To find the interest rate ($r$) in compound interest, you can rearrange the formula for compound interest:

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

Where:

• $A$ = Final amount
• $P$ = Principal amount (initial investment)
• $r$ = Interest rate per period (what we need to find)
• $n$ = Number of times interest is compounded per year
• $t$ = Time in years

Step-by-Step Process to Find $r$:

1. Rearrange the formula to isolate $r$. $\left(\frac{A}{P}\right) = \left(1 + \frac{r}{n}\right)^{nt}$

2. Take the $nt$-th root on both sides: $\left(\frac{A}{P}\right)^{\frac{1}{nt}} = 1 + \frac{r}{n}$

3. Solve for $r$: $r = n \left(\left(\frac{A}{P}\right)^{\frac{1}{nt}} - 1\right)$

Example:

Suppose you invest $1000, and after 2 years, the amount becomes $1210. Interest is compounded annually ($n = 1$).

Given:
$A = 1210$, $P = 1000$, $t = 2$, and $n = 1$.

1. Plug values into the formula: $\left(\frac{1210}{1000}\right)^{\frac{1}{2 \cdot 1}} = 1 + \frac{r}{1}$

2. Simplify: $(1.21)^{0.5} = 1 + r$

3. Calculate: $1.1 = 1 + r \implies r = 0.1 \text{ or } 10\%$

The interest rate is 10% per year.

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Do you need further clarification or have questions?

Here are five related questions you might explore next:

1. How does the compounding frequency affect the final amount?
2. What is the difference between nominal and effective interest rates?
3. How do you calculate continuous compound interest?
4. How can you find the time required to reach a target amount with compound interest?
5. What happens to the interest rate if the principal amount changes?

Tip: If you know the future amount and principal, logarithms can also help find the interest rate efficiently.