At what rate per cent will a sum of Rs 4,000 yield 1,324 as compound interest in 3 years?

Answer

To solve the problem of finding the rate of interest at which a sum of Rs 4,000 will yield Rs 1,324 as compound interest in 3 years, we can follow these steps:

### Step 1: Identify the given values
- Principal amount (P) = Rs 4,000
- Compound Interest (CI) = Rs 1,324
- Time (t) = 3 years

### Step 2: Calculate the total amount (A)
The total amount (A) after 3 years can be calculated using the formula:
\[ A = P + CI \]
Substituting the values:
\[ A = 4000 + 1324 = 5324 \]

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Substituting the known values into the formula:
\[ 5324 = 4000 \left(1 + \frac{r}{100}\right)^3 \]

### Step 4: Isolate the term with r
To isolate the term with r, divide both sides by 4000:
\[ \frac{5324}{4000} = \left(1 + \frac{r}{100}\right)^3 \]
Calculating the left side:
\[ 1.331 = \left(1 + \frac{r}{100}\right)^3 \]

### Step 5: Take the cube root
Next, take the cube root of both sides to eliminate the exponent:
\[ 1 + \frac{r}{100} = \sqrt[3]{1.331} \]

### Step 6: Calculate the cube root
Calculating the cube root:
\[ 1 + \frac{r}{100} \approx 1.1 \]

### Step 7: Solve for r
Now, subtract 1 from both sides:
\[ \frac{r}{100} = 1.1 - 1 = 0.1 \]
Multiplying both sides by 100 gives:
\[ r = 10 \]

### Final Answer
The rate of interest (r) is 10%.

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