Question

Smith wishes to accumulate exactly $15,000 in a fund that credits interest at a nominal rate of 6% compounded semiannually. In which range is the amount of the final deposit?

a) $10,000 - $15,000
b) $15,000 - $20,000
c) $20,000 - $25,000
d) $25,000 - $30,000

Answer 1

To accumulate $15,000 in a fund with a 6% interest rate compounded semiannually, the final deposit will be in the range of $10,000 - $15,000.

Step-by-step explanation:

To accumulate exactly $15,000 in a fund that credits interest at a nominal rate of 6% compounded semiannually, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

• A is the final amount
• P is the initial deposit
• r is the nominal interest rate (as a decimal)
• n is the number of times the interest is compounded per year
• t is the number of years.

In this case, we have:

• A = $15,000
• P is what we need to find
• r = 0.06
• n = 2 (since the interest is compounded semiannually)
• t = 1 (since we want to accumulate the amount in 1 year)

Plugging these values into the formula:

$15,000 = P(1 + 0.06/2)^(2*1)

Simplifying the equation:

$15,000 = P(1.03)^2

Next, divide both sides of the equation by (1.03)^2 to isolate P:

P = $15,000 / (1.03)^2

Calculating this gives us approximately:

P ≈ $14,563.11

Therefore, the amount of the final deposit is in the range a) $10,000 - $15,000.