Jane needs $30,000 to buy another car in eight years. How much should she deposit at the end of each half year into an account that earns 6% interest compounded semiannually?

1.03 per period

16 periods

30,000 = x [(1.03)^16 -1] /0.03
30,000 = 20.157 x
x = 1488

Well, Jane is clearly planning ahead! Now, let me grab my calculator and clown wig, and we can figure this out together.

To begin, we need to find out how many deposits Jane will make over the eight-year period. Since she is depositing at the end of each half year, that means she will make 16 deposits (since there are 2 half-years in a year, and 8 years in total).

Now, let's focus on the interest rate. We know that the account earns 6% interest, compounded semiannually. That means the interest is applied twice a year. So, we need to divide the interest rate by 2 to get 3%.

Next, we can plug in the numbers into the formula for the future value of an ordinary annuity:

FV = P * [(1 + r)^n - 1] / r

Where:
FV is the future value (in this case, $30,000)
P is the deposit amount
r is the interest rate per period (3% in this case)
n is the number of periods (16 deposits in total)

Now, we can solve for P.

So, $30,000 = P * [(1 + 0.03)^16 - 1] / 0.03

After crunching some numbers, the deposit amount (P) comes out to be approximately $868.69.

Therefore, Jane should deposit around $868.69 at the end of each half year to save up $30,000 for her new car in eight years. And don't worry, this answer is no laughing matter!

To calculate the amount Jane should deposit at the end of each half year, we will use the formula for compound interest:

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

Where:
A = Future value (the amount Jane needs, which is $30,000)
P = Principal amount (the amount to be deposited at the end of each half year)
r = Annual interest rate (6% or 0.06)
n = Number of compounding periods per year (semiannually, so 2)
t = Number of years (8)

Substituting the values into the formula, we get:

$30,000 = P(1 + 0.06/2)^(2*8)

Simplifying further:

$30,000 = P(1 + 0.03)^16

Now, let's solve for P:

P = $30,000 / (1.03)^16

Using a calculator, we find:

P ≈ $17,465.55

Therefore, Jane should deposit approximately $17,465.55 at the end of each half year into the account to accumulate $30,000 in 8 years.

To find out how much Jane should deposit at the end of each half year, we can use the formula for compound interest:

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

Where:
A is the future value of the investment (in this case, $30,000)
P is the principal (the amount to be deposited at the end of each half-year)
r is the annual interest rate (6% in this case)
n is the number of times the interest is compounded per year (semiannually, so n = 2)
t is the number of years (8 in this case)

Now, let's plug in the values and solve for P:

$30,000 = P(1 + 0.06/2)^(2*8)

$30,000 = P(1 + 0.03)^16

$30,000 = P(1.03)^16

Divide both sides of the equation by (1.03)^16:

P = $30,000 / (1.03)^16

Using a calculator, we can calculate (1.03)^16 ≈ 1.6010323:

P = $30,000 / 1.6010323

P ≈ $18,736.32

Therefore, Jane should deposit approximately $18,736.32 at the end of each half year into the account that earns 6% interest compounded semiannually in order to reach her goal of $30,000 in eight years.