You want to refurnish your home in 3 years. The interior decorator informs you that this will cost $25,000. What should be your payment into an account that gives 6% compounded quarterly?
25000(1+.06)^4*3
I think that is how you would set it up.
You may want to wait for a math tutor to verify.
the quarterly rate will be .06/4 = .015
let the payment you make now be $x
so
x(1.015)^12 = 25000
x = 25000/(1.015^12
= 20909.69
or you could use the formula
PV = Amt(1+i)^-n
present value = 25000(1.015)^-12
= 20909.69
To calculate the payment needed for an account that compounds interest quarterly, we can use the future value of an annuity formula. The formula is:
FV = P * ((1 + r/n)^(n*t) - 1) / (r/n)
Where:
FV = Future Value (the amount you want to accumulate)
P = Payment (the amount you need to deposit)
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
t = Number of years
In this case, the future value (FV) is $25,000, the annual interest rate (r) is 6% (or 0.06), the compounding periods per year (n) is 4 (since it's compounded quarterly), and the number of years (t) is 3.
By substituting these values into the formula, we can solve for the payment (P).
$25,000 = P * ((1 + 0.06/4)^(4*3) - 1) / (0.06/4)
To solve this equation, we can simplify it step by step.
First, calculate the interest rate per compounding period: 0.06/4 = 0.015 (or 1.5%).
Next, calculate the number of compounding periods: 4*3 = 12.
Then, calculate the value inside the brackets: (1 + 0.015)^12 - 1 = 0.19609 (or 19.61%).
Finally, rearrange the equation to solve for P:
P = $25,000 / 0.19609
P ≈ $127,415.80
Therefore, you should make a payment of approximately $127,415.80 into an account that gives a 6% interest rate compounded quarterly to accumulate $25,000 in 3 years.