In 3 years Rs. 2000 amounts to Rs. 2662 at r% compound interest, compounded annually. Find the value of r. 5% 12% 10% 8%

1. Break Down the Problem

We have the principal amount (P) of Rs. 2000, the amount (A) after 3 years is Rs. 2662, and we need to find the rate of interest (r) under compound interest. The formula for compound interest is:

$A = P \left(1 + \frac{r}{100}\right)^n$

Where:

• $A$ = Amount after time $n$
• $P$ = Principal amount
• $r$ = Rate of interest per annum
• $n$ = Number of years

2. Relevant Concepts

Using the formula $A = P \left(1 + \frac{r}{100}\right)^n$, we can rearrange it to find $r$.

3. Analysis and Detail

Substituting the known values into the compound interest formula:

$2662 = 2000 \left(1 + \frac{r}{100}\right)^3$

Now, we will isolate $(1 + \frac{r}{100})^3$:

$\frac{2662}{2000} = \left(1 + \frac{r}{100}\right)^3$

Calculating the left-hand side:

$\frac{2662}{2000} = 1.331$

So we have:

$1.331 = \left(1 + \frac{r}{100}\right)^3$

Next, we take the cube root of both sides:

$1 + \frac{r}{100} = 1.331^{\frac{1}{3}}$

Calculating $1.331^{\frac{1}{3}}$:

$1 + \frac{r}{100} \approx 1.1$

Therefore:

$\frac{r}{100} \approx 1.1 - 1 = 0.1$

Multiplying both sides by 100 to find $r$:

$r \approx 0.1 \times 100 = 10$

4. Verify and Summarize

The calculations appear correct. We can also verify the results by plugging $r = 10\%$ back into the compound interest formula to ensure $A$ equals Rs. 2662:

$A = 2000 \left(1 + \frac{10}{100}\right)^3 = 2000 \left(1.1\right)^3 = 2000 \times 1.331 = 2662$

Final Answer

The value of $r$ is $\boxed{10} \%$.