At what compound rate of interest a sum of 20000 will amount to 29282 in 4 years?

To find the compound rate of interest that makes a sum of $ 20,000 $ amount to $ 29,282 $ in 4 years, we can use the compound interest formula:

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

where:
- $ A $ is the amount of money accumulated after n years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times interest is compounded per year.
- $ t $ is the time the money is invested for in years.

In this case, the interest is compounded annually ($ n = 1 $), so the formula simplifies to:

\[ A = P \left(1 + r\right)^t \]

Given:
- $ A = 29,282 $
- $ P = 20,000 $
- $ t = 4 $

We need to solve for $ r $. Plugging in the known values:

\[ 29,282 = 20,000 \left(1 + r\right)^4 \]

To isolate $ (1 + r) $, divide both sides by 20,000:

\[ \frac{29,282}{20,000} = (1 + r)^4 \]

\[ 1.4641 = (1 + r)^4 \]

Now, take the fourth root of both sides to solve for $ 1 + r $:

\[ 1 + r = \sqrt[4]{1.4641} \]

\[ 1 + r = 1.1 \]

Subtract 1 from both sides to solve for $ r $:

\[ r = 0.1 \]

So, the annual compound interest rate is $ 10\% $.

Do you want details or have any questions? Here are some questions you might want to ask:

1. How is compound interest different from simple interest?
2. Can you explain the compound interest formula in detail?
3. What is the effect of increasing the number of compounding periods per year?
4. How do you calculate the compound interest for non-annual compounding periods?
5. What are some practical applications of compound interest?

**Tip:** Remember to convert the interest rate to a decimal (e.g., 10% becomes 0.10) when using it in formulas.

Learn how to calculate compound interest to determine the annual rate needed for $20,000 to grow to $29,282 in 4 years. This problem involves using the compound interest formula with annual compounding. Suitable for Grades 9-12 students and anyone interested in financial mathematics.

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