Lee is buying apples to make pies. The apples cost $1 per pound and each pie requires 3 cups of...

1. Translate the problem requirements: We need to find the total cost in dollars for enough apples to make p pies. Each pie needs 3 cups of peeled apples, apples cost $1 per pound, when peeled apples lose \(\frac{1}{5}\) of weight (keeping \(\frac{4}{5}\)), and 2 cups of peeled apples weigh 1 pound.
2. Calculate total peeled apple requirements: Determine how many cups and pounds of peeled apples are needed for p pies.
3. Work backwards to find original apple weight: Since peeled apples are \(\frac{4}{5}\) of the original weight, calculate how many pounds of original apples we need to buy.
4. Convert to final cost: Multiply the pounds of original apples by $1 per pound to get the total cost.

Execution of Strategic Approach

1. Translate the problem requirements

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

• We want to make p pies
• Each pie needs 3 cups of peeled and cored apples
• Apples cost $1 per pound (this is the price for whole, unpeeled apples)
• When we peel and core apples, they lose \(\frac{1}{5}\) of their weight, so they keep \(\frac{4}{5}\) of their original weight
• 2 cups of peeled apples weigh 1 pound

Our goal is to find the total cost in dollars for the original apples we need to buy.

Process Skill: TRANSLATE - Converting all the given information into clear, usable facts

2. Calculate total peeled apple requirements

Let's figure out how much peeled apple we need in total:

• For p pies, we need: \(p \times 3 = 3p\) cups of peeled apples

Now let's convert this to pounds. Since 2 cups of peeled apples weigh 1 pound:
• \(3p\) cups of peeled apples = \(3p \div 2 = \frac{3p}{2}\) pounds of peeled apples

So we need \(\frac{3p}{2}\) pounds of peeled and cored apples.

3. Work backwards to find original apple weight

Here's the key insight: the peeled apples we calculated above represent only \(\frac{4}{5}\) of the original apple weight (since \(\frac{1}{5}\) is lost during peeling).

Think of it this way: if peeled apples are \(\frac{4}{5}\) of the original weight, then:
Original weight \(\times \frac{4}{5}\) = Peeled weight

So: Original weight = Peeled weight \(\div \frac{4}{5}\) = Peeled weight \(\times \frac{5}{4}\)

We need \(\frac{3p}{2}\) pounds of peeled apples, so:
Original apple weight = \(\frac{3p}{2} \times \frac{5}{4} = \frac{15p}{8}\) pounds

Process Skill: INFER - Understanding that we must work backwards from the final requirement to the initial purchase

4. Convert to final cost

Since apples cost $1 per pound, and we need \(\frac{15p}{8}\) pounds of original apples:

Total cost = \(\frac{15p}{8}\) pounds × $1/pound = \(\frac{15p}{8}\) dollars

Looking at our answer choices, this matches choice (E): \(\frac{15p}{8}\)

Final Answer

The cost of apples for p pies is \(\frac{15p}{8}\) dollars, which corresponds to answer choice (E).

To verify: We need \(3p\) cups of peeled apples → \(\frac{3p}{2}\) pounds of peeled apples → \(\frac{15p}{8}\) pounds of original apples → $\(\frac{15p}{8}\) total cost.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the weight loss during peeling: Students often confuse "loses \(\frac{1}{5}\) of its weight" with "retains \(\frac{1}{5}\) of its weight." The correct interpretation is that apples retain \(\frac{4}{5}\) of their original weight after peeling, but many students incorrectly assume they retain only \(\frac{1}{5}\).

2. Confusion about what price applies to: The $1 per pound price is for whole, unpeeled apples, but students may mistakenly think this price applies to the peeled apples. This leads to incorrect cost calculations since they don't account for needing to buy more original weight.

3. Missing the need to work backwards: Students may try to work forward from cups to cost without realizing they need to determine the original apple weight first. They might directly calculate cost from peeled apple requirements, missing the crucial step of converting back to original weight.

Errors while executing the approach

1. Incorrect fraction arithmetic when converting cups to pounds: When converting \(3p\) cups to pounds using the ratio "2 cups = 1 pound," students may incorrectly calculate \(3p \times 2 = 6p\) pounds instead of the correct \(3p \div 2 = \frac{3p}{2}\) pounds.

2. Wrong multiplication when finding original weight: Students may incorrectly multiply by \(\frac{4}{5}\) instead of \(\frac{5}{4}\) when working backwards from peeled weight to original weight. They might calculate \(\frac{3p}{2} \times \frac{4}{5} = \frac{6p}{5}\) instead of \(\frac{3p}{2} \times \frac{5}{4} = \frac{15p}{8}\).

3. Computational errors with fraction multiplication: When calculating \(\frac{3p}{2} \times \frac{5}{4}\), students may make arithmetic mistakes such as getting \(\frac{15p}{6}\) instead of \(\frac{15p}{8}\), or incorrectly simplifying the final fraction.

Errors while selecting the answer

1. Selecting an intermediate calculation result: Students might select \(\frac{3p}{2}\) (answer choice D) which represents the pounds of peeled apples needed, rather than continuing to find the cost of original apples required.

2. Choosing the reciprocal or related fraction: Students who get confused during fraction operations might select \(\frac{6p}{5}\) (answer choice C) or other similar-looking fractions that result from computational errors in the final steps.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for p

Let's set p = 8 pies (chosen because it works well with the fractions involved)

Step 2: Calculate peeled apple requirements

• Total cups of peeled apples needed = 8 pies × 3 cups per pie = 24 cups

• Convert to pounds: Since 2 cups = 1 pound, then 24 cups = 24 ÷ 2 = 12 pounds of peeled apples

Step 3: Work backwards to original apple weight

• When apples are peeled, they lose \(\frac{1}{5}\) of weight, so peeled apples = \(\frac{4}{5}\) of original weight

• If peeled apples weigh 12 pounds, then: Original weight × \(\frac{4}{5}\) = 12

• Original weight = \(12 \div \frac{4}{5} = 12 \times \frac{5}{4} = 15\) pounds

Step 4: Calculate total cost

• Cost = 15 pounds × $1 per pound = $15

Step 5: Verify with answer choices

Substituting p = 8 into each choice:

1. \(\frac{3p}{10} = \frac{3(8)}{10} = 2.4\)
2. \(\frac{4p}{10} = \frac{4(8)}{10} = 3.2\)
3. \(\frac{6p}{5} = \frac{6(8)}{5} = 9.6\)
4. \(\frac{3p}{2} = \frac{3(8)}{2} = 12\)
5. \(\frac{15p}{8} = \frac{15(8)}{8} = 15\) ✓

Only choice E gives us $15, confirming our answer.