A certain manufacturer sells its products to stores in 113 different regions worldwide, with an average (arithmetic mean) of 181...

1. Translate the problem requirements: We need to find the total units sold worldwide, which equals \((\mathrm{number\ of\ regions}) \times (\mathrm{average\ stores\ per\ region}) \times (\mathrm{average\ units\ sold\ per\ store})\). The answer choices are orders of magnitude apart, so we can use approximation.
2. Set up the multiplication structure: Identify the three values we need to multiply: 113 regions, 181 stores per region, and 51,752 units per store.
3. Apply strategic approximation: Round each number to make calculations manageable while maintaining accuracy sufficient to distinguish between answer choices that differ by factors of 10.
4. Execute the calculation and match to order of magnitude: Multiply the approximated values and compare the result to the given powers of 10 to identify the closest match.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break this down in plain English first. We have a manufacturer that sells products through stores around the world. Think of it like this: imagine you're trying to count all the apples sold by a grocery chain that has stores in different cities.

Here's what we know:
• The manufacturer sells to stores in 113 different regions
• Each region has an average of 181 stores
• Each store sold an average of 51,752 units last year

To find the total units sold worldwide, we need to multiply all these together:
Total units = \((\mathrm{Number\ of\ regions}) \times (\mathrm{Stores\ per\ region}) \times (\mathrm{Units\ per\ store})\)

The key insight here is that the answer choices are powers of 10 (like \(10^6\), \(10^7\), etc.), which means they differ by factors of 10. This tells us we don't need to be perfectly precise - we can use approximation to get close enough to pick the right answer.

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

2. Set up the multiplication structure

Now let's identify our three numbers clearly:
• Regions: 113
• Stores per region: 181
• Units per store: 51,752

So we need to calculate: \(113 \times 181 \times 51,752\)

Looking at these numbers, doing this multiplication exactly would be quite tedious. But since our answer choices are so far apart (each one is 10 times bigger than the previous), we can round these numbers to make our life easier.

3. Apply strategic approximation

Let's round each number to something manageable while keeping enough accuracy:

• \(113 \approx 100\) (this is slightly smaller, so our answer will be slightly underestimated)
• \(181 \approx 200\) (this is slightly larger, helping to balance out the previous underestimate)
• \(51,752 \approx 50,000\) (this is slightly smaller)

Now our calculation becomes much simpler:
\(100 \times 200 \times 50,000\)

Let's think about this step by step:
• \(100 \times 200 = 20,000\)
• \(20,000 \times 50,000 = ?\)

To multiply \(20,000 \times 50,000\), let's think of it as:
\(20,000 \times 50,000 = (2 \times 10^4) \times (5 \times 10^4) = 2 \times 5 \times 10^4 \times 10^4 = 10 \times 10^8 = 10^9\)

Process Skill: SIMPLIFY - Using strategic rounding to make calculations manageable

4. Execute the calculation and match to order of magnitude

Our approximation gave us \(10^9\), which matches answer choice D exactly.

Let's do a quick sanity check: our approximations were pretty close to the original numbers, and some of our rounding went up while some went down, so they partially cancel out. This gives us confidence that \(10^9\) is the right order of magnitude.

To double-check our thinking: if we wanted a more precise estimate, we could note that:
• 113 is about 13% larger than 100
• 181 is about 10% smaller than 200
• 51,752 is about 3% larger than 50,000

These small adjustments would still keep us very close to \(10^9\), and definitely wouldn't move us to \(10^8\) or \(10^{10}\).

Final Answer

The total number of units sold worldwide last year is closest to \(10^9\).

Answer: D) \(10^9\)

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the multiplication structure: Students may get confused about what needs to be multiplied together. They might think they need to add the numbers instead of multiply, or they might miss one of the three components (regions, stores per region, or units per store). The key insight is recognizing this is a three-step multiplication: regions × stores per region × units per store.

2. Overthinking the precision requirement: Students may not recognize that the answer choices are powers of 10 that differ by factors of 10. This means they might attempt exact calculations instead of using strategic approximation, making the problem unnecessarily difficult and time-consuming.

Errors while executing the approach

1. Poor approximation choices: Students might round numbers too aggressively (like rounding 51,752 to 100,000) or not aggressively enough, leading to calculations that are either too far from the actual answer or still too complex to handle efficiently.

2. Arithmetic errors in powers of 10: When calculating \(20,000 \times 50,000\), students may make mistakes with the exponents. For example, they might incorrectly calculate \((2 \times 10^4) \times (5 \times 10^4)\) as \(10 \times 10^6\) instead of \(10 \times 10^8\), leading to \(10^7\) instead of \(10^9\).

3. Calculation sequence errors: Students might multiply the numbers in a different order or make basic arithmetic mistakes when calculating \(100 \times 200 = 20,000\), which would throw off their entire final calculation.

Errors while selecting the answer

1. Misreading the power notation: Students might confuse the answer choices and select \(10^8\) (choice C) instead of \(10^9\) (choice D), especially if they made a small error in their approximation or calculation that brought them close to the boundary between these two orders of magnitude.