeters.

Perimeter of rectangle R = \(2(\mathrm{length} + \mathrm{width}) = 2(3\mathrm{k} + 2\mathrm{k}) = 2(5\mathrm{k}) = 10\mathrm{k}\)

Perimeter of square S = \(4(\mathrm{side\ length}) = 4\mathrm{s}\)

Since the perimeters are equal:
\(10\mathrm{k} = 4\mathrm{s}\)

Solving for s: \(\mathrm{s} = \frac{10\mathrm{k}}{4} = \frac{5\mathrm{k}}{2}\)

Process Skill: APPLY CONSTRAINTS - Using the equal perimeter condition to establish the relationship between variables

4. Calculate and compare areas

Now we can find both areas and compare them.

Area of rectangle R = \(\mathrm{length} \times \mathrm{width} = 3\mathrm{k} \times 2\mathrm{k} = 6\mathrm{k}^2\)

Area of square S = \(\mathrm{side} \times \mathrm{side} = \mathrm{s}^2 = \left(\frac{5\mathrm{k}}{2}\right)^2 = \frac{25\mathrm{k}^2}{4}\)

To find the ratio of area of R to area of S:
Ratio = \(\frac{\mathrm{Area\ of\ R}}{\mathrm{Area\ of\ S}} = 6\mathrm{k}^2 \div \frac{25\mathrm{k}^2}{4}\)

When dividing by a fraction, we multiply by its reciprocal:
\(= 6\mathrm{k}^2 \times \frac{4}{25\mathrm{k}^2} = \frac{24\mathrm{k}^2}{25\mathrm{k}^2} = \frac{24}{25}\)

Final Answer

The ratio of the area of R to the area of S is \(24:25\).

This matches answer choice B: \(24:25\).

Common Faltering Points

Errors while devising the approach

• Misinterpreting the ratio constraint: Students may think the \(2:3\) ratio means the rectangle has fixed sides of length 2 and 3, rather than understanding it represents a proportional relationship (\(2\mathrm{k}\) and \(3\mathrm{k}\)). This leads to incorrect setup and wrong final answer.
• Confusing which ratio is being asked: Students may set up to find the ratio of area of S to area of R instead of area of R to area of S, leading them to look for the reciprocal of the correct answer in the choices.
• Missing the equal perimeter constraint: Students may focus only on the ratio of sides and forget to apply the crucial condition that both shapes have equal perimeters, making it impossible to establish the relationship between the square's side length and the rectangle's dimensions.

Errors while executing the approach

• Arithmetic errors in perimeter calculation: Students may incorrectly calculate the rectangle's perimeter as \(2\mathrm{k} + 3\mathrm{k} = 5\mathrm{k}\) instead of \(2(2\mathrm{k} + 3\mathrm{k}) = 10\mathrm{k}\), forgetting that perimeter requires adding all four sides.
• Mistakes when squaring fractions: When calculating the area of the square as \(\left(\frac{5\mathrm{k}}{2}\right)^2\), students may incorrectly compute this as \(\frac{25\mathrm{k}^2}{2}\) instead of \(\frac{25\mathrm{k}^2}{4}\), forgetting to square the denominator.
• Division by fraction errors: When finding the ratio \(6\mathrm{k}^2 \div \frac{25\mathrm{k}^2}{4}\), students may incorrectly divide instead of multiplying by the reciprocal, getting \(6\mathrm{k}^2 \times \frac{25\mathrm{k}^2}{4}\) or making other computational mistakes.

Errors while selecting the answer

• Selecting the reciprocal ratio: Students who correctly calculate \(\frac{24}{25}\) may mistakenly select answer choice A (\(25:16\)) or look for \(25:24\) among the choices, having confused the direction of the ratio during their solution process.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers based on the ratio

Since the rectangle's sides are in the ratio \(2:3\), let's choose concrete values that make this ratio easy to work with. Let the sides of rectangle R be:

• Length = 6 units (representing the "3" part of the ratio)
• Width = 4 units (representing the "2" part of the ratio)

Notice that \(4:6 = 2:3\), satisfying our given ratio.

Step 2: Calculate the perimeter of rectangle R

Perimeter of R = \(2(\mathrm{length} + \mathrm{width}) = 2(6 + 4) = 2(10) = 20\) units

Step 3: Find the side length of square S

Since the perimeters are equal, square S also has perimeter = 20 units

For a square: Perimeter = \(4 \times \mathrm{side\ length}\)

So: \(4 \times \mathrm{side\ length} = 20\)

Side length of square S = 5 units

Step 4: Calculate the areas

Area of rectangle R = \(\mathrm{length} \times \mathrm{width} = 6 \times 4 = 24\) square units

Area of square S = \((\mathrm{side\ length})^2 = 5^2 = 25\) square units

Step 5: Find the ratio

Ratio of area of R to area of S = \(24:25\)

Why this approach works: The key insight is that we can choose any specific numbers that satisfy the \(2:3\) ratio constraint. The final ratio of areas will be the same regardless of which specific numbers we choose, as long as they maintain the \(2:3\) relationship. This makes the problem much more concrete and easier to follow.