What amount, in dollars, invested for one year at a simple annual interest of 2% would earn the same amount...

1. Translate the problem requirements: We need to find what principal amount at 2% simple interest for one year would earn the same interest as $10,000 at 4% annual rate compounded semiannually for one year.
2. Calculate the benchmark interest earned: Find the actual interest earned by $10,000 under compound interest conditions to establish our target.
3. Set up the simple interest equation: Use the target interest amount to determine what principal at 2% simple interest would generate the same return.
4. Solve for the unknown principal: Apply simple interest formula to find the required investment amount.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to do in plain English.

We have two different investment scenarios:

Scenario 1: Some unknown amount of money (let's call it X dollars) invested at 2% simple interest for one year

Scenario 2: $10,000 invested at 4% annual interest compounded semiannually for one year

The problem wants us to find the value of X such that both scenarios earn the exact same amount of interest money.

Think of it like this: if your friend earned a certain amount from their investment, how much would you need to invest in a different type of account to earn the same interest amount?

Process Skill: TRANSLATE - Converting the problem statement into a clear mathematical comparison

2. Calculate the benchmark interest earned

Now we need to figure out exactly how much interest the $10,000 earns under compound interest conditions. This will be our target amount.

When we say "4% annual interest compounded semiannually," this means:

• The 4% annual rate gets split into two 6-month periods
• Each period has a 2% interest rate (4% ÷ 2 = 2%)
• The interest earned in the first period also earns interest in the second period

Let's calculate step by step:

After 6 months (first compounding period):
Interest earned = $10,000 × 2% = $10,000 × 0.02 = $200
New balance = $10,000 + $200 = $10,200

After 12 months (second compounding period):
Interest earned = $10,200 × 2% = $10,200 × 0.02 = $204
Final balance = $10,200 + $204 = $10,404

Total interest earned = $10,404 - $10,000 = $404

So our benchmark is $404 in interest earnings.

3. Set up the simple interest equation

Now we know that whatever amount we invest at 2% simple interest must earn exactly $404 in interest over one year.

For simple interest, the relationship is straightforward:
\(\mathrm{Interest\ Earned} = \mathrm{Principal} \times \mathrm{Interest\ Rate} \times \mathrm{Time}\)

In our case:

• Interest Earned = $404 (our target from step 2)
• Interest Rate = 2% = 0.02
• Time = 1 year
• Principal = Unknown (this is what we're solving for)

So our equation becomes:
\(\$404 = \mathrm{Principal} \times 0.02 \times 1\)
\(\$404 = \mathrm{Principal} \times 0.02\)

4. Solve for the unknown principal

We have the equation: \(\$404 = \mathrm{Principal} \times 0.02\)

To find the Principal, we divide both sides by 0.02:

\(\mathrm{Principal} = \$404 \div 0.02\)
\(\mathrm{Principal} = \$404 \div \frac{2}{100}\)
\(\mathrm{Principal} = \$404 \times \frac{100}{2}\)
\(\mathrm{Principal} = \$404 \times 50\)
\(\mathrm{Principal} = \$20,200\)

Let's verify this makes sense:
If we invest $20,200 at 2% simple interest for 1 year:
Interest = $20,200 × 0.02 × 1 = $404 ✓

This matches exactly with the interest earned by the $10,000 compound interest scenario.

4. Final Answer

The amount that needs to be invested is $20,200.

Looking at our answer choices, this corresponds to Answer Choice E: $20,200.

Common Faltering Points

Errors while devising the approach

1. Confusing simple interest vs. compound interest setup: Students may incorrectly assume both investments use the same type of interest calculation, missing that one scenario uses simple interest while the other uses compound interest. This leads to setting up identical formulas for both sides of the equation.

2. Misinterpreting "same amount of interest": Students might think they need to find when the final balances are equal, rather than when just the interest earned amounts are equal. This changes the entire problem setup from comparing interest earnings to comparing total investment values.

3. Misunderstanding "compounded semiannually": Students may think this means the interest rate becomes 4% every six months (totaling 8% annually), rather than understanding it means the 4% annual rate is split into 2% per six-month period.

Errors while executing the approach

1. Arithmetic errors in compound interest calculation: When calculating the second period's interest, students often forget to apply the 2% rate to the new balance ($10,200) and instead apply it again to the original principal ($10,000), getting $200 + $200 = $400 instead of the correct $200 + $204 = $404.

2. Division errors when solving for principal: When dividing $404 by 0.02, students may make calculation mistakes, especially if they don't convert to $404 ÷ (2/100) = $404 × 50. Common errors include getting $20,020 or $2,020 due to misplaced decimal points.

3. Using wrong time period: Students may forget that "semiannually" means two compounding periods in one year, and incorrectly calculate compound interest for only one period or use the wrong time frame in their calculations.

Errors while selecting the answer

1. Selecting a close but incorrect value: Given the answer choices are very close to each other ($20,000, $20,050, $20,100, $20,150, $20,200), students who made small arithmetic errors might select $20,150 (choice D) instead of the correct $20,200, thinking their calculation is "close enough."

2. Forgetting to verify the answer: Students may arrive at the correct calculation but fail to double-check by plugging $20,200 back into the simple interest formula to confirm it yields $404 in interest, potentially second-guessing themselves and choosing a different answer.