A certain electronic component is sold in boxes of 54 for $16.20 and in boxes of 27 for $13.20. A...

1. Translate the problem requirements: We need to find how much extra the customer paid per component when forced to buy 2 boxes of 27 components instead of 1 box of 54 components for the same total quantity.
2. Calculate cost per component for each box size: Determine the unit cost when buying the 54-component box versus the 27-component box.
3. Find the total cost difference for 54 components: Compare what the customer actually paid (2 boxes of 27) versus what they would have paid (1 box of 54).
4. Calculate the extra cost per component: Divide the total extra cost by the number of components (54) to find the per-component difference.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what happened here in everyday terms. The customer needed exactly 54 components for their project. Normally, they could buy one box of 54 components, but that size wasn't available. So they had to buy two boxes of 27 components instead (\(27 + 27 = 54\) components total).

We need to figure out how much extra they paid per component because of this inconvenience. Think of it like buying in bulk versus buying smaller packages - usually smaller packages cost more per unit.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical comparison

2. Calculate cost per component for each box size

Let's find the cost per component for each box size using simple division.

For the 54-component box:

- Cost: $16.20 for 54 components
- Cost per component = \(\$16.20 ÷ 54 = \$0.30\) per component

For the 27-component box:

- Cost: $13.20 for 27 components
- Cost per component = \(\$13.20 ÷ 27 = \$0.49\) per component (approximately)

Already we can see that the smaller boxes cost more per component, which makes sense.

3. Find the total cost difference for 54 components

Now let's compare what the customer actually paid versus what they should have paid for 54 components.

What the customer actually paid:

- 2 boxes of 27 components = \(2 × \$13.20 = \$26.40\)

What the customer would have paid with the larger box:

- 1 box of 54 components = $16.20

Total extra amount paid:

- \(\$26.40 - \$16.20 = \$10.20\) extra for the same 54 components

4. Calculate the extra cost per component

To find how much extra the customer paid per component, we divide the total extra cost by the number of components.

Extra cost per component = Total extra cost ÷ Number of components
Extra cost per component = \(\$10.20 ÷ 54\)
Extra cost per component = $0.189...

Rounding to the nearest cent, this is approximately $0.19 per component.

Final Answer

The customer paid approximately $0.19 more per component due to the unavailability of the larger boxes.

This matches answer choice (B) $0.19.

To verify: The customer paid $0.49 per component (small boxes) instead of $0.30 per component (large box), and \(\$0.49 - \$0.30 = \$0.19\), confirming our answer.

Common Faltering Points

Errors while devising the approach

Students often get confused about what exactly they need to find. The question asks for "how much more did the customer pay for each component" - some students might calculate the total extra amount paid ($10.20) and stop there, forgetting to divide by the number of components to get the per-component difference.

Some students might misread that the customer needed 54 components and had to buy 2 boxes of 27. They might think the customer bought extra components (more than 54) and try to factor that into their calculation, when in fact \(2 × 27 = 54\) components exactly.

Errors while executing the approach

Students frequently make arithmetic mistakes when dividing \(\$13.20 ÷ 27\) or \(\$16.20 ÷ 54\). For example, they might calculate \(\$13.20 ÷ 27\) as $0.40 instead of approximately $0.489, leading to incorrect final answers.

Some students round the cost per component for the 27-box to $0.49 early in their calculation, then use \(\$0.49 - \$0.30 = \$0.19\). While this gives the right answer by coincidence, it's better practice to work with the exact division result (\(\$10.20 ÷ 54\)) to avoid potential rounding errors.

Students might subtract in the wrong order, calculating what the customer would have saved rather than what extra they paid. For instance, calculating \(\$16.20 - \$26.40 = -\$10.20\) and getting confused by the negative result.

Errors while selecting the answer

After calculating that the customer paid $10.20 extra total, some students might look for an answer choice close to $10.20 rather than remembering they need the per-component extra cost of approximately $0.19.