What annual instalment will discharge a debt of Rs 6450 due in 4 years at 5 % simple interset ? (5 प्रतिशत साधारण ब्याज पर 4 वर्षों में देय 6450 रुपये के ऋण को कौन सी वार्षिक किस्त से चुकाया जाएगा?)

Options

A. Rs 1500
B. Rs 1835
C. Rs 1935
D. Rs 1950

Answer

To solve the problem of finding the annual installment that will discharge a debt of Rs 6450 due in 4 years at 5% simple interest, we can follow these steps:

### Step 1: Understand the Problem
We need to find the annual installment (let's denote it as \( x \)) that will pay off a total debt of Rs 6450 over 4 years with a simple interest rate of 5%.

### Step 2: Set Up the Simple Interest Formula
The formula for simple interest is:
\[ \text{Simple Interest} (SI) = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

### Step 3: Calculate the Total Amount for Each Year
For each year, the installment \( x \) incurs interest based on how long it has been outstanding.

1. **First Year**:
- Principal = \( x \)
- Interest = \( \frac{x \times 5 \times 1}{100} = \frac{5x}{100} = \frac{x}{20} \)

2. **Second Year**:
- Principal = \( x \)
- Interest = \( \frac{x \times 5 \times 2}{100} = \frac{10x}{100} = \frac{x}{10} \)

3. **Third Year**:
- Principal = \( x \)
- Interest = \( \frac{x \times 5 \times 3}{100} = \frac{15x}{100} = \frac{3x}{20} \)

4. **Fourth Year**:
- Principal = \( x \)
- Interest = \( \frac{x \times 5 \times 4}{100} = \frac{20x}{100} = \frac{x}{5} \)

### Step 4: Sum Up All Installments and Interest
The total amount paid after 4 years will be:
\[ \text{Total Amount} = x + \left(\frac{x}{20}\right) + x + \left(\frac{x}{10}\right) + x + \left(\frac{3x}{20}\right) + x + \left(\frac{x}{5}\right) \]
This simplifies to:
\[ = 4x + \left(\frac{x}{20} + \frac{x}{10} + \frac{3x}{20} + \frac{x}{5}\right) \]

### Step 5: Find a Common Denominator
The common denominator for \( 20, 10, 20, 5 \) is \( 20 \):
\[ = 4x + \left(\frac{x + 2x + 3x + 4x}{20}\right) \]
\[ = 4x + \left(\frac{10x}{20}\right) = 4x + \frac{x}{2} = 4x + 0.5x = 4.5x \]

### Step 6: Set the Equation Equal to the Total Debt
Now we set the total amount equal to the debt:
\[ 4.5x = 6450 \]

### Step 7: Solve for \( x \)
To find \( x \):
\[ x = \frac{6450}{4.5} \]
Calculating this gives:
\[ x = 1433.33 \]

### Conclusion
The annual installment that will discharge the debt of Rs 6450 in 4 years at 5% simple interest is approximately Rs 1433.33.

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