X invested an amount of Rs. 12000 at the simple interest rate of 10% per annum and another amount at the simple interest rate of 20% per annum. The total interest earned at the end of one year on the total amount invested became 14% per annum. Find the total amount invested invested at 20%.

Answer

To solve the problem step by step, we will use the concept of simple interest and the allegation method.

### Step 1: Understand the given information
X invested Rs. 12,000 at a simple interest rate of 10% per annum and another amount at a simple interest rate of 20% per annum. The total interest earned at the end of one year on the total amount invested became 14% per annum.

### Step 2: Set up the variables
Let the amount invested at 20% per annum be denoted as \( P_2 \). The total amount invested is:
\[ P_1 + P_2 = 12000 + P_2 \]

### Step 3: Calculate the interest from both investments
- Interest from the first investment (at 10%):
\[ \text{Interest}_1 = \frac{P_1 \times \text{Rate}_1 \times \text{Time}}{100} = \frac{12000 \times 10 \times 1}{100} = 1200 \]

- Interest from the second investment (at 20%):
\[ \text{Interest}_2 = \frac{P_2 \times \text{Rate}_2 \times \text{Time}}{100} = \frac{P_2 \times 20 \times 1}{100} = \frac{20P_2}{100} = 0.2P_2 \]

### Step 4: Total interest earned
The total interest earned from both investments at the end of one year is:
\[ \text{Total Interest} = \text{Interest}_1 + \text{Interest}_2 = 1200 + 0.2P_2 \]

### Step 5: Set up the equation for total interest
According to the problem, the total interest earned is 14% of the total amount invested:
\[ \text{Total Interest} = 14\% \text{ of } (12000 + P_2) \]
\[ 1200 + 0.2P_2 = \frac{14}{100} \times (12000 + P_2) \]

### Step 6: Simplify the equation
Convert the percentage to a decimal:
\[ 1200 + 0.2P_2 = 0.14 \times (12000 + P_2) \]
Expanding the right side:
\[ 1200 + 0.2P_2 = 1680 + 0.14P_2 \]

### Step 7: Rearranging the equation
Now, rearranging the equation to isolate \( P_2 \):
\[ 1200 - 1680 = 0.14P_2 - 0.2P_2 \]
\[ -480 = -0.06P_2 \]
Dividing both sides by -0.06:
\[ P_2 = \frac{480}{0.06} = 8000 \]

### Step 8: Conclusion
Thus, the total amount invested at 20% per annum is:
\[ \boxed{8000} \]