find the number of years it will take for a borrowed amount of $3000 to double at a compound interest of 14%

To determine the number of years it will take for a borrowed amount of $3,000 to double at a compound interest rate of 14%, we can use the formula for compound interest:

\[
A = P \times (1 + \frac{r}{100})^n
\]

where:
- $ A $ is the final amount,
- $ P $ is the principal (initial amount),
- $ r $ is the annual interest rate,
- $ n $ is the number of years.

Since we want the amount to double, $ A = 2 \times P $. So, substituting, we get:

\[
2P = P \times (1 + \frac{r}{100})^n
\]

We can simplify by canceling $ P $ from both sides:

\[
2 = (1 + \frac{r}{100})^n
\]

Now, substitute $ r = 14 $ and solve for $ n $:

\[
2 = (1 + \frac{14}{100})^n
\]
\[
2 = (1.14)^n
\]

To solve for $ n $, take the natural logarithm of both sides:

\[
\ln(2) = n \times \ln(1.14)
\]

Now, isolate $ n $:

\[
n = \frac{\ln(2)}{\ln(1.14)}
\]

Calculating this:

1. $\ln(2) \approx 0.6931$
2. $\ln(1.14) \approx 0.1310$

So,

\[
n = \frac{0.6931}{0.1310} \approx 5.29
\]

Therefore, it will take approximately **5.29 years** for the amount to double.

---

Would you like more details on the steps? 

### Related Questions
1. How does the formula for compound interest differ from that for simple interest?
2. How would the time change if the interest rate were compounded semi-annually instead of annually?
3. What effect does increasing the interest rate have on the doubling time?
4. Can we use the Rule of 72 to estimate the doubling time at other interest rates?
5. How is logarithmic calculation applied in other financial growth scenarios?

#### Tip:
The Rule of 72 is a quick way to estimate the time needed to double an investment. Divide 72 by the interest rate to get an approximate doubling time.

Find the number of years it will take for a borrowed amount of $3000 to double at a compound interest of 14%.

Calculate the time required for a $3000 investment to double with a 14% compound interest rate. This guide uses the compound interest formula and logarithmic functions for a step-by-step solution. Suitable for high school students and anyone studying exponential growth in finance.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation