A couple plans to save for their child's college education. What principal must be deposited by the parents when their child is born in order to have $41,000 when the child reaches the age of 18? Assume the money earns 9% interest, compounded quarterly. (Round your answer to two decimal places.)
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To find the principal that must be deposited by the parents when their child is born, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value (desired amount), which is $41,000
P = the principal (initial deposit)
r = the interest rate, which is 9% or 0.09
n = the number of times interest is compounded per year, which is 4 in this case (quarterly)
t = the number of years, which is 18
Substituting the given values into the formula:
$41,000 = P(1 + 0.09/4)^(4*18)
Simplifying:
$41,000 = P(1 + 0.0225)^(72)
Now, divide both sides of the equation by (1 + 0.0225)^(72):
$41,000 / (1 + 0.0225)^(72) = P
P ≈ $5,997.35
Therefore, the principal that must be deposited by the parents when their child is born to have $41,000 when the child reaches the age of 18 is approximately $5,997.35.
To determine the principal amount that needs to be deposited by the parents when their child is born, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = future value (in this case, $41,000)
P = principal amount (unknown)
r = annual interest rate (9%, or 0.09 as a decimal)
n = number of times compounding occurs per year (quarterly, so n = 4)
t = number of years (18 in this case)
Now let's substitute the known values into the formula and solve for P:
$41,000 = P(1 + 0.09/4)^(4*18)
To solve this, we can divide both sides of the equation by (1 + 0.09/4)^(4*18):
$41,000 / (1 + 0.09/4)^(4*18) = P
Using this equation, we can calculate the value of P to determine the principal amount that needs to be deposited by the parents.