QuestionHow long will it take for a $\$ 5000$ investment to grow to $\$ 6455$ at an annual rate of $9.4 \%$ , compounded quarterly? Assume that no withdrawals are made. Do not round any intermediate computations, and round your answer to the nearest hundredth.

Studdy Solution

STEP 1

1. The initial investment (principal) is \$5000.
2. The future value of the investment is \$6455.
3. The annual interest rate is 9.4%.
4. The interest is compounded quarterly.
5. No withdrawals are made during the investment period.
6. We need to find the time it takes for the investment to grow to the desired amount.

STEP 2

1. Identify the formula for compound interest.
2. Substitute the known values into the formula.
3. Solve for the time variable.
4. Round the answer to the nearest hundredth.

STEP 3

Identify the formula for compound interest. The formula is:
$A = P \left(1 + \frac{r}{n}\right)^{nt}$
Where: - $A$ is the future value of the investment. - $P$ is the principal amount (initial investment). - $r$ is the annual interest rate (as a decimal). - $n$ is the number of times the interest is compounded per year. - $t$ is the time in years.

STEP 4

Substitute the known values into the formula. We have:
- $A = 6455$ - $P = 5000$ - $r = 0.094$ (since 9.4% as a decimal is 0.094) - $n = 4$ (quarterly compounding)
Substitute these values into the formula:
$6455 = 5000 \left(1 + \frac{0.094}{4}\right)^{4t}$

STEP 5

Solve for the time variable $t$ .
First, simplify the expression inside the parentheses:
$6455 = 5000 \left(1 + 0.0235\right)^{4t}$ $6455 = 5000 \times (1.0235)^{4t}$
Divide both sides by 5000 to isolate the exponential term:
$\frac{6455}{5000} = (1.0235)^{4t}$ $1.291 = (1.0235)^{4t}$
Take the natural logarithm of both sides to solve for $t$ :
$\ln(1.291) = \ln((1.0235)^{4t})$
Apply the logarithmic identity $\ln(a^b) = b \ln(a)$ :
$\ln(1.291) = 4t \cdot \ln(1.0235)$
Solve for $t$ :
$t = \frac{\ln(1.291)}{4 \cdot \ln(1.0235)}$

STEP 6

Calculate the value of $t$ using a calculator:
$t = \frac{\ln(1.291)}{4 \cdot \ln(1.0235)}$ $t \approx \frac{0.2553}{4 \times 0.0232}$ $t \approx \frac{0.2553}{0.0928}$ $t \approx 2.75$
Round the answer to the nearest hundredth:
The time it will take for the investment to grow to \$6455 is approximately \( \boxed{2.75} \) years.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms