If a sum of Rs. 13040 is to paid back in two equal annual installments at `3(3)/(4)`% per annum, what is the amount of each installment?

Answer

To solve the problem of finding the amount of each installment when a sum of Rs. 13040 is to be paid back in two equal annual installments at a rate of 3(3/4)% per annum, we can follow these steps:

### Step 1: Convert the interest rate to a decimal
The interest rate given is 3(3/4)%. First, we need to convert this mixed fraction into an improper fraction and then into a decimal.

\[ 3(3/4) = 3 + \frac{3}{4} = \frac{12}{4} + \frac{3}{4} = \frac{15}{4} \text{ percent} \]

Now, convert this to a decimal:

\[ \frac{15}{4} \div 100 = \frac{15}{400} = 0.0375 \]

### Step 2: Set up the equation for the installments
Let the amount of each installment be \( P \). Since there are two installments, the first installment will incur interest for one year, and the second installment will incur interest for two years. The total amount paid back can be expressed as:

\[ P(1 + r) + P(1 + r)^2 = 13040 \]

Where \( r = 0.0375 \).

### Step 3: Substitute \( r \) into the equation
Substituting \( r \) into the equation gives:

\[ P(1 + 0.0375) + P(1 + 0.0375)^2 = 13040 \]

Calculating \( 1 + 0.0375 \):

\[ 1 + 0.0375 = 1.0375 \]

Now, calculate \( (1.0375)^2 \):

\[ (1.0375)^2 = 1.0375 \times 1.0375 = 1.075140625 \]

### Step 4: Substitute back into the equation
Now substitute back into the equation:

\[ P(1.0375) + P(1.075140625) = 13040 \]

Combine the terms:

\[ P(1.0375 + 1.075140625) = 13040 \]

Calculating \( 1.0375 + 1.075140625 \):

\[ 1.0375 + 1.075140625 = 2.112640625 \]

So the equation becomes:

\[ P(2.112640625) = 13040 \]

### Step 5: Solve for \( P \)
Now, solve for \( P \):

\[ P = \frac{13040}{2.112640625} \]

Calculating the value:

\[ P \approx 6175 \]

### Conclusion
The amount of each installment is approximately Rs. 6175.

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