1.03 per period
16 periods
30,000 = x [(1.03)^16 -1] /0.03
30,000 = 20.157 x
x = 1488
16 periods
30,000 = x [(1.03)^16 -1] /0.03
30,000 = 20.157 x
x = 1488
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!
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.
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.