Rs 18,600 was divided among three parts and each part was lent out at 10% per annum rate of simple interest for 2 years, 4 years and 5 years respectively. If the total amount (interest + principal) is same in all the three conditions, Find each part.

Options

A. 5600,6000, 7000
B. 6000, 7000, 5600
C. 7000,6000, 5600
D. 5600, 7000, 6000

Answer

To solve the problem of dividing Rs 18,600 into three parts such that each part, when lent out at a 10% per annum simple interest rate for different durations, yields the same total amount (principal + interest), we can follow these steps:

### Step 1: Define the Parts
Let the three parts be:
- Part A = Principal amount lent for 2 years
- Part B = Principal amount lent for 4 years
- Part C = Principal amount lent for 5 years

### Step 2: Write the Total Amount for Each Part
The total amount (Principal + Interest) for each part can be calculated using the formula for simple interest:
\[ \text{Total Amount} = \text{Principal} + \text{Simple Interest} \]
Where Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Here, \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.

For each part, we can express the total amounts as follows:
1. For Part A (2 years):
\[ \text{Total Amount A} = A + \frac{A \times 10 \times 2}{100} = A + 0.2A = 1.2A \]

2. For Part B (4 years):
\[ \text{Total Amount B} = B + \frac{B \times 10 \times 4}{100} = B + 0.4B = 1.4B \]

3. For Part C (5 years):
\[ \text{Total Amount C} = C + \frac{C \times 10 \times 5}{100} = C + 0.5C = 1.5C \]

### Step 3: Set Up the Equation
Since the total amounts are the same, we can set the equations equal to each other:
\[ 1.2A = 1.4B = 1.5C \]

### Step 4: Express in Terms of a Common Variable
Let’s express \( A \), \( B \), and \( C \) in terms of a common variable \( k \):
\[ 1.2A = 1.4B \implies A = \frac{1.4}{1.2}B = \frac{7}{6}B \]
\[ 1.4B = 1.5C \implies B = \frac{1.5}{1.4}C = \frac{15}{14}C \]

Now substituting \( B \) in terms of \( C \) into the equation for \( A \):
\[ A = \frac{7}{6} \left(\frac{15}{14}C\right) = \frac{105}{84}C = \frac{5}{4}C \]

### Step 5: Express All Parts in Terms of C
Now we have:
- \( A = \frac{5}{4}C \)
- \( B = \frac{15}{14}C \)
- \( C = C \)

### Step 6: Substitute into the Total Amount
The total amount of all parts is given as Rs 18,600:
\[ A + B + C = 18600 \]
Substituting the values of \( A \) and \( B \):
\[ \frac{5}{4}C + \frac{15}{14}C + C = 18600 \]

### Step 7: Find a Common Denominator and Solve
The common denominator for 4, 14, and 1 is 28:
\[ \frac{35}{28}C + \frac{30}{28}C + \frac{28}{28}C = 18600 \]
Combining the fractions:
\[ \frac{93}{28}C = 18600 \]
Now, solve for \( C \):
\[ C = 18600 \times \frac{28}{93} = 5600 \]

### Step 8: Find A and B
Now substitute \( C \) back to find \( A \) and \( B \):
\[ B = \frac{15}{14}C = \frac{15}{14} \times 5600 = 6000 \]
\[ A = \frac{5}{4}C = \frac{5}{4} \times 5600 = 7000 \]

### Final Parts
Thus, the three parts are:
- Part A = Rs 7,000
- Part B = Rs 6,000
- Part C = Rs 5,600