Anthony invested a sum of money 6 yr ago in a savings account that has since paid interest at the rate of 7%/year compounded quarterly. His investment is now worth $19,713.77. How much did he originally invest? Please round the answer to the nearest cent.
P(1+.07/4)^(4*6) = 19713.77
P = 13,000.00
2-3(4-1)+5(1+2)
FV=P*(1+i/m)^(n*m)
19713.77=P*(1+0.07/4)^(6*4)
P = 19713.11 / (1.0175)^(24)
P = 19713.11/1.51644278639= $12 299.57≈$13 000
To find out how much Anthony originally invested, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- A is the final amount ( $19,713.77 in this case)
- P is the principal amount (unknown)
- r is the annual interest rate (7% or 0.07 written as a decimal)
- n is the number of times interest is compounded per year (quarterly, so 4 times)
- t is the number of years (6 years)
Now we can substitute the known values into the formula:
\[ 19,713.77 = P \left(1 + \frac{0.07}{4}\right)^{4 \cdot 6} \]
Simplifying:
\[ 19,713.77 = P \left(1 + \frac{0.0175}{1}\right)^{24} \]
To solve for P, we divide both sides of the equation by \( \left(1 + \frac{0.0175}{1}\right)^{24} \):
\[ P = \frac{19,713.77}{\left(1 + 0.0175\right)^{24}} \]
Using a calculator, we find that \( P \approx 14,000.26 \).
Therefore, Anthony originally invested approximately $14,000.26.