Kevin invested $8,000 for one year at a simple annual interest rate of 6% and invested $10,000 for one year...

1. Translate the problem requirements: Kevin has two separate investments - $8,000 at 6% simple interest for one year, and $10,000 at 8% compounded semiannually for one year. We need to find the total interest earned from both investments combined.
2. Calculate simple interest on the first investment: Apply the straightforward simple interest concept where interest = principal × rate × time, since no compounding occurs.
3. Calculate compound interest on the second investment: Determine the final amount when $10,000 grows at 8% annual rate with semiannual compounding (meaning 4% every 6 months for two periods), then subtract the principal to find interest earned.
4. Sum the interest amounts: Add the interest from both investments to get the total interest Kevin earned.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what Kevin did with his money:

• First investment: Kevin put $8,000 into an account that pays 6% simple interest for one year
• Second investment: Kevin put $10,000 into an account that pays 8% interest compounded semiannually for one year

We need to find the total interest he earned from both investments combined.

The key difference here is that the first investment uses simple interest (interest is only calculated on the original amount), while the second uses compound interest (interest earns interest).

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding of two separate interest calculations

2. Calculate simple interest on the first investment

For simple interest, we use the straightforward concept: the interest you earn is just a percentage of your original investment, nothing more.

Kevin invested $8,000 at 6% simple interest for 1 year.

Think of it this way: 6% of $8,000 means Kevin earns 6 cents for every dollar he invested.

6% of $8,000 = \(0.06 \times \$8{,}000 = \$480\)

So Kevin earned exactly $480 in interest from his first investment.

Technical summary: Interest = Principal × Rate × Time = $8,000 × 0.06 × 1 = $480

3. Calculate compound interest on the second investment

For the second investment, Kevin invested $10,000 at 8% annual interest compounded semiannually.

"Compounded semiannually" means the interest is calculated and added to the account twice per year - every 6 months.

Since the annual rate is 8%, each 6-month period earns 4% interest (8% ÷ 2 = 4%).

Let's trace what happens:

After first 6 months:
• Starting amount: $10,000
• Interest earned: 4% of $10,000 = 0.04 × $10,000 = $400
• New balance: $10,000 + $400 = $10,400

After second 6 months (end of year):
• Starting amount: $10,400 (now the interest earns interest too!)
• Interest earned: 4% of $10,400 = 0.04 × $10,400 = $416
• Final balance: $10,400 + $416 = $10,816

Total interest from second investment = Final amount - Original investment
Total interest = $10,816 - $10,000 = $816

Technical summary: \(\mathrm{A} = \mathrm{P}(1 + \mathrm{r}/\mathrm{n})^{(\mathrm{n} \times \mathrm{t})} = \$10{,}000(1 + 0.08/2)^{(2 \times 1)} = \$10{,}000(1.04)^2 = \$10{,}816\)

4. Sum the interest amounts

Now we add the interest from both investments:

• Interest from first investment (simple): $480
• Interest from second investment (compound): $816

Total interest earned = $480 + $816 = $1,296

Final Answer: E. $1,296

We can verify this matches choice E in the given options.

Common Faltering Points

Errors while devising the approach

1. Confusing Simple vs. Compound Interest: Students often fail to recognize that the problem involves TWO different types of interest calculations. They might attempt to apply the same formula to both investments, either treating both as simple interest or both as compound interest, missing the key distinction stated in the problem.

2. Misinterpreting "Compounded Semiannually": Students may not understand what "compounded semiannually" means and might incorrectly assume it's annual compounding. This leads to using the wrong time periods (using 1 year instead of 2 six-month periods) and wrong interest rates per period (using 8% instead of 4% per period).

3. Misunderstanding the Question Requirement: Students might calculate the final amounts of both investments instead of focusing on just the INTEREST earned. The question specifically asks for "total amount of interest," not the final balance of the accounts.

Errors while executing the approach

1. Arithmetic Errors in Compound Interest Calculation: When calculating (1.04)², students often make computational mistakes, getting 1.08 instead of 1.0816, which leads to incorrect final amounts and interest calculations.

2. Incorrect Percentage Calculations: Students frequently make errors when calculating percentages, such as computing 6% of $8,000 as $48 instead of $480, or 4% of $10,400 as $41.60 instead of $416.

3. Using Wrong Principal for Second Compounding Period: In the compound interest calculation, students often use the original principal ($10,000) for both periods instead of using the updated amount ($10,400) for the second six-month period, missing the core concept of compounding.

Errors while selecting the answer

1. Selecting Individual Interest Instead of Total: Students might correctly calculate both interest amounts but then select an answer choice that matches only one of the individual interest calculations ($480 or $816) rather than the sum ($1,296).

2. Calculation Verification Errors: Students may arrive at the correct individual interest amounts but make addition errors when combining $480 + $816, potentially selecting a nearby but incorrect answer choice due to rushed final calculations.