On a certain sum of money lent out at `16%` p.a. the difference between the compound interest for 1 year, payable half yearly, and the simple interest for 1 year is `Rs. 56`. The sum is

Options

A. `Rs. 1080`
B. `Rs. 7805`
C. `Rs. 8750 `
D. `Rs. 5780 `

Answer

To solve the problem step by step, we will first find the simple interest (SI) and compound interest (CI) for the given conditions and then use the information provided to find the sum of money.

### Step 1: Calculate Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount (the sum of money)
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

Given:
- \(R = 16\%\)
- \(T = 1\) year

So, the Simple Interest for 1 year is:
\[ SI = \frac{P \times 16 \times 1}{100} = \frac{16P}{100} = 0.16P \]

### Step 2: Calculate Compound Interest (CI)
Since the interest is compounded half-yearly, we need to adjust the rate and time:
- Half-yearly rate = \( \frac{16\%}{2} = 8\%\)
- Number of compounding periods in 1 year = 2

The formula for Compound Interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^n - P \]
Where:
- \(n\) = number of compounding periods

Substituting the values:
\[ CI = P \left(1 + \frac{8}{100}\right)^2 - P \]
Calculating \( \left(1 + \frac{8}{100}\right)^2 \):
\[ = P \left(1.08\right)^2 - P = P(1.1664) - P = 0.1664P \]

### Step 3: Find the Difference between CI and SI
According to the problem, the difference between the Compound Interest and Simple Interest is given as Rs. 56:
\[ CI - SI = 56 \]
Substituting the values we found:
\[ 0.1664P - 0.16P = 56 \]
Simplifying:
\[ 0.0064P = 56 \]

### Step 4: Solve for Principal (P)
Now, we can solve for \(P\):
\[ P = \frac{56}{0.0064} \]
Calculating:
\[ P = 8750 \]

### Final Answer
The sum of money is Rs. 8750.

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