The difference between Cl and Sl for 3 years @ 20% p.a. is Rs. 152 . What is the principal lent in each case ?

Options

A. 1200
B. 155
C. 1187.5
D. none of these

Answer

To solve the problem of finding the principal amount when the difference between Compound Interest (CI) and Simple Interest (SI) for 3 years at a rate of 20% per annum is Rs. 152, we can follow these steps:

### Step 1: Understand the formulas for CI and SI
- The formula for **Simple Interest (SI)** is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.

- The formula for **Compound Interest (CI)** is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

### Step 2: Calculate Simple Interest for 3 years
Given:
- Rate \( R = 20\% \)
- Time \( T = 3 \) years

Using the SI formula:
\[ SI = \frac{P \times 20 \times 3}{100} = \frac{60P}{100} = 0.6P \]

### Step 3: Calculate Compound Interest for 3 years
Using the CI formula:
\[ CI = P \left(1 + \frac{20}{100}\right)^3 - P \]
Calculating \( \left(1 + \frac{20}{100}\right)^3 \):
\[ = P \left(1.2\right)^3 - P = P \left(1.728\right) - P = 1.728P - P = 0.728P \]

### Step 4: Find the difference between CI and SI
According to the problem:
\[ CI - SI = 152 \]
Substituting the values we found:
\[ 0.728P - 0.6P = 152 \]
This simplifies to:
\[ 0.128P = 152 \]

### Step 5: Solve for Principal \( P \)
To find \( P \):
\[ P = \frac{152}{0.128} \]
Calculating \( P \):
\[ P = 1187.5 \]

### Final Answer
The principal lent in each case is **Rs. 1187.5**.

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