On a certain sum of money, the difference between the compound interest for a year,payable half-yearly, and the simple interest for a year is रु 180/-. Find the sum lent out, if the rate of interest in both the cases is 10% per annum.

Explanation

To solve the problem step by step, we will follow the process of calculating the simple interest and compound interest, and then use the information given to find the sum lent out.

### Step 1: Define the variables
Let the sum of money (principal) be \( P \).

### Step 2: Calculate Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R \) is the rate of interest (10% per annum)
- \( T \) is the time (1 year)

Substituting the values:
\[ SI = \frac{P \times 10 \times 1}{100} = \frac{P \times 10}{100} = \frac{P}{10} \]

### Step 3: Calculate Compound Interest (CI) for half-yearly compounding
Since the interest is compounded half-yearly, the rate for each half year is:
\[ \text{Half-yearly rate} = \frac{10}{2} = 5\% \]

For the first half year:
\[ CI_{1} = \frac{P \times 5 \times 0.5}{100} = \frac{P \times 5}{200} = \frac{P}{40} \]

After the first half year, the amount becomes:
\[ A_{1} = P + CI_{1} = P + \frac{P}{40} = P \left(1 + \frac{1}{40}\right) = P \left(\frac{41}{40}\right) \]

For the second half year, we calculate the interest on the new amount:
\[ CI_{2} = A_{1} \times \frac{5}{100} = P \left(\frac{41}{40}\right) \times \frac{5}{100} = \frac{41P \times 5}{4000} = \frac{205P}{4000} = \frac{P}{20} \]

### Step 4: Total Compound Interest for the year
\[ CI = CI_{1} + CI_{2} = \frac{P}{40} + \frac{P}{20} \]
To add these fractions, we need a common denominator:
\[ CI = \frac{P}{40} + \frac{2P}{40} = \frac{3P}{40} \]

### Step 5: Find the difference between Compound Interest and Simple Interest
According to the problem, the difference between CI and SI is given as:
\[ CI - SI = 180 \]
Substituting the values we calculated:
\[ \frac{3P}{40} - \frac{P}{10} = 180 \]

### Step 6: Simplify the equation
To simplify, convert \(\frac{P}{10}\) to a fraction with a denominator of 40:
\[ \frac{P}{10} = \frac{4P}{40} \]
Now, substituting back:
\[ \frac{3P}{40} - \frac{4P}{40} = 180 \]
\[ \frac{-P}{40} = 180 \]

### Step 7: Solve for \( P \)
Multiplying both sides by -40:
\[ P = -180 \times 40 = -7200 \]
Since we are dealing with money, we take the absolute value:
\[ P = 7200 \]

### Final Answer
The sum lent out is **₹ 7200**.

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