Divide ₹12000 onto two parts such that the simple interest on the first part for 2 years at 6% per annum is equal to the simple interest on the second part for 3 years at 8% per annum.

Answer

To solve the problem of dividing ₹12000 into two parts such that the simple interest on the first part for 2 years at 6% per annum is equal to the simple interest on the second part for 3 years at 8% per annum, we can follow these steps:

### Step 1: Define the two parts
Let the first part be \( P_1 \) and the second part be \( P_2 \). According to the problem, we have:
\[ P_1 + P_2 = 12000 \]

### Step 2: Write the formula for Simple Interest
The formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

### Step 3: Calculate the Simple Interest for both parts
For the first part \( P_1 \):
- Rate \( R_1 = 6\% \)
- Time \( T_1 = 2 \) years
So, the Simple Interest on \( P_1 \) is:
\[ SI_1 = \frac{P_1 \times 6 \times 2}{100} = \frac{12P_1}{100} = 0.12P_1 \]

For the second part \( P_2 \):
- Rate \( R_2 = 8\% \)
- Time \( T_2 = 3 \) years
So, the Simple Interest on \( P_2 \) is:
\[ SI_2 = \frac{P_2 \times 8 \times 3}{100} = \frac{24P_2}{100} = 0.24P_2 \]

### Step 4: Set the Simple Interests equal
According to the problem, the Simple Interest on both parts is equal:
\[ SI_1 = SI_2 \]
Substituting the expressions we found:
\[ 0.12P_1 = 0.24P_2 \]

### Step 5: Express \( P_2 \) in terms of \( P_1 \)
From the equation \( 0.12P_1 = 0.24P_2 \), we can express \( P_2 \) in terms of \( P_1 \):
\[ P_2 = \frac{0.12P_1}{0.24} = \frac{1}{2}P_1 \]

### Step 6: Substitute \( P_2 \) back into the total amount equation
Now substitute \( P_2 \) in the equation \( P_1 + P_2 = 12000 \):
\[ P_1 + \frac{1}{2}P_1 = 12000 \]
This simplifies to:
\[ \frac{3}{2}P_1 = 12000 \]

### Step 7: Solve for \( P_1 \)
To find \( P_1 \):
\[ P_1 = 12000 \times \frac{2}{3} = 8000 \]

### Step 8: Find \( P_2 \)
Now substitute \( P_1 \) back to find \( P_2 \):
\[ P_2 = 12000 - P_1 = 12000 - 8000 = 4000 \]

### Final Result
Thus, the two parts are:
- First part \( P_1 = ₹8000 \)
- Second part \( P_2 = ₹4000 \)