Question

How much money should be deposited today in an account that eams 4% compounded semiannually so that it will accumulate to $14,000 in three years?

Answer 1

To accumulate $14,000 in three years in an account that earns 4% compounded semiannually, approximately $12,425.17 should be deposited today.

Step-by-step explanation:

To find the amount of money that should be deposited today in an account that earns 4% compounded semiannually and accumulates to $14,000 in three years, we can use the formula for compound interest:

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

Where:

• A is the final amount ($14,000)
• P is the principal amount we want to find
• r is the annual interest rate (4%)
• n is the number of times interest is compounded per year (2 in this case since it's compounded semiannually)
• t is the number of years (3)

Plugging in the values, we get:

$14,000 = P(1 + 0.04/2)^(2*3)

Simplifying the equation:

Dividing both sides of the equation by (1 + 0.04/2)^(2*3), we get:

P = $14,000 / (1 + 0.04/2)^(2*3)

Calculating the expression on the right, we find:

P ≈ $12,425.17

Answer 2

To find the initial deposit needed to reach $14,000 in 3 years with a 4% interest rate compounded semiannually, we use the compound interest formula, resulting in an initial deposit of approximately $12,306.21.

Step-by-step explanation:

To determine how much money should be deposited today in an account that earns 4% compounded semiannually to accumulate to $14,000 in three years, we utilize the formula for compound interest:

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

Where:

• A = the future value of the investment/loan, including interest
• P = the principal investment amount (initial deposit)
• r = the annual interest rate (decimal)
• n = the number of times interest is compounded per year
• t = the number of years the money is invested/borrowed for

Given:

• A = $14,000
• r = 4% or 0.04 (as a decimal)
• n = 2 (because the interest is compounded semiannually)
• t = 3 years

We need to solve for P. Rearranging the formula gives us:

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

Now, we plug in the values:

P =
`$14,000 / (1 + 0.04/2)^(2*3)`

P =
`$14,000 /(1 + 0.02)^(6)`

P =
`$14,000 / (1.02)^(6)`

P is approximately $12,306.21, which is the amount that needs to be deposited today to reach the goal of $14,000 in three years with the given interest conditions.