Rs. 12800 is payable after 3 years. If is to be paid in 3 installments each year. First installment is half of the second & one-third of the third installment. Find each installment if rate of interest is 10% per annum.

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Define the Installments
Let the first installment be \( M \). According to the problem:
- The second installment is \( 2M \).
- The third installment is \( 3M \).

### Step 2: Calculate the Amount Due After Each Year
The total amount payable after 3 years is Rs. 12,800, and the interest rate is 10% per annum.

#### Year 1:
- Amount due at the end of Year 1 = Principal + Interest - First Installment
- Interest for Year 1 = \( 10\% \) of Rs. 12,800 = \( \frac{10}{100} \times 12800 = 1280 \)
- Amount due = \( 12800 + 1280 - M = 14080 - M \)

#### Year 2:
- Interest for Year 2 = \( 10\% \) of (Amount due at end of Year 1) = \( 10\% \) of \( (14080 - M) \)
- Interest = \( \frac{10}{100} \times (14080 - M) = 1408 - 0.1M \)
- Amount due at the end of Year 2 = Amount due from Year 1 + Interest - Second Installment
- Amount due = \( (14080 - M) + (1408 - 0.1M) - 2M = 15488 - 3.1M \)

#### Year 3:
- Interest for Year 3 = \( 10\% \) of (Amount due at end of Year 2) = \( 10\% \) of \( (15488 - 3.1M) \)
- Interest = \( \frac{10}{100} \times (15488 - 3.1M) = 1548.8 - 0.31M \)
- Amount due at the end of Year 3 = Amount due from Year 2 + Interest - Third Installment
- Amount due = \( (15488 - 3.1M) + (1548.8 - 0.31M) - 3M = 14036.8 - 6.41M \)

### Step 3: Set Up the Equation
At the end of Year 3, the amount due must equal the third installment:
\[ 14036.8 - 6.41M = 0 \]

### Step 4: Solve for M
Rearranging the equation gives:
\[ 6.41M = 14036.8 \]
\[ M = \frac{14036.8}{6.41} \]
\[ M \approx 2183.25 \]

### Step 5: Calculate Each Installment
- First Installment: \( M \approx 2183.25 \)
- Second Installment: \( 2M \approx 2 \times 2183.25 \approx 4366.50 \)
- Third Installment: \( 3M \approx 3 \times 2183.25 \approx 6549.75 \)

### Final Answer
- First Installment: Rs. 2183.25
- Second Installment: Rs. 4366.50
- Third Installment: Rs. 6549.75

---