Can someone please explain how they got the answer for this problem its in my book:
Directions: Find the amount that should be invested now to accumulate the following amounts, if the money is coupounded as indicated.
$7500 at 12% compounded quarterly for 9 yrs.
answer: $2587.74
X * 1.03^36 = $7500.
X * 2.89828 = 7500
X = ?
? I have no idea
To find the amount that should be invested now to accumulate $7500 at 12% compounded quarterly for 9 years, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($7500 in this case)
P = the principal amount (amount to be invested now)
r = the annual interest rate (12% or 0.12)
n = the number of times interest is compounded per year (quarterly means it is compounded 4 times a year)
t = the number of years (9 years in this case)
Using this formula, we can rewrite the equation as:
$7500 = P(1 + 0.12/4)^(4*9)
Simplifying it further:
$7500 = P(1 + 0.03)^36
To solve for P, we need to isolate it. Divide both sides of the equation by (1 + 0.03)^36:
P = $7500 / (1 + 0.03)^36
Evaluating this expression:
P = $7500 / 2.89828
P ≈ $2587.74
Therefore, the amount that should be invested now to accumulate $7500 at 12% compounded quarterly for 9 years is approximately $2587.74.