If the difference between CI and SI on some amount for 1.5 years is 60.40 then find the principle amount if interest is compounded semi-annually?

Options

A. ₹30000
B. ₹50000
C. ₹60000
D. ₹40000

Answer

To solve the problem, we need to find the principal amount (P) given that the difference between Compound Interest (CI) and Simple Interest (SI) for 1.5 years is 60.40, and the interest is compounded semi-annually.

### Step 1: Understand the formula for the difference between CI and SI
The difference between Compound Interest and Simple Interest for a time period can be calculated using the formula:
\[ \text{Difference} = \text{CI} - \text{SI} = P \left( \left(1 + \frac{r}{100}\right)^n - 1 - \frac{rn}{100} \right) \]
where:
- \( P \) = Principal amount
- \( r \) = Rate of interest per annum
- \( n \) = Time in years

### Step 2: Determine the time and compounding frequency
Given that the interest is compounded semi-annually, we need to adjust the time and the rate:
- Time (n) = 1.5 years = 3 half-years
- The effective rate for semi-annual compounding is \( \frac{r}{2} \).

### Step 3: Set up the equation
From the problem, we know that the difference is 60.40. Therefore, we can write:
\[ 60.40 = P \left( \left(1 + \frac{r}{200}\right)^3 - 1 - \frac{3r}{200} \right) \]

### Step 4: Simplify the equation
Let’s denote \( x = \frac{r}{200} \). Then the equation becomes:
\[ 60.40 = P \left( (1 + x)^3 - 1 - 3x \right) \]

### Step 5: Expand \( (1 + x)^3 \)
Using the binomial expansion:
\[ (1 + x)^3 = 1 + 3x + 3x^2 + x^3 \]
So, we can substitute this back into our equation:
\[ 60.40 = P \left( 1 + 3x + 3x^2 + x^3 - 1 - 3x \right) \]
This simplifies to:
\[ 60.40 = P \left( 3x^2 + x^3 \right) \]

### Step 6: Solve for P
Rearranging gives:
\[ P = \frac{60.40}{3x^2 + x^3} \]

### Step 7: Determine the value of r
To find the principal amount, we need the value of \( r \). However, since the problem does not provide a specific rate, we can assume a reasonable rate for calculation. Let’s assume \( r = 10\% \):
- Thus, \( x = \frac{10}{200} = 0.05 \)

### Step 8: Substitute x back into the equation
Substituting \( x = 0.05 \):
\[ 3(0.05)^2 + (0.05)^3 = 3(0.0025) + (0.000125) = 0.0075 + 0.000125 = 0.007625 \]

### Step 9: Calculate P
Now substituting back into the equation for P:
\[ P = \frac{60.40}{0.007625} \approx 7925.49 \]

### Conclusion
The principal amount is approximately **7925.49**.

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