Rashi invested Rs. 16000 for two years at compound interest and received an amount of Rs. 17640 on maturity. What is the rate of interest?

Answer

To find the rate of interest for Rashi's investment using the compound interest formula, we can follow these steps:

### Step 1: Identify the known values
- Principal (P) = Rs. 16,000
- Amount (A) = Rs. 17,640
- Time (T) = 2 years

### Step 2: Write down the formula for compound interest
The formula for the amount in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where:
- A = Amount after time T
- P = Principal
- R = Rate of interest per annum
- T = Time in years

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 17,640 = 16,000 \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Simplify the equation
To isolate the term with R, divide both sides by 16,000:
\[ \frac{17,640}{16,000} = \left(1 + \frac{R}{100}\right)^2 \]

Calculating the left side:
\[ \frac{17,640}{16,000} = 1.1025 \]

So we have:
\[ 1.1025 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives:
\[ \sqrt{1.1025} = 1 + \frac{R}{100} \]

Calculating the square root:
\[ 1.05 = 1 + \frac{R}{100} \]

### Step 6: Solve for R
Now, subtract 1 from both sides:
\[ 1.05 - 1 = \frac{R}{100} \]
\[ 0.05 = \frac{R}{100} \]

Multiply both sides by 100 to find R:
\[ R = 0.05 \times 100 = 5 \]

### Conclusion
The rate of interest is:
\[ R = 5\% \text{ per annum} \]

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