Judy bought a quantity of pens in packages of 5 for $0.80 per package. She sold all of the pens...

1. Translate the problem requirements: Judy buys pens in groups of 5 for $0.80 per group, sells them in groups of 3 for $0.60 per group, makes $8.00 total profit. We need to find the total number of pens.
2. Set up profit per pen calculation: Calculate how much profit Judy makes on each individual pen by finding the difference between her selling price per pen and buying price per pen.
3. Apply total profit to find quantity: Use the profit per pen to determine how many pens she must have bought and sold to achieve $8.00 total profit.
4. Verify with answer choices: Check that our calculated answer matches one of the given options and makes logical sense.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what Judy is doing in plain English:

• Buying: Judy purchases pens in packages of 5 for $0.80 per package
• Selling: She sells all these pens in packages of 3 for $0.60 per package
• Profit: Her total profit is $8.00
• Question: How many individual pens did she buy and sell?

The key insight here is that she buys and sells the SAME number of pens - she's just repackaging them. When she buys 5 pens at once, she later sells those same 5 pens by grouping them differently (in groups of 3).

Process Skill: TRANSLATE - Converting the problem's business scenario into clear mathematical relationships

2. Set up profit per pen calculation

To find profit per pen, we need to determine:

• How much she pays per individual pen when buying
• How much she receives per individual pen when selling

Cost per pen when buying:

She pays $0.80 for 5 pens, so cost per pen = \(\$0.80 \div 5 = \$0.16\) per pen

Revenue per pen when selling:

She receives $0.60 for 3 pens, so revenue per pen = \(\$0.60 \div 3 = \$0.20\) per pen

Profit per pen:

Profit per pen = Revenue per pen - Cost per pen = \(\$0.20 - \$0.16 = \$0.04\) per pen

This means Judy makes 4 cents profit on every pen she buys and resells.

3. Apply total profit to find quantity

Now we can use the relationship between total profit and profit per pen:

If Judy makes $0.04 profit per pen, and her total profit is $8.00, then:

Number of pens = Total profit ÷ Profit per pen

Number of pens = \(\$8.00 \div \$0.04 = 200\) pens

Let's verify this makes sense:

• She bought 200 pens in packages of 5: \(200 \div 5 = 40\) packages
• Cost: \(40 \times \$0.80 = \$32.00\)
• She sold 200 pens in packages of 3: \(200 \div 3 = 66\frac{2}{3}\) packages
• Revenue: \(66\frac{2}{3} \times \$0.60 = \$40.00\)
• Profit: \(\$40.00 - \$32.00 = \$8.00\) ✓

4. Verify with answer choices

Our calculated answer is 200 pens, which matches choice D exactly.

Let's double-check by testing another answer choice to confirm our method:

Testing choice C (100 pens):

• Profit would be: \(100 \times \$0.04 = \$4.00\)
• This doesn't match the required $8.00 profit

Final Answer: D. 200

Judy bought and sold 200 pens, making a profit of 4 cents per pen to achieve her total profit of $8.00.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what quantity to find: Students may think the question asks for the number of packages bought or sold, rather than the total number of individual pens. This happens because the problem mentions "packages of 5" and "packages of 3," leading students to focus on package counts instead of pen counts.

2. Not recognizing the repackaging concept: Students might incorrectly assume Judy buys some pens and sells different pens, rather than understanding she buys a certain number of pens and then sells those same exact pens (just repackaged differently). This fundamental misunderstanding would lead to setting up incorrect equations.

3. Attempting to work directly with packages instead of per-pen analysis: Students may try to set up equations using the package information directly without converting to per-pen costs and revenues. This approach becomes unnecessarily complex because 5 and 3 don't have a simple relationship, making the profit calculation much harder.

Errors while executing the approach

1. Arithmetic errors in division: When calculating cost per pen (\(\$0.80 \div 5\)) or revenue per pen (\(\$0.60 \div 3\)), students may make simple division mistakes. For example, they might calculate \(\$0.60 \div 3\) as $0.25 instead of $0.20, leading to an incorrect profit per pen.

2. Incorrect profit calculation: Students might subtract in the wrong order (cost - revenue instead of revenue - cost) or make errors when subtracting decimals (\(\$0.20 - \$0.16\)). Some may also forget that profit = revenue - cost and instead add the two values.

3. Final division error: When dividing total profit by profit per pen (\(\$8.00 \div \$0.04\)), students may struggle with dividing by a decimal or make computational errors, potentially getting 20, 2000, or other incorrect values instead of 200.

Errors while selecting the answer

1. Selecting number of packages instead of pens: After correctly calculating that 40 packages were bought (\(200 \div 5\)), students might select choice A (40) thinking this is the answer the question wants, forgetting that the question specifically asks for the number of pens.

2. Confusing verification calculations with the final answer: During verification, students calculate various intermediate values (like 40 packages bought, $32 total cost, $40 total revenue). They might accidentally select one of these intermediate values if it matches an answer choice, rather than the actual number of pens (200).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number
Since Judy buys pens in packages of 5 and sells in packages of 3, we need a number divisible by both 5 and 3. Let's use 15 pens as our starting point (LCM of 3 and 5).

Step 2: Calculate profit for 15 pens
• Buying: 15 pens = 3 packages of 5 pens each
Cost = \(3 \times \$0.80 = \$2.40\)
• Selling: 15 pens = 5 packages of 3 pens each
Revenue = \(5 \times \$0.60 = \$3.00\)
• Profit for 15 pens = \(\$3.00 - \$2.40 = \$0.60\)

Step 3: Scale up to target profit
We need $8.00 total profit, and 15 pens gives $0.60 profit.
Multiplier needed = \(\$8.00 \div \$0.60 = \frac{80}{6} = \frac{40}{3} \approx 13.33\)

Step 4: Try multiples of 15
Since we need whole packages, let's try \(15 \times 13 = 195\) pens:
• Buying: \(195 \div 5 = 39\) packages, Cost = \(39 \times \$0.80 = \$31.20\)
• Selling: \(195 \div 3 = 65\) packages, Revenue = \(65 \times \$0.60 = \$39.00\)
• Profit = \(\$39.00 - \$31.20 = \$7.80\) (close, but not exactly $8.00)

Step 5: Try 200 pens (closest answer choice)
• Buying: \(200 \div 5 = 40\) packages, Cost = \(40 \times \$0.80 = \$32.00\)
• Selling: \(200 \div 3 = 66.67\) packages... This doesn't work as we can't sell fractional packages.

Step 6: Systematic check with answer choices
Let's check 200 pens differently - maybe the problem allows partial package sales:
• Cost per pen when buying: \(\$0.80 \div 5 = \$0.16\) per pen
• Revenue per pen when selling: \(\$0.60 \div 3 = \$0.20\) per pen
• Profit per pen = \(\$0.20 - \$0.16 = \$0.04\)
• For $8.00 profit: \(\$8.00 \div \$0.04 = 200\) pens

Answer: D. 200