Directions: Select the correct alternative for the question given below.

A business allocates a principal sum into a commercial investment account that accrues interest compounded annually. The interest earned during the first year totals $3,000, and the interest accrued during the second year totals $3,450. Determine 40% of the original principal sum invested.

The correct option is $8,000.

Step 1: Define variables and set up the first-year interest equation.

Let $P$ be the original principal sum invested, and $r$ be the annual compound interest rate (in decimal form).

The interest earned during the first year ($I_1$) is given as $3,000. Since interest for the first year is calculated solely on the initial principal $P$, we have:

$I_1=P\times r=3000$

Step 2: Determine the interest rate from the second-year interest.

In compound interest, the principal for the second year is the original principal plus the interest accrued in the first year. Therefore, the total balance at the start of the second year is $P+3000$.

The interest accrued during the second year ($I_2$) is given as $3,450:

$I_2=(P+3000)\times r=3450$

Expanding this equation gives:

$P\times r+3000\times r=3450$

Substitute the value of $P\times r=3000$ from Step 1 into the equation:

$3000+3000\times r=3450$

Subtract 3,000 from both sides:

$3000\times r=450$

Solve for $r$:

$r=\frac{450}{3000}=0.15$

Thus, the annual interest rate is 15%.

Step 3: Calculate the original principal sum ($P$).

Substitute $r=0.15$ back into the equation from Step 1:

$P\times 0.15=3000$

$P=\frac{3000}{0.15}=20000$

The original principal sum invested was $20,000.

Step 4: Find 40% of the original principal sum.

Calculate 40% of $20,000:

$40\%\text{of}P=0.40\times 20000=8000$

Therefore, 40% of the original principal sum invested is $8,000.