If interest being compound half yearly then what sum (in Rs) will amount to Rs 38416 in 2 years at the rate of 80% per annum at compound interest?

Options

A. 14000
B. 15000
C. 10000
D. 12000

Answer

To solve the problem of finding the principal sum (P) that will amount to Rs. 38,416 in 2 years at a compound interest rate of 80% per annum, compounded half-yearly, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Amount (A) = Rs. 38,416
- Rate of interest (R) = 80% per annum
- Time (T) = 2 years

2. **Convert the Rate and Time for Half-Yearly Compounding:**
- Since the interest is compounded half-yearly, we need to adjust the rate and time:
- Half-yearly rate = R/2 = 80% / 2 = 40%
- Total number of compounding periods in 2 years = 2 * 2 = 4 periods

3. **Use the Compound Interest Formula:**
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
where:
- A = final amount
- P = principal amount
- R = rate of interest per period
- n = number of compounding periods

Plugging in the values:
\[ 38416 = P \left(1 + \frac{40}{100}\right)^4 \]
Simplifying:
\[ 38416 = P \left(1 + 0.4\right)^4 \]
\[ 38416 = P (1.4)^4 \]

4. **Calculate \( (1.4)^4 \):**
\[ (1.4)^4 = 1.4 \times 1.4 \times 1.4 \times 1.4 \]
Calculating step-by-step:
- \( 1.4 \times 1.4 = 1.96 \)
- \( 1.96 \times 1.4 = 2.744 \)
- \( 2.744 \times 1.4 = 3.8416 \)

So, \( (1.4)^4 = 3.8416 \).

5. **Substitute Back to Find P:**
Now we can substitute back into the equation:
\[ 38416 = P \times 3.8416 \]
To find P, divide both sides by 3.8416:
\[ P = \frac{38416}{3.8416} \]

6. **Calculate P:**
Performing the division:
\[ P = 10,000 \]

### Final Answer:
The principal sum (P) that will amount to Rs. 38,416 in 2 years at the rate of 80% per annum compounded half-yearly is **Rs. 10,000**.