A sum of Rs 6,400 earsn a compound intersec of Rs 1,008.80 in 18 months when the interset is reckoned half-yearly. Find the reate of intersect.

Answer

To solve the problem step by step, we will follow the given information and apply the compound interest formula.

### Step 1: Identify the given values
- Principal (P) = Rs 6,400
- Compound Interest (CI) = Rs 1,008.80
- Time (T) = 18 months = 1.5 years = 3 half-years (since interest is compounded half-yearly)

### Step 2: Calculate the Amount (A)
The total amount (A) after earning compound interest can be calculated as:
\[ A = P + CI \]
Substituting the values:
\[ A = 6400 + 1008.80 = 7408.80 \]

### Step 3: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = Total amount after interest
- \( P \) = Principal amount
- \( r \) = Rate of interest per half-year
- \( n \) = Number of half-years

Substituting the known values into the formula:
\[ 7408.80 = 6400 \left(1 + \frac{r}{100}\right)^3 \]

### Step 4: Simplify the Equation
Dividing both sides by 6400:
\[ \frac{7408.80}{6400} = \left(1 + \frac{r}{100}\right)^3 \]
Calculating the left side:
\[ \frac{7408.80}{6400} = 1.157 \]

### Step 5: Set Up the Equation
Now we have:
\[ \left(1 + \frac{r}{100}\right)^3 = 1.157 \]

### Step 6: Take the Cube Root
Taking the cube root of both sides:
\[ 1 + \frac{r}{100} = \sqrt[3]{1.157} \]
Calculating the cube root:
\[ 1 + \frac{r}{100} \approx 1.05 \]

### Step 7: Solve for r
Now, we can solve for \( r \):
\[ \frac{r}{100} = 1.05 - 1 \]
\[ \frac{r}{100} = 0.05 \]
Multiplying both sides by 100:
\[ r = 5 \]

### Step 8: Conclusion
The rate of interest per half-year is 5%. Therefore, the annual rate of interest is:
\[ \text{Annual Rate} = 5\% \times 2 = 10\% \]

### Final Answer
The rate of interest is **5% per half-year** or **10% per annum**.

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