if you have R5000 and you invest it with a compound interest of 4.5 percent compounded annually, how long will it take for the amount to double (give years and round off to 2 decimal places only at the end) 

To find out how long it will take for an investment to double with compound interest, we use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- \( A \) is the future value of the investment/loan, including interest.
- \( P \) is the principal investment amount (the initial deposit or loan amount).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed for.

Given:
- \( P = 5000 \) (initial investment).
- \( r = 4.5\% = 0.045 \).
- \( n = 1 \) (compounded annually).
- \( A = 2P = 10000 \) (the amount doubles).

We want to solve for \( t \). The formula simplifies to:

\[
10000 = 5000 \left(1 + \frac{0.045}{1}\right)^{1 \cdot t}
\]

Simplify this equation:

\[
2 = \left(1.045\right)^t
\]

To solve for \( t \), take the natural logarithm (ln) of both sides:

\[
\ln(2) = \ln\left(1.045^t\right)
\]

\[
\ln(2) = t \cdot \ln(1.045)
\]

\[
t = \frac{\ln(2)}{\ln(1.045)}
\]

Let's calculate the value of \( t \).It will take approximately **15.75 years** for the amount to double with a compound interest rate of 4.5% compounded annually.

Would you like more details or have any questions? Here are some related questions:

1. How would the time change if the interest rate were higher?
2. What if the interest were compounded semi-annually instead of annually?
3. How do you calculate the doubling time for different compounding frequencies?
4. What is the effect of continuous compounding on the time required?
5. How would the formula change if additional contributions were made each year?

**Tip:** For quick estimates, you can use the Rule of 72, which approximates the doubling time by dividing 72 by the annual interest rate (in percentage terms).

If you have R5000 and you invest it with a compound interest of 4.5 percent compounded annually, how long will it take for the amount to double (give years and round off to 2 decimal places only at the end)?

Learn how to calculate the time it takes for an investment of R5000 to double with a 4.5% compound interest rate, compounded annually. This detailed solution involves compound interest formulas and logarithmic functions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation