Which of the following is closest to \(100((1+0.03)^4-1)\)?

1. Translate the problem requirements: We need to calculate 100 times the result of \((1.03)^4\) minus 1. This represents finding what happens when something grows by 3% for 4 periods, then expressing the total growth as a percentage.
2. Calculate the compound growth factor: Compute \((1.03)^4\) by multiplying 1.03 by itself four times to find the total growth factor after 4 periods.
3. Find the net growth amount: Subtract 1 from the growth factor to isolate just the growth portion (removing the original principal).
4. Convert to percentage form: Multiply by 100 to express the decimal growth as a percentage, matching the format requested in the problem.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this expression is asking for in everyday terms. We have \(100((1+0.03)^4-1)\), which looks intimidating but is actually describing a common real-world situation.

Think of it this way: imagine you invest $1 and it grows by 3% each year for 4 years. The \((1+0.03)\) part represents your money after one year - you have your original dollar plus 3% growth. The exponent 4 means this happens four times in a row. The minus 1 removes your original investment, leaving just the profit. Finally, multiplying by 100 converts this profit from decimal form to percentage form.

So we're finding: What percentage profit do you make when something grows by 3% per year for 4 years?

Process Skill: TRANSLATE - Converting the mathematical expression into a concrete scenario

2. Calculate the compound growth factor

Now let's calculate \((1.03)^4\) step by step. We need to multiply 1.03 by itself four times.

Starting amount after 0 years: 1.00
After 1 year: \(1.00 \times 1.03 = 1.03\)
After 2 years: \(1.03 \times 1.03 = 1.0609\)
After 3 years: \(1.0609 \times 1.03 = 1.092727\)
After 4 years: \(1.092727 \times 1.03 = 1.12550881\)

So \((1.03)^4 \approx 1.1255\)

This means that after 4 years of 3% annual growth, you'd have about 1.1255 times your original amount.

3. Find the net growth amount

Now we subtract 1 to find just the growth portion, removing the original principal.

Net growth = \(1.1255 - 1 = 0.1255\)

This means the pure profit or growth is about 0.1255 times the original amount. In other words, you gained about 12.55% of what you started with.

4. Convert to percentage form

Finally, we multiply by 100 to express this as a percentage:

\(100 \times 0.1255 = 12.55\)

So \(100((1+0.03)^4-1) \approx 12.55\)

Looking at our answer choices, 12.600 is the closest value to our calculated result of 12.55.

Final Answer

The answer is C: 12.600.

This makes intuitive sense - when something grows by 3% per year for 4 years, the total growth is slightly more than \(4 \times 3\% = 12\%\) due to compounding effects. Our calculated value of approximately 12.6% confirms this reasoning.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the expression structure

Students might focus only on the exponent and think this is simply asking for \(3\% \times 4 = 12\%\), missing that this is a compound growth problem. They fail to recognize that the \((1+0.03)^4\) structure represents compound interest where each year's growth builds on the previous year's total.

2. Confusion about what the formula represents

Students may not understand that \(100((1+r)^n-1)\) is the standard compound interest formula for finding percentage growth. They might attempt to use simple interest calculations or other inappropriate formulas instead of recognizing this as a compound growth scenario.

Errors while executing the approach

1. Computational errors in calculating \((1.03)^4\)

Students often make arithmetic mistakes when multiplying 1.03 by itself four times, especially in the later steps where decimals become longer. Common errors include rounding too early or making multiplication errors like calculating \(1.03^2\) as 1.06 instead of 1.0609.

2. Forgetting to subtract 1 before multiplying by 100

Students may calculate \((1.03)^4\) correctly to get approximately 1.1255, then multiply directly by 100 to get 112.55, forgetting that the formula requires subtracting 1 first to isolate just the growth portion.

3. Order of operations confusion

Students might incorrectly apply the operations, such as calculating \(100(1.03)^4 - 100\) instead of \(100((1.03)^4 - 1)\), leading to a completely different result.

Errors while selecting the answer

1. Selecting an answer that differs significantly in magnitude

After getting a result around 12.55, students might select 112.600 (choice E) if they made the error of not subtracting 1 before multiplying by 100, or they might select 0.126 (choice B) if they forgot to multiply by 100 at the end.