- Translate the problem requirements: We have a carpet that is x feet by y feet, costs $50 total, and we need to find the cost per square yard (not per square foot)
- Calculate the carpet's area in square feet: Find the total area using the given dimensions
- Convert area from square feet to square yards: Use the given conversion factor to express area in the required units
- Apply the unit rate formula: Divide total cost by area in square yards to get cost per square yard
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know and what we need to find:
What we have:
- A rectangular carpet that measures x feet by y feet
- The total cost is $50
- We know that \(9\) square feet = \(1\) square yard
What we need to find:
- The cost per square yard (not per square foot!)
Think of this like buying fabric or flooring - we often get a total price for a piece, but we want to know the rate per unit area to compare with other options.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
2. Calculate the carpet's area in square feet
For any rectangle, the area is simply length times width.
Since our carpet measures x feet by y feet:
Area in square feet = \(\mathrm{x} \times \mathrm{y} = \mathrm{xy}\) square feet
This is straightforward - if you had a carpet that was 6 feet by 8 feet, the area would be \(6 \times 8 = 48\) square feet.
3. Convert area from square feet to square yards
Now we need to convert our area from square feet to square yards using the given conversion.
We're told that \(9\) square feet = \(1\) square yard
So if our carpet has an area of xy square feet, how many square yards is that?
Think of it this way: if 9 square feet makes 1 square yard, then xy square feet makes \(\mathrm{xy} \div 9\) square yards.
Area in square yards = \(\mathrm{xy} \div 9 = \frac{\mathrm{xy}}{9}\) square yards
For example, if we had 36 square feet, that would be \(36 \div 9 = 4\) square yards.
4. Apply the unit rate formula
Now we can find the cost per square yard using the basic unit rate formula:
Cost per square yard = Total cost ÷ Total area in square yards
We have:
- Total cost = $50
- Total area in square yards = \(\frac{\mathrm{xy}}{9}\)
So: Cost per square yard = \(50 \div \left(\frac{\mathrm{xy}}{9}\right)\)
When we divide by a fraction, we multiply by its reciprocal:
\(50 \div \left(\frac{\mathrm{xy}}{9}\right) = 50 \times \frac{9}{\mathrm{xy}} = \frac{450}{\mathrm{xy}}\)
Therefore, the cost per square yard is \(\frac{450}{\mathrm{xy}}\) dollars.
5. Final Answer
The cost of the carpet is \(\frac{450}{\mathrm{xy}}\) dollars per square yard.
This matches answer choice E. \(\frac{450}{\mathrm{xy}}\).
To verify this makes sense: if the carpet dimensions were larger (bigger x or y), the cost per square yard should be smaller since we're spreading the same $50 over more area. Our answer \(\frac{450}{\mathrm{xy}}\) does get smaller as x or y increases, which confirms our logic is correct.
Common Faltering Points
Errors while devising the approach
- Misunderstanding the unit conversion requirement: Students often miss that the question asks for cost per square YARD, not per square foot. They might set up their calculation to find cost per square foot (which would be \(\frac{50}{\mathrm{xy}}\)) instead of converting to square yards first.
- Confusing what needs to be calculated: Some students might think they need to find the total area or the area per dollar, rather than understanding they need to find the rate of dollars per unit area (specifically per square yard).
- Misinterpreting the conversion factor: Students might incorrectly think that since 9 square feet = 1 square yard, they should multiply by 9 instead of divide by 9 when converting from square feet to square yards.
Errors while executing the approach
- Incorrect division by fraction: When calculating \(50 \div \left(\frac{\mathrm{xy}}{9}\right)\), students often make the error of treating this as \(\frac{50 \times \mathrm{xy}}{9}\) instead of correctly applying the rule that dividing by a fraction means multiplying by its reciprocal: \(50 \times \frac{9}{\mathrm{xy}} = \frac{450}{\mathrm{xy}}\).
- Area conversion errors: Students might incorrectly convert xy square feet to square yards by multiplying by 9 instead of dividing by 9, leading them to think the area is \(9\mathrm{xy}\) square yards instead of \(\frac{\mathrm{xy}}{9}\) square yards.
Errors while selecting the answer
- Selecting the intermediate calculation: Students who correctly calculate the area in square yards as \(\frac{\mathrm{xy}}{9}\) might mistakenly select answer choice C (\(\frac{\mathrm{xy}}{9}\)) thinking this is the final answer, when this is actually just the area, not the cost per square yard.
- Choosing the reciprocal: Some students might arrive at the correct calculation structure but select answer choice D (\(\frac{\mathrm{xy}}{50}\)) instead of E (\(\frac{450}{\mathrm{xy}}\)), confusing which value should be in the numerator versus denominator.
Alternate Solutions
Smart Numbers Approach
Step 1: Choose convenient values for the carpet dimensions
Let's select x = 3 feet and y = 3 feet. These values are chosen because:
- They create a 9 square foot area (\(3 \times 3 = 9\)), which converts cleanly to exactly 1 square yard
- This eliminates complex fractions and makes the calculation straightforward
- The numbers satisfy all given constraints in the problem
Step 2: Calculate the carpet's area
Area = \(\mathrm{x} \times \mathrm{y} = 3 \times 3 = 9\) square feet
Step 3: Convert to square yards
Since 9 square feet = 1 square yard, our carpet area = 1 square yard
Step 4: Calculate cost per square yard
Cost per square yard = Total cost ÷ Area in square yards = \(\$50 \div 1 = \$50\) per square yard
Step 5: Check which answer choice gives us $50
Substituting x = 3 and y = 3 into each choice:
- \(50\mathrm{xy} = 50(3)(3) = 450\) ❌
- \(450\mathrm{xy} = 450(3)(3) = 4,050\) ❌
- \(\frac{\mathrm{xy}}{9} = \frac{(3)(3)}{9} = 1\) ❌
- \(\frac{\mathrm{xy}}{50} = \frac{(3)(3)}{50} = 0.18\) ❌
- \(\frac{450}{\mathrm{xy}} = \frac{450}{3 \times 3} = \frac{450}{9} = 50\) ✓
Answer: E
