Ruchi invested? 1600 at the rate of compound interest for 2 yr. She got ? 1764 after the specified period. Find the rate of interest.

Answer

To find the rate of compound interest that Ruchi earned on her investment, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = ₹1600
- Amount (A) after 2 years = ₹1764
- Time (N) = 2 years

### Step 2: Use the formula for compound interest
The formula for the amount (A) in compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- A = final amount
- P = principal amount
- r = rate of interest
- n = number of years

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula gives:
\[ 1764 = 1600 \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Divide both sides by the principal
To isolate the term with the rate, divide both sides by 1600:
\[ \frac{1764}{1600} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Simplify the left side
Calculating the left side:
\[ \frac{1764}{1600} = 1.1025 \]
So, we have:
\[ 1.1025 = \left(1 + \frac{r}{100}\right)^2 \]

### Step 6: Take the square root of both sides
To remove the square, take the square root of both sides:
\[ \sqrt{1.1025} = 1 + \frac{r}{100} \]

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

### Step 7: Solve for r
Now, isolate r:
\[ \frac{r}{100} = 1.05 - 1 \]
\[ \frac{r}{100} = 0.05 \]

### Step 8: Multiply by 100 to find r
Multiply both sides by 100 to find the rate:
\[ r = 0.05 \times 100 = 5 \]

### Final Answer
The rate of interest (r) is **5%**.

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