An amount was lent for one year at the rate of `10%` per annum compounding annually had the compounding been done half yearly, the interest would have increased by `Rs. 80`. What was the amount (in Rs.) lent?

Options

A. `16000 `
B. `32000`
C. `48000 `
D. `64000`

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the Compounding Interest
When an amount is compounded annually at a rate of 10%, the interest for one year is simply 10% of the principal amount (P).

**Hint:** Remember that annual compounding means the interest is calculated once at the end of the year.

### Step 2: Calculate the Half-Yearly Compounding Rate
If the compounding is done half-yearly, the annual interest rate of 10% is divided by 2, giving us a half-yearly rate of 5%. Since there are two compounding periods in a year, we will calculate the effective interest rate for half-yearly compounding.

**Hint:** The formula for effective interest rate when compounded multiple times in a year is:
\[ \text{Effective Rate} = \left(1 + \frac{r}{n}\right)^n - 1 \]
where \( r \) is the annual interest rate and \( n \) is the number of compounding periods.

### Step 3: Calculate the Effective Interest Rate for Half-Yearly Compounding
Using the formula:
\[ \text{Effective Rate} = \left(1 + \frac{10\%}{2}\right)^2 - 1 = \left(1 + 0.05\right)^2 - 1 = 1.1025 - 1 = 0.1025 \text{ or } 10.25\% \]

**Hint:** This shows that with half-yearly compounding, you earn slightly more interest than with annual compounding.

### Step 4: Find the Difference in Interest Earned
The difference in interest earned between half-yearly and annual compounding is:
\[ 10.25\% - 10\% = 0.25\% \]

**Hint:** This difference in interest is crucial for finding the principal amount.

### Step 5: Set Up the Equation for Extra Interest
According to the problem, the extra interest earned by compounding half-yearly instead of annually is Rs. 80. We can express this as:
\[ 0.25\% \text{ of } P = 80 \]

**Hint:** Write the percentage as a decimal when setting up the equation.

### Step 6: Solve for Principal Amount (P)
Convert 0.25% to decimal:
\[ 0.25\% = \frac{0.25}{100} = 0.0025 \]
Now, substituting into the equation:
\[ 0.0025P = 80 \]
To find \( P \):
\[ P = \frac{80}{0.0025} = 32000 \]

**Hint:** When dividing by a decimal, it may help to multiply both the numerator and denominator by 10000 to simplify.

### Final Answer
The amount lent (P) is Rs. 32,000.