- Translate the problem requirements: We need to find how many grapefruit are in a combination box, given that each fruit has the same unit price regardless of box type, and combination boxes contain twice as many oranges as grapefruit.
- Calculate unit prices from single-fruit boxes: Use the orange-only and grapefruit-only boxes to determine the price per individual orange and grapefruit.
- Set up the combination box composition: Express the contents of a combination box in terms of the number of grapefruit, using the 2:1 orange-to-grapefruit ratio.
- Apply price consistency principle: The total cost of fruits in a combination box must equal $8.50 when calculated using the unit prices from step 2.
- Solve for the number of grapefruit: Use the price equation to find the exact number of grapefruit in each combination box.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in everyday language:
- Orange-only boxes: 32 oranges cost $8.00
- Grapefruit-only boxes: 20 grapefruit cost $7.00
- Combination boxes: cost $8.50 and contain twice as many oranges as grapefruit
- Each fruit costs the same per piece, no matter which box it's in
We need to find how many grapefruit are in each combination box.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
2. Calculate unit prices from single-fruit boxes
Since each fruit has the same unit price regardless of box type, we can find the cost per individual fruit:
For oranges: If 32 oranges cost $8.00, then each orange costs \(\$8.00 \div 32 = \$0.25\) per orange
For grapefruit: If 20 grapefruit cost $7.00, then each grapefruit costs \(\$7.00 \div 20 = \$0.35\) per grapefruit
These unit prices must be the same whether the fruits are in single-fruit boxes or combination boxes.
3. Set up the combination box composition
Let's say there are g grapefruit in each combination box.
Since combination boxes contain twice as many oranges as grapefruit, there must be 2g oranges in each combination box.
So each combination box contains:
4. Apply price consistency principle
The total cost of all fruits in a combination box must equal $8.50.
Using our unit prices:
- Cost of g grapefruit = \(g \times \$0.35 = \$0.35g\)
- Cost of 2g oranges = \(2g \times \$0.25 = \$0.50g\)
Total cost = \(\$0.35g + \$0.50g = \$0.85g\)
Since the combination box costs $8.50:
\(\$0.85g = \$8.50\)
5. Solve for the number of grapefruit
To find g, we divide both sides by $0.85:
\(g = \$8.50 \div \$0.85 = 10\)
Let's verify: If there are 10 grapefruit and 20 oranges in each combination box:
- Cost of 10 grapefruit = \(10 \times \$0.35 = \$3.50\)
- Cost of 20 oranges = \(20 \times \$0.25 = \$5.00\)
- Total cost = \(\$3.50 + \$5.00 = \$8.50\) ✓
Final Answer
There must be 10 grapefruit in each combination box.
The answer is D.
Common Faltering Points
Errors while devising the approach
- Misinterpreting the constraint "twice as many oranges as grapefruit": Students might confuse this relationship and think there are twice as many grapefruit as oranges, leading them to set up the equation as 2g grapefruit and g oranges instead of g grapefruit and 2g oranges.
- Overlooking the "unit price independence" principle: Students may not realize that the cost per individual orange and grapefruit must be the same regardless of which type of box they're sold in. This leads them to try complex approaches instead of simply calculating unit prices from the single-fruit boxes.
- Misunderstanding what needs to be found: Some students might confuse the question and try to find the total number of fruits in the combination box or the number of oranges, rather than specifically focusing on finding the number of grapefruit.
Errors while executing the approach
- Unit price calculation errors: Students often make arithmetic mistakes when dividing \(\$8.00 \div 32 = \$0.25\) or \(\$7.00 \div 20 = \$0.35\), possibly getting $0.20 for oranges or $0.30 for grapefruit, which would lead to incorrect final answers.
- Incorrect equation setup: Even with the right approach, students might incorrectly write the total cost equation as \(\$0.25g + \$0.35(2g) = \$8.50\) instead of \(\$0.35g + \$0.25(2g) = \$8.50\), mixing up which fruit has which unit price.
- Division errors in the final step: Students may struggle with the decimal division \(\$8.50 \div \$0.85 = 10\), potentially getting 8.5 or making other computational mistakes that lead to non-integer answers.
Errors while selecting the answer
No likely faltering points.
