From January 1, 1991, to January 1, 1993, the number of people enrolled in health maintenance organizations increased by 15...

1. Translate the problem requirements: We need to find the enrollment in 1991, knowing that it increased by \(15\%\) over two years to reach 45 million in 1993. This is a reverse percentage calculation.
2. Set up the relationship between initial and final values: Express the 1993 enrollment in terms of the unknown 1991 enrollment using the percentage increase.
3. Solve for the 1991 enrollment: Use algebraic manipulation to find the initial value.
4. Round to the nearest million: Apply the rounding requirement to match the answer format.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know and what we need to find in everyday terms.

We know that:

• On January 1, 1993, there were 45 million people enrolled
• From 1991 to 1993, enrollment increased by \(15\%\)
• We need to find how many people were enrolled on January 1, 1991

Think of it this way: imagine you had some amount of money in 1991, and by 1993 it grew by \(15\%\) to become $45. What was your original amount?

This is a "working backwards" problem. We're going from the final result (45 million in 1993) back to the starting point (unknown enrollment in 1991).

Process Skill: TRANSLATE - Converting the percentage increase language into a clear mathematical relationship

2. Set up the relationship between initial and final values

Let's call the 1991 enrollment "x million people" - this is what we're looking for.

When something increases by \(15\%\), the new amount becomes the original amount plus \(15\%\) of the original amount.

In plain English:

• Original amount = \(\mathrm{x}\)
• Increase = \(15\%\) of \(\mathrm{x}\) = \(0.15\mathrm{x}\)
• Final amount = Original + Increase = \(\mathrm{x} + 0.15\mathrm{x}\)

Since \(\mathrm{x} + 0.15\mathrm{x} = 1\mathrm{x} + 0.15\mathrm{x} = 1.15\mathrm{x}\), we can say:
Final amount = \(1.15 \times \mathrm{Original\,amount}\)

So: \(45\,\mathrm{million} = 1.15 \times (1991\,\mathrm{enrollment})\)
Or: \(45 = 1.15\mathrm{x}\)

3. Solve for the 1991 enrollment

To find x, we need to "undo" the multiplication by 1.15. We do this by dividing both sides by 1.15:

\(\mathrm{x} = 45 \div 1.15\)

Let's calculate this step by step:
\(45 \div 1.15 = 45 \div \frac{115}{100} = 45 \times \frac{100}{115}\)

To make this easier, let's simplify:
\(45 \times \frac{100}{115} = \frac{4500}{115}\)

Now \(4500 \div 115\):
\(115 \times 39 = 4485\)
\(115 \times 40 = 4600\)

Since 4500 is between 4485 and 4600, and closer to 4485, we get approximately 39.1 million.

4. Round to the nearest million

Our calculation gave us approximately 39.1 million people.

Since we need to round to the nearest million:
39.1 million rounds to 39 million.

Let's verify: If 39 million people were enrolled in 1991, then after a \(15\%\) increase:
\(39 \times 1.15 = 39 \times 1.15 = 44.85\,\mathrm{million}\)

This is very close to 45 million (the small difference is due to rounding), confirming our answer.

Final Answer

The number of people enrolled in health maintenance organizations on January 1, 1991, was 39 million.

The answer is (B) 39.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the direction of percentage change

Students may confuse which value is the starting point and which is the ending point. They might think that 45 million was the 1991 enrollment that increased to some unknown 1993 value, rather than understanding that 45 million is the final result after the increase.

2. Setting up the wrong equation for percentage increase

Students often write the equation as "\(\mathrm{x} + 15 = 45\)" instead of "\(\mathrm{x} + 0.15\mathrm{x} = 45\)" or "\(1.15\mathrm{x} = 45\)". They treat the \(15\%\) as a simple addition of 15 rather than \(15\%\) of the original amount.

3. Confusing percentage increase with percentage of total

Some students might think that if enrollment increased by \(15\%\), then the 1991 enrollment was \(15\%\) of 45 million, leading them to calculate \(0.15 \times 45\) instead of recognizing that 45 million represents \(115\%\) of the original amount.

Errors while executing the approach

1. Arithmetic errors when dividing by 1.15

Students may struggle with the division \(45 \div 1.15\), especially when converting to fractions \((45 \div \frac{115}{100})\) or when performing long division, leading to incorrect intermediate calculations.

2. Incorrect conversion between decimal and percentage

When setting up \(1.15\mathrm{x} = 45\), students might use 1.015 instead of 1.15, confusing the decimal representation of \(15\%\) (which is 0.15) with the multiplier for a \(15\%\) increase (which is 1.15).

3. Rounding errors during intermediate steps

Students may round too early in their calculations or use imprecise approximations when dividing, leading to a final answer that's off by 1-2 million from the correct value.

Errors while selecting the answer

1. Selecting the unrounded decimal value

If students calculate 39.1 million but don't notice that the question asks to round "to the nearest million," they might look for 39.1 in the answer choices or become confused when they don't see their exact calculated value.

2. Rounding in the wrong direction

Students might incorrectly round 39.1 to 40 instead of 39, either due to careless errors or confusion about rounding rules, especially since 39.1 is relatively close to the boundary between answer choices.