Question: $\newline$ Suppose that $\$ 12,000$ is invested in a bond fund and the account grows to $\$ 14,309.26$ in $4$ yrs. $\newline$ Use the model $A=P e^{r t}$ to determine $\newline$ b. How long will it take the investment to reach $\$ 20,000$ if the rate of return continues? Round to the nearest tenth of a year.

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Grade 8

Compound interest

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Q. Question:

\newline

Suppose that

\$ 12,000

is invested in a bond fund and the account grows to

\$ 14,309.26

4

yrs.

\newline

Use the model

A=P e^{r t}

to determine

\newline

b. How long will it take the investment to reach

\$ 20,000

if the rate of return continues? Round to the nearest tenth of a year.

Find Rate of Return: First, we need to find the rate of return using the given model $A=Pe^{rt}$ where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount, $r$ is the rate of interest per year, and $t$ is the time in years.
Calculate Rate of Return: We know that $A=\$14,309.26$ , $P=\$12,000$ , and $t=4$ years. We plug these values into the equation to solve for $r$ . $\$14,309.26 = \$12,000 \times e^{r\times 4}$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ . $\frac{\$14,309.26}{\$12,000} = e^{r*4}$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$
Calculate Time to Reach $\$20,000$ : Divide both sides by $\$12,000$ to isolate $e^{r*4}$ .
$\frac{\$14,309.26}{\$12,000} = e^{r*4}$ Calculate the left side of the equation.
$\frac{\$14,309.26}{\$12,000} = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1$ . $192438333$ .
$\ln(1.192438333) \approx 0.1778$
Calculate Time to Reach $\$20,000$ : Divide both sides by $\$12,000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1$ . $192438333$ .
$\ln(1.192438333) \approx 0.1778$ Divide both sides by $\$12,000$ $0$ to solve for $\$12,000$ $1$ .
$\$12,000$ $2$
Calculate Time to Reach $\$20,000$ : Divide both sides by $\$12,000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1$ . $192438333$ .
$\ln(1.192438333) \approx 0.1778$ Divide both sides by $4$ to solve for $\$12,000$ $0$ .
$\$12,000$ $1$ Calculate $\$12,000$ $0$ .
$\$12,000$ $3$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ . $\$$ $14$ , $309$ . $26$ / $\$$ $12$ , $000$ = $e^{r*4}$ Calculate the left side of the equation. $\$$ $14$ , $309$ . $26$ / $\$$ $12$ , $000$ = $1$ . $192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ . $\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\$$ $0$ to simplify the right side of the equation. $\$$ $1$ Calculate the natural logarithm of $1$ . $192438333$ . $\$$ $2$ Divide both sides by $4$ to solve for $\$$ $3$ . $\$$ $4$ Calculate $\$$ $3$ . $\$$ $6$ Now we have the rate of return, $\$$ $7$ . We use this rate to find out how long it will take for the investment to reach $\$$ $20$ , $000$ . We set $\$$ $9$ to $\$$ $20$ , $000$ and solve for $e^{r*4}$ $1$ using the same model $e^{r*4}$ $2$ . $\$$ $20$ , $000$ = $\$$ $12$ , $000$ * $e^{r*4}$ $5$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1$ . $192438333$ .
$\ln(1.192438333) \approx 0.1778$ Divide both sides by $4$ to solve for $\$$ $0$ .
$\$$ $1$ Calculate $\$$ $0$ .
$\$$ $3$ Now we have the rate of return, $\$$ $4$ . We use this rate to find out how long it will take for the investment to reach $\$$ $20$ , $000$ .
We set $\$$ $6$ to $\$$ $20$ , $000$ and solve for $\$$ $8$ using the same model $\$$ $9$ .
$e^{r*4}$ $0$ Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ $2$ .
