A sum when invested for a year at 20% compounding annually would fetch Rs. 10 less when compounding is done half-yearly. Find the sum invested.

Options

A. Rs. 850
B. Rs. 950
C. Rs. 1000
D. Rs. 1100

Answer

To solve the problem, we need to find the principal amount (sum invested) based on the given conditions about compound interest. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We know that when a sum is invested at 20% compounded annually for one year, it yields Rs. 10 less than when it is compounded half-yearly.

### Step 2: Define the Variables
Let the principal amount (sum invested) be \( P \).

### Step 3: Calculate the Amount with Annual Compounding
When compounded annually at 20%, the amount after one year is calculated as:
\[ A_1 = P \times (1 + r)^t \]
where \( r = 0.20 \) (20% as a decimal) and \( t = 1 \) year.
\[ A_1 = P \times (1 + 0.20)^1 = P \times 1.20 \]

### Step 4: Calculate the Amount with Half-Yearly Compounding
When compounded half-yearly, the rate for each half-year is half of 20%, which is 10% (or 0.10 as a decimal). The time period is 2 half-years.
\[ A_2 = P \times (1 + r/n)^{nt} \]
where \( n = 2 \) (number of compounding periods per year) and \( t = 1 \) year.
\[ A_2 = P \times (1 + 0.10)^{2} = P \times (1.10)^2 \]
Calculating \( (1.10)^2 \):
\[ (1.10)^2 = 1.21 \]
Thus,
\[ A_2 = P \times 1.21 \]

### Step 5: Set Up the Equation Based on the Problem Statement
According to the problem, the amount from half-yearly compounding is Rs. 10 more than the amount from annual compounding:
\[ A_2 = A_1 + 10 \]
Substituting the expressions we found:
\[ P \times 1.21 = P \times 1.20 + 10 \]

### Step 6: Simplify the Equation
Rearranging the equation gives:
\[ P \times 1.21 - P \times 1.20 = 10 \]
\[ P \times (1.21 - 1.20) = 10 \]
\[ P \times 0.01 = 10 \]

### Step 7: Solve for \( P \)
Now, divide both sides by 0.01:
\[ P = \frac{10}{0.01} = 1000 \]

### Conclusion
The sum invested is Rs. 1000.