Loading...
From January 1, 1991, to January 1, 1993, the number of people enrolled in health maintenance organizations increased by 15 percent. The enrollment on January 1, 1993, was 45 million. How many million people, to the nearest million, were enrolled in health maintenance organizations on January 1, 1991 ?
Let's start by understanding what we know and what we need to find in everyday terms.
We know that:
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
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:
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: \(45 = 1.15\mathrm{x}\)
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.
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.
The number of people enrolled in health maintenance organizations on January 1, 1991, was 39 million.
The answer is (B) 39.
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.
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.
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.
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.
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).
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.
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.
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.