The difference between CI and SI for 3 yr at the rate of 20% pa is ? 152. What is the principal lent ?

Answer

To find the principal amount given the difference between compound interest (CI) and simple interest (SI) for 3 years at a rate of 20% per annum is 152, we can follow these steps:

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

- **Compound Interest (CI)** can be calculated using the formula:
\[ CI = A - P \]
where \( A \) is the total amount after interest, calculated as:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

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

Substituting these values into the SI formula:
\[ SI = \frac{P \times 20 \times 3}{100} = \frac{60P}{100} = \frac{3P}{5} \]

### Step 3: Calculate Compound Interest for 3 years
Using the CI formula:
\[ A = P \left(1 + \frac{20}{100}\right)^3 = P \left(1 + 0.2\right)^3 = P \left(1.2\right)^3 \]
Calculating \( (1.2)^3 \):
\[ (1.2)^3 = 1.728 \]
Thus,
\[ A = 1.728P \]
Now, substituting into the CI formula:
\[ CI = A - P = 1.728P - P = 0.728P \]

### Step 4: Find the difference between CI and SI
We know from the problem statement:
\[ CI - SI = 152 \]
Substituting the values we calculated:
\[ 0.728P - \frac{3P}{5} = 152 \]

### Step 5: Convert \(\frac{3P}{5}\) to a decimal
To combine the terms, convert \(\frac{3P}{5}\) to a decimal:
\[ \frac{3P}{5} = 0.6P \]
Now the equation becomes:
\[ 0.728P - 0.6P = 152 \]
This simplifies to:
\[ 0.128P = 152 \]

### Step 6: Solve for P
Now, we can solve for \( P \):
\[ P = \frac{152}{0.128} \]
Calculating this gives:
\[ P = 1187.5 \]

### Final Answer
The principal amount \( P \) is:
\[ \boxed{1187.5} \]