During a certain year, 80 percent of the visitors to the Smithsonian Institution were United States citizens, 50 percent of...

1. Translate the problem requirements: Break down the visitor composition - 80% are US citizens, then among the remaining 20%, half are Canadian/Mexican, and among those who aren't US/Canadian/Mexican, half are European. We need to find the ratio of US citizens to European citizens.
2. Set up a concrete total to work with: Choose a convenient total number of visitors that makes percentage calculations straightforward.
3. Calculate each visitor group systematically: Work through the percentages step by step to find the number of US citizens and European citizens.
4. Determine the ratio: Compare the number of US citizens to European citizens to find how many times larger the US group is.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening with the visitors to the Smithsonian step by step:

• 80% of all visitors are US citizens
• This leaves 20% of visitors who are NOT US citizens
• Among these non-US visitors, 50% are Canadian or Mexican citizens
• Among the visitors who are NOT US, Canadian, OR Mexican, 50% are European citizens

We need to find: How many times larger is the US citizen group compared to the European citizen group?

Process Skill: TRANSLATE - Converting the nested percentage language into clear mathematical relationships

2. Set up a concrete total to work with

To make our calculations easy, let's assume there were 100 total visitors to the Smithsonian that year.

Using 100 as our total makes percentage calculations straightforward:

• We can directly convert percentages to actual numbers
• We avoid complex fractions in our intermediate steps
• The final ratio will be the same regardless of the total we choose

3. Calculate each visitor group systematically

Now let's work through each group of visitors:

US Citizens:
\(80\% \text{ of } 100 \text{ total visitors} = 80 \text{ US citizens}\)

Non-US Citizens:
\(100 - 80 = 20 \text{ non-US citizens}\)

Canadian/Mexican Citizens:
\(50\% \text{ of the } 20 \text{ non-US citizens} = 50\% \times 20 = 10 \text{ Canadian/Mexican citizens}\)

Citizens who are NOT US, Canadian, or Mexican:
\(20 \text{ non-US citizens} - 10 \text{ Canadian/Mexican citizens} = 10 \text{ citizens from other countries}\)

European Citizens:
\(50\% \text{ of the } 10 \text{ "other country" citizens} = 50\% \times 10 = 5 \text{ European citizens}\)

Process Skill: SIMPLIFY - Working with convenient numbers and tracking each subset clearly

4. Determine the ratio

Now we can find how many times larger the US citizen group is compared to the European citizen group:

US citizens: 80
European citizens: 5

Ratio = \(\text{US citizens} \div \text{European citizens} = 80 \div 5 = 16\)

This means the number of US citizens was 16 times the number of European citizens.

Final Answer

The number of United States citizens who visited the Smithsonian was approximately 16 times the number of citizens of European countries who visited.

The answer is D. 16

Common Faltering Points

Errors while devising the approach

1. Misinterpreting nested percentage calculations
Students often struggle with the phrase "50 percent of the remaining visitors" and "50 percent of the visitors who were not citizens of the United States, Canada, or Mexico." They may incorrectly apply these percentages to the total visitor count rather than understanding that each percentage applies to a progressively smaller subset. For example, they might calculate Canadian/Mexican citizens as 50% of all 100 visitors instead of 50% of the 20 non-US visitors.

2. Confusion about what the question is asking for
The question asks "how many times" the US citizen count compares to the European citizen count, which requires finding a ratio. Some students may misinterpret this as asking for a difference (subtraction) or may try to find what percentage European citizens represent, rather than setting up the correct ratio of US citizens ÷ European citizens.

Errors while executing the approach

1. Applying percentages to wrong base numbers
Even when students understand the nested structure, they may make calculation errors by applying percentages to incorrect base numbers. For instance, when calculating European citizens, they might apply 50% to the 20 non-US citizens instead of applying it to the 10 citizens who are not from US, Canada, or Mexico.

2. Arithmetic errors in percentage calculations
Students may make simple computational mistakes, such as calculating 50% of 20 as 15 instead of 10, or incorrectly computing 50% of 10 as 6 instead of 5. These errors compound through the sequential calculations and lead to an incorrect final ratio.

Errors while selecting the answer

1. Inverting the ratio
After correctly calculating that there are 80 US citizens and 5 European citizens, students may incorrectly compute the ratio as European ÷ US \((5 \div 80 = 0.0625)\) instead of US ÷ European \((80 \div 5 = 16)\). This would lead them to select a much smaller answer choice or become confused when their decimal result doesn't match any of the given options.

Alternate Solutions

Smart Numbers Approach

This problem is perfect for the smart numbers approach because we're dealing with percentages and need to find a ratio. We can choose a convenient total number of visitors that makes our calculations straightforward.

Step 1: Choose a Smart Total

Let's use 1,000 total visitors. This number is chosen because:

• It makes percentage calculations easy \((80\% \text{ of } 1,000 = 800)\)
• It's large enough to handle multiple percentage breakdowns without fractions
• The successive 50% splits will work cleanly

Step 2: Calculate US Citizens

US citizens = 80% of total visitors
US citizens = \(0.80 \times 1,000 = 800 \text{ visitors}\)

Step 3: Calculate Non-US Visitors

Non-US visitors = 100% - 80% = 20% of total
Non-US visitors = \(0.20 \times 1,000 = 200 \text{ visitors}\)

Step 4: Calculate Canadian/Mexican Citizens

Canadian/Mexican = 50% of non-US visitors
Canadian/Mexican = \(0.50 \times 200 = 100 \text{ visitors}\)

Step 5: Calculate Remaining Non-US/Canadian/Mexican Visitors

Remaining visitors = Non-US visitors - Canadian/Mexican
Remaining visitors = \(200 - 100 = 100 \text{ visitors}\)

Step 6: Calculate European Citizens

European citizens = 50% of remaining visitors
European citizens = \(0.50 \times 100 = 50 \text{ visitors}\)

Step 7: Find the Ratio

Ratio = US citizens ÷ European citizens
Ratio = \(800 \div 50 = 16\)

Answer: D. 16

Key insight: The smart number 1,000 was strategically chosen because it makes all percentage calculations result in whole numbers, eliminating calculation errors and making the problem much more manageable.