When Greg was born, his grandmother put $5000 in a savings account for his college expenses. It was compounded at 2.5 percent semiannually, and left in the account 18 years. How much was in the account? Round to the nearest dollar.
Use the formula:
A=P(1+i)^n
A=?
P=$5000
i=2.5/2
=1.25/100
=0.0125
n=2*18=36
now we substitute:
A=5000(1+0.0125)^36
=5000(1.0125)^36
=$7819.72
To calculate the final amount in the account after 18 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount in the account
P = the principal amount (initial deposit)
r = annual interest rate (2.5% or 0.025)
n = number of times the interest is compounded per year (2 times per year, so n = 2)
t = number of years (18 years)
Plugging in the values, we have:
A = 5000(1 + 0.025/2)^(2*18)
Simplifying inside the parentheses:
A = 5000(1.0125)^(36)
Evaluating the exponent:
A ≈ 5000(1.472202)
Calculating the final amount:
A ≈ $7361.01
Therefore, after 18 years, there would be approximately $7,361 in the savings account.
To calculate the final amount in the savings account, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, Greg's grandmother initially deposited $5000 into the savings account (P). The interest is compounded semiannually, so n = 2. The annual interest rate is 2.5%, which can be written as 0.025 in decimal form (r). Greg left the money in the account for 18 years (t).
Now, let's plug in these values into the formula and calculate the final amount:
A = 5000(1 + 0.025/2)^(2*18)
First, calculate the value inside the parentheses:
1 + 0.025/2 = 1.0125
Next, raise this value to the power of (2*18) to account for the compounding periods over 18 years:
(1.0125)^(2*18) ≈ 1.4557
Finally, multiply the principal amount by the value obtained above:
A = 5000 * 1.4557 ≈ $7,278.50
Therefore, the approximate amount in the savings account after 18 years is $7,278.