Bernie invested a sum of money 8 yr ago in a savings account, which has since paid interest at the rate of 3%/year, compounded monthly. His investment is now worth $39,396.92. How much did he originally invest?
$ ?
P = Po(1+r)^n.
P = $39,396.92.
Po = ?.
r = (3%/12) / 100% = 0.0025 = Monthly %
rate expressed as a decimal.
n = 12Comp/yr * 8yrs = 96 Compounding
periods.
P = Po(1.0025)^96 = $39,396.92
Po*1.27087 = 39,396.92
Po = $31,000.00 = Initial deposit.
To find out how much Bernie initially invested, we can use the formula for the future value of an investment with monthly compounding:
\[ A = P \left(1 + \frac{r}{n}\right)^{n*t} \]
Where:
A = Future value of the investment ($39,396.92 in this case)
P = Principal amount (initial investment)
r = Annual interest rate (3% or 0.03)
n = Number of times interest is compounded per year (12 in this case since it is compounded monthly)
t = Number of years (8 in this case)
By plugging in the given values:
\[ 39,396.92 = P \left(1 + \frac{0.03}{12}\right)^{12*8} \]
To solve for P, we can rearrange the formula as:
\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{n*t}} \]
Substituting the values:
\[ P = \frac{39,396.92}{\left(1 + \frac{0.03}{12}\right)^{12*8}} \]
Calculating this expression gives us:
\[ P \approx \$27,000.00 \]
Therefore, Bernie initially invested approximately $27,000.00.
To find out how much Bernie originally invested, we need to solve for the principal amount of money.
We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (in this case, $39,396.92)
P = the principal amount (the amount Bernie originally invested)
r = the annual interest rate (3% or 0.03 as a decimal)
n = the number of times the interest is compounded per year (monthly compounding, so n = 12)
t = the number of years
Substituting the given values into the formula, we have:
39,396.92 = P(1 + 0.03/12)^(12*8)
Now we can solve for P by rearranging the equation. We'll divide both sides of the equation by [(1 + 0.03/12)^(12*8)]:
P = 39,396.92 / [(1 + 0.03/12)^(12*8)]
Now we can calculate the value of P using a calculator or a spreadsheet:
P ≈ $29,643.11
Therefore, Bernie originally invested approximately $29,643.11.