$e^{r*4}$ $3$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ . $\$$ $14$ , $309$ . $26$ / $\$$ $12$ , $000$ = $e^{r*4}$ Calculate the left side of the equation. $\$$ $14$ , $309$ . $26$ / $\$$ $12$ , $000$ = $1$ . $192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ . $\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\$$ $0$ to simplify the right side of the equation. $\$$ $1$ Calculate the natural logarithm of $1$ . $192438333$ . $\$$ $2$ Divide both sides by $4$ to solve for $\$$ $3$ . $\$$ $4$ Calculate $\$$ $3$ . $\$$ $6$ Now we have the rate of return, $\$$ $7$ . We use this rate to find out how long it will take for the investment to reach $\$$ $20$ , $000$ . We set $\$$ $9$ to $\$$ $20$ , $000$ and solve for $e^{r*4}$ $1$ using the same model $e^{r*4}$ $2$ . $\$$ $20$ , $000$ = $\$$ $12$ , $000$ * $e^{r*4}$ $5$ Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ $5$ . $\$$ $20$ , $000$ / $\$$ $12$ , $000$ = $e^{r*4}$ $5$ Calculate the left side of the equation. $\$$ $20$ , $000$ / $\$$ $12$ , $000$ = $1$ . $666666667$
Calculate Time to Reach $\$20,000$ : Divide both sides by $\$12,000$ to isolate $e^{(r*4)}$ .
$\$14,309.26 / \$12,000 = e^{(r*4)}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{(r*4)})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1.192438333$ .
$\$12,000$ $0$ Divide both sides by $\$12,000$ $1$ to solve for $\$12,000$ $2$ .
$\$12,000$ $3$ Calculate $\$12,000$ $2$ .
$\$12,000$ $5$ Now we have the rate of return, $\$12,000$ $6$ . We use this rate to find out how long it will take for the investment to reach $\$20,000$ .
We set $\$12,000$ $8$ to $\$20,000$ and solve for $e^{(r*4)}$ $0$ using the same model $e^{(r*4)}$ $1$ .
$e^{(r*4)}$ $2$ Divide both sides by $\$12,000$ to isolate $e^{(r*4)}$ $4$ .
$e^{(r*4)}$ $5$ Calculate the left side of the equation.
$e^{(r*4)}$ $6$ Take the natural logarithm (ln) of both sides to solve for $e^{(r*4)}$ $7$ .
$e^{(r*4)}$ $8$
Calculate Time to Reach $\$20,000$ : Divide both sides by $\$12,000$ to isolate $e^{r\cdot 4}$ .
$\frac{\$14,309.26}{\$12,000} = e^{r\cdot 4}$ Calculate the left side of the equation.
$\frac{\$14,309.26}{\$12,000} = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r\cdot 4$ .
$\ln(1.192438333) = \ln(e^{r\cdot 4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r\cdot 4$ Calculate the natural logarithm of $1.192438333$ .
$\$12,000$ $0$ Divide both sides by $\$12,000$ $1$ to solve for $\$12,000$ $2$ .
$\$12,000$ $3$ Calculate $\$12,000$ $2$ .
$\$12,000$ $5$ Now we have the rate of return, $\$12,000$ $6$ . We use this rate to find out how long it will take for the investment to reach $\$20,000$ .
We set $\$12,000$ $8$ to $\$20,000$ and solve for $e^{r\cdot 4}$ $0$ using the same model $e^{r\cdot 4}$ $1$ .
$e^{r\cdot 4}$ $2$ Divide both sides by $\$12,000$ to isolate $e^{r\cdot 4}$ $4$ .
$e^{r\cdot 4}$ $5$ Calculate the left side of the equation.
$e^{r\cdot 4}$ $6$ Take the natural logarithm (ln) of both sides to solve for $e^{r\cdot 4}$ $7$ .
$e^{r\cdot 4}$ $8$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\frac{\$14,309.26}{\$12,000} = e^{r\cdot 4}$ $0$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1.192438333$ .
$\$$ $0$ Divide both sides by $\$$ $1$ to solve for $\$$ $2$ .
$\$$ $3$ Calculate $\$$ $2$ .
$\$$ $5$ Now we have the rate of return, $\$$ $6$ . We use this rate to find out how long it will take for the investment to reach $\$$ $20$ , $000$ .
We set $\$$ $8$ to $\$$ $20$ , $000$ and solve for $e^{r*4}$ $0$ using the same model $e^{r*4}$ $1$ .
$e^{r*4}$ $2$ Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ $4$ .
$e^{r*4}$ $5$ Calculate the left side of the equation.
$e^{r*4}$ $6$ Take the natural logarithm (ln) of both sides to solve for $e^{r*4}$ $7$ .
$e^{r*4}$ $8$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\$14,309.26 / \$12,000 = e^{r*4}$ $0$ Calculate the natural logarithm of $\$14,309.26 / \$12,000 = e^{r*4}$ $1$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $2$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1$ . $192438333$ .
