- 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 R's area to S's area.
- Express both shapes using the rectangle's dimensions: Use the \(2:3\) ratio to define rectangle's sides, then use equal perimeters to find the square's side length.
- Calculate and compare the areas: Find the area of rectangle R and square S, then form their ratio.
- Simplify the ratio to match answer choices: Reduce the fraction to its simplest form and identify the corresponding answer choice.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in everyday terms:
- We have a square S and a rectangle R
- Both shapes have the same perimeter (distance around the outside)
- The rectangle's sides are in the ratio \(2:3\), meaning if one side is 2 units, the other is 3 units
- We need to find how their areas compare
Process Skill: TRANSLATE
2. Express both shapes using the rectangle's dimensions
Let's use concrete numbers to make this easier to understand. Since the rectangle's sides are in ratio \(2:3\), let's say:
- Rectangle's shorter side = \(2\mathrm{k}\) (where k is some number)
- Rectangle's longer side = \(3\mathrm{k}\)
- Rectangle's perimeter = \(2(2\mathrm{k}) + 2(3\mathrm{k}) = 4\mathrm{k} + 6\mathrm{k} = 10\mathrm{k}\)
Since the square has the same perimeter:
- Square's perimeter = \(10\mathrm{k}\)
- Each side of square = \(10\mathrm{k} ÷ 4 = 2.5\mathrm{k}\)
Now we can see both shapes clearly in terms of the same variable k.
Process Skill: SIMPLIFY
3. Calculate and compare the areas
Now let's find the areas using our expressions:
Rectangle R's area:
- Area = length × width = \(3\mathrm{k} × 2\mathrm{k} = 6\mathrm{k}^2\)
Square S's area:
- Area = side × side = \(2.5\mathrm{k} × 2.5\mathrm{k} = 6.25\mathrm{k}^2\)
Ratio of R's area to S's area:
- Ratio = \(6\mathrm{k}^2 : 6.25\mathrm{k}^2\)
- The \(\mathrm{k}^2\) cancels out: \(6 : 6.25\)
4. Simplify the ratio to match answer choices
We need to convert \(6 : 6.25\) to a simple fraction:
- \(6 : 6.25 = \frac{6}{6.25}\)
- To eliminate the decimal, multiply both by 4: \(\frac{6 × 4}{6.25 × 4} = \frac{24}{25}\)
- So the ratio is \(24:25\)
Let's verify: \(24 ÷ 25 = 0.96\), and \(6 ÷ 6.25 = 0.96\) ✓
Final Answer
The ratio of the area of rectangle R to the area of square S is \(24:25\).
This matches answer choice B. 24:25.
This makes intuitive sense: when two shapes have the same perimeter, the square will have the larger area, so the rectangle's area should be smaller than the square's area, giving us a ratio less than 1, which \(\frac{24}{25} = 0.96\) confirms.
Common Faltering Points
Errors while devising the approach
- Misinterpreting the ratio constraint: Students might think the sides are exactly 2 and 3 units instead of understanding that \(2:3\) means the sides are \(2\mathrm{k}\) and \(3\mathrm{k}\) for some value k. This leads to using fixed values rather than expressions with variables.
- Confusing which ratio is asked: The question asks for the ratio of area of R to area of S, but students might set up the calculation for area of S to area of R instead, leading to the reciprocal of the correct answer.
- Misunderstanding equal perimeters: Students might incorrectly assume that equal perimeters means the shapes have equal areas, missing the fundamental concept that among shapes with equal perimeters, the square has the maximum area.
Errors while executing the approach
- Arithmetic errors in perimeter calculation: When calculating the rectangle's perimeter as \(2(2\mathrm{k}) + 2(3\mathrm{k})\), students might make errors like getting \(2(2\mathrm{k} + 3\mathrm{k}) = 2(5\mathrm{k}) = 10\mathrm{k}\) correctly, but then incorrectly calculating the square's side length from the perimeter.
- Decimal calculation mistakes: When finding the square's side length as \(10\mathrm{k} ÷ 4 = 2.5\mathrm{k}\), students might make errors, or when calculating \((2.5\mathrm{k})^2 = 6.25\mathrm{k}^2\), they might get \(6.5\mathrm{k}^2\) or other incorrect values due to decimal multiplication errors.
- Variable cancellation errors: When forming the ratio \(6\mathrm{k}^2 : 6.25\mathrm{k}^2\), students might forget to cancel out the \(\mathrm{k}^2\) terms or make errors in the cancellation process, leading to unnecessarily complex expressions.
Errors while selecting the answer
- Fraction conversion mistakes: When converting \(6:6.25\) to a simple ratio, students might incorrectly multiply by different factors or make arithmetic errors, getting ratios like \(12:12.5\) instead of \(24:25\).
- Selecting the reciprocal: Students who correctly calculate the ratio as \(24:25\) might mistakenly select an answer choice that represents \(25:24\), especially if they set up the initial ratio incorrectly but performed the calculations correctly.
Alternate Solutions
Smart Numbers Approach
Step 1: Choose smart numbers for the rectangle's dimensions
Since the rectangle's sides are in ratio \(2:3\), let's use the smallest convenient values:
• Shorter side = 2 units
• Longer side = 3 units
This gives us concrete dimensions that maintain the required \(2:3\) ratio.
Step 2: Calculate the rectangle's perimeter
Perimeter of rectangle R = \(2(\text{length} + \text{width}) = 2(3 + 2) = 10\) units
Step 3: Find the square's side length
Since the perimeters are equal:
Perimeter of square S = 10 units
Side of square = \(10 ÷ 4 = 2.5\) units
Step 4: Calculate both areas
• Area of rectangle R = length × width = \(3 × 2 = 6\) square units
• Area of square S = \((\text{side})^2 = (2.5)^2 = 6.25\) square units
Step 5: Find the ratio
Ratio of area of R to area of S = \(6 : 6.25 = \frac{6}{6.25}\)
To convert to integers: \(\frac{6}{6.25} = \frac{6}{\frac{25}{4}} = 6 × \frac{4}{25} = \frac{24}{25}\)
Therefore, the ratio is \(24:25\)
Why this works: The \(2:3\) ratio constraint allows us to pick the simplest values (2 and 3) that satisfy this relationship. All other valid rectangles with sides in \(2:3\) ratio are just scaled versions of this one, and scaling both shapes proportionally won't change the area ratio.
