For a shipment of n bottles of a certain product, a mail order firm charges a price of $40.00 per...

1. Translate the problem requirements: We need to find the smallest number of bottles where the total cost per bottle becomes less than \(\$42.50\). The total cost includes: bottle price (\(\$40\) each) + shipping fee (\(5\%\) of total bottle cost) + handling fee (\(\$5\) per shipment).
2. Set up the cost per bottle expression: Express the total cost for n bottles, then divide by n to get cost per bottle.
3. Create and solve the inequality: Set up an inequality where cost per bottle < \(\$42.50\) and solve for n.
4. Verify with answer choices: Test the calculated value against the given options to confirm the least value that satisfies our condition.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're dealing with in plain English first. A customer orders n bottles, and the mail order firm charges them in three parts:

• Each bottle costs \(\$40.00\) (this is straightforward)
• There's a shipping fee of \(5\%\) calculated on the total cost of all bottles
• There's a one-time handling fee of \(\$5.00\) per shipment (regardless of how many bottles)

We want to find the smallest number of bottles where the average cost per bottle drops below \(\$42.50\). Notice that as we order more bottles, the \(\$5\) handling fee gets spread across more bottles, making each bottle effectively cheaper.

Process Skill: TRANSLATE - Converting the word problem into mathematical understanding of the cost structure

2. Set up the cost per bottle expression

Let's think about this step by step. For n bottles:

• Cost of bottles = \(\mathrm{n} \times \$40.00 = \$40\mathrm{n}\)
• Shipping fee = \(5\%\) of bottle cost = \(0.05 \times \$40\mathrm{n} = \$2\mathrm{n}\)
• Handling fee = \(\$5.00\) (fixed amount)

So the total cost for n bottles = \(\$40\mathrm{n} + \$2\mathrm{n} + \$5 = \$42\mathrm{n} + \$5\)

To get the cost per bottle, we divide the total cost by the number of bottles:

Cost per bottle = \(\frac{42\mathrm{n} + 5}{\mathrm{n}} = 42 + \frac{5}{\mathrm{n}}\)

This makes intuitive sense! Each bottle effectively costs \(\$42\) (the \(\$40\) bottle price plus \(\$2\) shipping), plus we have to split the \(\$5\) handling fee among all bottles.

3. Create and solve the inequality

We want the cost per bottle to be less than \(\$42.50\):

\(42 + \frac{5}{\mathrm{n}} < 42.50\)

Subtracting 42 from both sides:

\(\frac{5}{\mathrm{n}} < 0.50\)

To solve this, we can multiply both sides by n (since n is positive):

\(5 < 0.50\mathrm{n}\)

Dividing both sides by 0.50:

\(\mathrm{n} > \frac{5}{0.50} = 10\)

So we need n > 10, which means the smallest whole number value is n = 11.

Process Skill: MANIPULATE - Carefully handling the inequality to isolate n

4. Verify with answer choices

Let's check our answer by testing n = 11:

Cost per bottle = \(42 + \frac{5}{11} = 42 + 0.4545... = \$42.45\)

Since \(\$42.45 < \$42.50\), our answer works!

Let's also verify that n = 10 doesn't work:

Cost per bottle = \(42 + \frac{5}{10} = 42 + 0.50 = \$42.50\)

This equals \(\$42.50\), but we need it to be less than \(\$42.50\), so n = 10 is not sufficient.

Therefore, the least value of n is 11, which corresponds to answer choice D.

Final Answer

The answer is D) 11. With 11 bottles, the cost per bottle becomes \(\$42.45\), which is the first whole number of bottles that brings the cost per bottle below the \(\$42.50\) threshold.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the shipping fee calculation base: Students might incorrectly calculate the \(5\%\) shipping fee on the total cost (including shipping and handling) rather than just on the bottle cost (\(\$40\mathrm{n}\)). This leads to a more complex equation and incorrect setup.

2. Confusion about what "charge per bottle" means: Some students might think they need to find the cost per bottle for just the bottles themselves (\(\$40\)) rather than understanding that they need the total average cost per bottle including all fees distributed across n bottles.

3. Setting up the wrong inequality direction: Students might set up the inequality as "greater than" instead of "less than" \(\$42.50\), especially if they misread the question or get confused about what threshold they're trying to achieve.

Errors while executing the approach

1. Arithmetic errors when solving \(\frac{5}{\mathrm{n}} < 0.50\): Students often make mistakes when manipulating fractions, particularly when dividing 5 by 0.50, incorrectly getting n > 5 instead of n > 10, or making sign errors when moving terms across the inequality.

2. Incorrectly handling the inequality when multiplying by n: Some students forget that since n is positive, the inequality direction stays the same when multiplying both sides by n, or they might incorrectly flip the inequality sign.

3. Calculation errors in the cost per bottle formula: Students might make errors when simplifying \(\frac{42\mathrm{n} + 5}{\mathrm{n}}\), either incorrectly distributing the division or making algebraic mistakes that lead to wrong expressions like \(42\mathrm{n} + \frac{5}{\mathrm{n}}\) instead of \(42 + \frac{5}{\mathrm{n}}\).

Errors while selecting the answer

1. Selecting n = 10 instead of n = 11: After correctly solving n > 10, students might choose n = 10 forgetting that they need the smallest integer that satisfies the strict inequality (greater than, not greater than or equal to).

2. Verification errors leading to wrong choice: Students might make arithmetic mistakes when checking their answer choices, incorrectly calculating the cost per bottle for n = 11 and thinking it doesn't work, then selecting a different option.