$\ln(1.192438333) \approx 0.1778$ Divide both sides by $4$ to solve for $\$$ $0$ .
$\$$ $1$ Calculate $\$$ $0$ .
$\$$ $3$ Now we have the rate of return, $\$$ $4$ . We use this rate to find out how long it will take for the investment to reach $\$$ $20$ , $000$ .
We set $\$$ $6$ to $\$$ $20$ , $000$ and solve for $\$$ $8$ using the same model $\$$ $9$ .
$e^{r*4}$ $0$ Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ $2$ .
$e^{r*4}$ $3$ Calculate the left side of the equation.
$e^{r*4}$ $4$ Take the natural logarithm (ln) of both sides to solve for $e^{r*4}$ $5$ .
$e^{r*4}$ $6$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$e^{r*4}$ $8$ Calculate the natural logarithm of $1$ . $666666667$ .
$e^{r*4}$ $9$ Divide both sides by $\$14,309.26 / \$12,000 = e^{r*4}$ $0$ to solve for $\$$ $8$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $2$
Calculate Time to Reach $\$$ $20$ , $000$ : Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1.192438333$ .
$\$$ $0$ Divide both sides by $\$$ $1$ to solve for $\$$ $2$ .
$\$$ $3$ Calculate $\$$ $2$ .
$\$$ $5$ Now we have the rate of return, $\$$ $6$ . We use this rate to find out how long it will take for the investment to reach $\$$ $20$ , $000$ .
We set $\$$ $8$ to $\$$ $20$ , $000$ and solve for $e^{r*4}$ $0$ using the same model $e^{r*4}$ $1$ .
$e^{r*4}$ $2$ Divide both sides by $\$$ $12$ , $000$ to isolate $e^{r*4}$ $4$ .
$e^{r*4}$ $5$ Calculate the left side of the equation.
$e^{r*4}$ $6$ Take the natural logarithm (ln) of both sides to solve for $e^{r*4}$ $7$ .
$e^{r*4}$ $8$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\$14,309.26 / \$12,000 = e^{r*4}$ $0$ Calculate the natural logarithm of $\$14,309.26 / \$12,000 = e^{r*4}$ $1$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $2$ Divide both sides by $\$14,309.26 / \$12,000 = e^{r*4}$ $3$ to solve for $e^{r*4}$ $0$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $5$ Calculate $e^{r*4}$ $0$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $7$
Calculate Time to Reach $\$20,000$ : Divide both sides by $\$12,000$ to isolate $e^{r*4}$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ Calculate the left side of the equation.
$\$14,309.26 / \$12,000 = 1.192438333$ Take the natural logarithm (ln) of both sides to solve for $r*4$ .
$\ln(1.192438333) = \ln(e^{r*4})$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\ln(1.192438333) = r*4$ Calculate the natural logarithm of $1.192438333$ .
$\$12,000$ $0$ Divide both sides by $\$12,000$ $1$ to solve for $\$12,000$ $2$ .
$\$12,000$ $3$ Calculate $\$12,000$ $2$ .
$\$12,000$ $5$ Now we have the rate of return, $\$12,000$ $6$ . We use this rate to find out how long it will take for the investment to reach $\$20,000$ .
We set $\$12,000$ $8$ to $\$20,000$ and solve for $e^{r*4}$ $0$ using the same model $e^{r*4}$ $1$ .
$e^{r*4}$ $2$ Divide both sides by $\$12,000$ to isolate $e^{r*4}$ $4$ .
$e^{r*4}$ $5$ Calculate the left side of the equation.
$e^{r*4}$ $6$ Take the natural logarithm (ln) of both sides to solve for $e^{r*4}$ $7$ .
$e^{r*4}$ $8$ Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation.
$\$14,309.26 / \$12,000 = e^{r*4}$ $0$ Calculate the natural logarithm of $\$14,309.26 / \$12,000 = e^{r*4}$ $1$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $2$ Divide both sides by $\$14,309.26 / \$12,000 = e^{r*4}$ $3$ to solve for $e^{r*4}$ $0$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $5$ Calculate $e^{r*4}$ $0$ .
$\$14,309.26 / \$12,000 = e^{r*4}$ $7$ Round to the nearest tenth of a year.
$\$14,309.26 / \$12,000 = e^{r*4}$ $8$ years