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To mail a package, the rate is \(\mathrm{x}\) cents for the first pound and \(\mathrm{y}\) cents for each additional pound, where \(\mathrm{x > y}\). Two packages weighing \(\mathrm{3}\) pounds and \(\mathrm{5}\) pounds, respectively, can be mailed separately or combined as one package. Which method is cheaper, and how much money is saved?
Let's start by understanding exactly what we're comparing. We have two packages - one weighs 3 pounds and another weighs 5 pounds. We can either:
Method 1: Mail them separately (as two different packages)
Method 2: Combine them into one 8-pound package
The shipping cost structure is:
• First pound costs \(\mathrm{x}\) cents
• Each additional pound costs \(\mathrm{y}\) cents
• We know that \(\mathrm{x} > \mathrm{y}\) (the first pound is more expensive than additional pounds)
Our goal is to figure out which method costs less and by how much.
Process Skill: TRANSLATE - Converting the word problem into a clear mathematical comparison
When we mail packages separately, each package gets charged the "first pound" rate plus additional pounds.
For the 3-pound package:
• First pound: \(\mathrm{x}\) cents
• Remaining 2 pounds: \(\mathrm{2y}\) cents
• Total for 3-pound package: \(\mathrm{x + 2y}\) cents
For the 5-pound package:
• First pound: \(\mathrm{x}\) cents
• Remaining 4 pounds: \(\mathrm{4y}\) cents
• Total for 5-pound package: \(\mathrm{x + 4y}\) cents
Total cost when mailing separately:
\(\mathrm{(x + 2y) + (x + 4y) = 2x + 6y}\) cents
Notice that we pay the expensive "first pound" rate twice - once for each package.
When we combine both packages, we create one 8-pound package.
For the combined 8-pound package:
• First pound: \(\mathrm{x}\) cents
• Remaining 7 pounds: \(\mathrm{7y}\) cents
• Total cost when combined: \(\mathrm{x + 7y}\) cents
Notice that we only pay the expensive "first pound" rate once, since it's now just one package.
Let's compare the two costs:
Separate mailing cost: \(\mathrm{2x + 6y}\) cents
Combined mailing cost: \(\mathrm{x + 7y}\) cents
To find which is cheaper, let's calculate the difference:
Separate cost - Combined cost = \(\mathrm{(2x + 6y) - (x + 7y)}\)
= \(\mathrm{2x + 6y - x - 7y}\)
= \(\mathrm{x - y}\) cents
Since \(\mathrm{x > y}\) (given in the problem), we know that \(\mathrm{x - y > 0}\).
This means: Separate cost - Combined cost = positive number
Therefore: Separate cost > Combined cost
Conclusion: Combined mailing is cheaper, and the savings is \(\mathrm{x - y}\) cents.
Process Skill: APPLY CONSTRAINTS - Using the given condition \(\mathrm{x > y}\) to determine which method is cheaper
Combined mailing is cheaper, with a savings of \(\mathrm{x - y}\) cents.
This matches answer choice (A): "Combined, with a savings of \(\mathrm{x - y}\) cents"
Why this makes intuitive sense: When mailing separately, we pay the high "first pound" rate twice. When combined, we only pay it once. The savings comes from avoiding one "first pound" charge (worth \(\mathrm{x}\) cents) but paying for one extra additional pound (worth \(\mathrm{y}\) cents), giving us a net savings of \(\mathrm{x - y}\) cents.
1. Misunderstanding the constraint \(\mathrm{x > y}\): Students might overlook or misinterpret that \(\mathrm{x > y}\) means the first pound is MORE expensive than additional pounds. This is crucial for determining which method is cheaper. Without recognizing this relationship, students cannot properly compare costs.
2. Confusion about what constitutes "separate" vs "combined" mailing: Students may misunderstand that mailing separately means each package gets its own "first pound" charge, while combined mailing treats both packages as one single package with only one "first pound" charge. This fundamental misunderstanding leads to incorrect cost calculations.
3. Unclear about what the question is asking: Students might focus only on calculating costs but miss that the question asks for both which method is cheaper AND the amount saved. Some students calculate correctly but forget to determine the savings amount.
1. Arithmetic errors in cost calculations: Students frequently make mistakes when adding up costs, especially when combining terms like \(\mathrm{(x + 2y) + (x + 4y) = 2x + 6y}\). Common errors include incorrectly adding coefficients or dropping terms during simplification.
2. Sign errors when calculating the difference: When computing \(\mathrm{(2x + 6y) - (x + 7y)}\), students often make mistakes with negative signs, potentially getting \(\mathrm{(x - y)}\) as \(\mathrm{(y - x)}\) or making errors in the subtraction process that leads to incorrect savings calculations.
3. Incorrectly counting additional pounds: Students may miscalculate how many "additional pounds" each package has. For example, thinking a 3-pound package has 3 additional pounds instead of 2, or an 8-pound combined package has 8 additional pounds instead of 7.
1. Choosing the wrong method despite correct calculations: Even when students correctly calculate that separate cost \(\mathrm{(2x + 6y)}\) is greater than combined cost \(\mathrm{(x + 7y)}\), they might still select "Separately" as the cheaper option due to misreading their own work or the answer choices.
2. Selecting the wrong savings amount: Students who correctly identify that combined mailing is cheaper might still choose answer choice (C) "Combined, with a savings of \(\mathrm{x}\) cents" instead of (A) "Combined, with a savings of \(\mathrm{x - y}\) cents" because they forget to account for the extra pound cost in the combined package.
We can solve this problem by choosing specific values for \(\mathrm{x}\) and \(\mathrm{y}\) that satisfy the given constraint \(\mathrm{x > y}\), then calculating costs with concrete numbers.
Let's choose values that make calculations simple:
• \(\mathrm{x = 50}\) cents (cost for first pound)
• \(\mathrm{y = 30}\) cents (cost for each additional pound)
Note: We ensure \(\mathrm{x > y}\) as required \(\mathrm{(50 > 30)}\)
3-pound package:
• First pound: 50 cents
• Additional 2 pounds: \(\mathrm{2 \times 30 = 60}\) cents
• Total: \(\mathrm{50 + 60 = 110}\) cents
5-pound package:
• First pound: 50 cents
• Additional 4 pounds: \(\mathrm{4 \times 30 = 120}\) cents
• Total: \(\mathrm{50 + 120 = 170}\) cents
Total cost for separate mailing: \(\mathrm{110 + 170 = 280}\) cents
8-pound combined package:
• First pound: 50 cents
• Additional 7 pounds: \(\mathrm{7 \times 30 = 210}\) cents
• Total: \(\mathrm{50 + 210 = 260}\) cents
Savings from combined mailing = \(\mathrm{280 - 260 = 20}\) cents
Notice that \(\mathrm{20 = 50 - 30 = x - y}\)
Conclusion: Combined mailing is cheaper, with savings of \(\mathrm{x - y}\) cents.
Answer: (A)