5) ryan invests a sum of money in a savings account with a fixed annual interest rate of 4.31% compounded 12…
5) ryan invests a sum of money in a savings account with a fixed annual interest rate of 4.31% compounded 12 times per year. after 10 years, the balance reaches $12,855.94. what was the amount of the initial investment? a = 12855.94\n6) ndiba invests a sum of money in a savings account with a fixed annual interest rate of 4.61% compounded 3 times per year. after 6 years, the balance reaches $5,485.85. what was the amount of the initial investment?\n7) john invests a sum of money in a retirement account with a fixed annual interest rate of 2.63% compounded continuously. after 15 years, the balance reaches $1,912.41. what was the amount of the initial investment?\n8) anjali invests a sum of money in a retirement account with a fixed annual interest rate of 6.79% compounded continuously. after 20 years, the balance reaches $14,037.16. what was the amount of the initial investment?\n9) adam invests $6,139 in a retirement account with a fixed annual interest rate compounded continuously. after 17 years, the balance reaches $8,624.97. what is the interest rate of the account?\n10) huong invests $8,589 in a retirement account with a fixed annual interest rate of 7% compounded continuously. how long will it take for the account balance to reach $21,337.85?
Answer
5)
Explanation:
Step1: Identify compound - interest formula
The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the annual interest rate (in decimal), $n$ is the number of times compounded per year, and $t$ is the number of years. Given $A = 12855.94$, $r=0.0431$, $n = 12$, and $t = 10$.
Step2: Rearrange the formula for $P$
$P=\frac{A}{(1 +\frac{r}{n})^{nt}}$. Substitute the values: $\frac{r}{n}=\frac{0.0431}{12}\approx0.003592$, $nt=12\times10 = 120$. Then $(1+\frac{r}{n})^{nt}=(1 + 0.003592)^{120}$. Using a calculator, $(1 + 0.003592)^{120}\approx1.5277$. So $P=\frac{12855.94}{1.5277}\approx8415$.
Answer:
$$8415$
6)
Explanation:
Step1: Use compound - interest formula
The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A = 5485.85$, $r = 0.0461$, $n=3$, and $t = 6$.
Step2: Rearrange for $P$
$P=\frac{A}{(1+\frac{r}{n})^{nt}}$. First, $\frac{r}{n}=\frac{0.0461}{3}\approx0.01537$, $nt=3\times6 = 18$. Then $(1+\frac{r}{n})^{nt}=(1 + 0.01537)^{18}$. Using a calculator, $(1 + 0.01537)^{18}\approx1.3177$. So $P=\frac{5485.85}{1.3177}\approx4163$.
Answer:
$$4163$
7)
Explanation:
Step1: Recall continuous - compounding formula
The continuous - compounding formula is $A=Pe^{rt}$, where $A = 1912.41$, $r = 0.0263$, and $t = 15$.
Step2: Rearrange for $P$
$P=\frac{A}{e^{rt}}$. Calculate $e^{rt}=e^{0.0263\times15}=e^{0.3945}$. Using a calculator, $e^{0.3945}\approx1.483$. So $P=\frac{1912.41}{1.483}\approx1290$.
Answer:
$$1290$
8)
Explanation:
Step1: Apply continuous - compounding formula
The continuous - compounding formula is $A = Pe^{rt}$, where $A=14037.16$, $r = 0.0679$, and $t = 20$.
Step2: Solve for $P$
$P=\frac{A}{e^{rt}}$. Calculate $e^{rt}=e^{0.0679\times20}=e^{1.358}$. Using a calculator, $e^{1.358}\approx3.887$. So $P=\frac{14037.16}{3.887}\approx3611$.
Answer:
$$3611$
9)
Explanation:
Step1: Use continuous - compounding formula
The continuous - compounding formula is $A = Pe^{rt}$, where $P = 6139$, $A = 8624.97$, and $t = 17$. We need to solve for $r$. Rearranging the formula gives $\frac{A}{P}=e^{rt}$.
Step2: Take natural logarithm of both sides
$\ln(\frac{A}{P})=rt$. Substitute the values: $\frac{A}{P}=\frac{8624.97}{6139}\approx1.405$. So $\ln(1.405)=rt$. Since $t = 17$, then $r=\frac{\ln(1.405)}{17}$. Using a calculator, $\ln(1.405)\approx0.34$ and $r=\frac{0.34}{17}=0.02$ or $2%$.
Answer:
$2%$
10)
Explanation:
Step1: Apply continuous - compounding formula
The continuous - compounding formula is $A = Pe^{rt}$, where $P = 8589$, $A = 21337.85$, and $r = 0.07$. We need to solve for $t$. Rearranging gives $\frac{A}{P}=e^{rt}$.
Step2: Take natural logarithm of both sides
$\ln(\frac{A}{P})=rt$. Substitute $\frac{A}{P}=\frac{21337.85}{8589}\approx2.484$. So $\ln(2.484)=0.07t$. Since $\ln(2.484)\approx0.909$, then $t=\frac{\ln(2.484)}{0.07}=\frac{0.909}{0.07}\approx13$.
Answer:
$13$ years