Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. She received Rs 1,860 as annual interest. However, had she interchanged the amount of investments in the two schemes, she would have received Rs 20 more as annual interest. How much money did she invest in each scheme?

Answer

To solve the problem, we need to set up a system of linear equations based on the information given.

### Step 1: Define the Variables
Let:
- \( x \) = amount invested in scheme A (at 8% per annum)
- \( y \) = amount invested in scheme B (at 9% per annum)

### Step 2: Set Up the Equations
From the problem, we have two pieces of information that can be translated into equations:

1. The total interest received from both investments is Rs 1860:
\[ 0.08x + 0.09y = 1860 \]

2. If the amounts invested in the two schemes were interchanged, the interest would have been Rs 20 more:
\[ 0.09x + 0.08y = 1860 + 20 = 1880 \]

### Step 3: Write the System of Equations
Now we have the following system of equations:
1. \( 0.08x + 0.09y = 1860 \) (Equation 1)
2. \( 0.09x + 0.08y = 1880 \) (Equation 2)

### Step 4: Multiply to Eliminate Decimals
To eliminate decimals, we can multiply both equations by 100:
1. \( 8x + 9y = 186000 \) (Equation 1)
2. \( 9x + 8y = 188000 \) (Equation 2)

### Step 5: Solve the System of Equations
We can solve these equations using the elimination method. Let's multiply Equation 1 by 9 and Equation 2 by 8:
1. \( 72x + 81y = 1674000 \) (Equation 3)
2. \( 72x + 64y = 1504000 \) (Equation 4)

Now, subtract Equation 4 from Equation 3:
\[ (72x + 81y) - (72x + 64y) = 1674000 - 1504000 \]
This simplifies to:
\[ 17y = 170000 \]
Thus,
\[ y = \frac{170000}{17} = 10000 \]

### Step 6: Substitute to Find \( x \)
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). Let's use Equation 1:
\[ 8x + 9(10000) = 186000 \]
This simplifies to:
\[ 8x + 90000 = 186000 \]
Subtracting 90000 from both sides:
\[ 8x = 186000 - 90000 \]
\[ 8x = 96000 \]
Thus,
\[ x = \frac{96000}{8} = 12000 \]

### Final Answer
The amounts invested in each scheme are:
- Amount invested in scheme A (x) = Rs 12,000
- Amount invested in scheme B (y) = Rs 10,000