A sum of Rs. 65000 was divided into 3 parts so that they yielded the same interest when they were lent for 2,3 and 4 years at 8% simple interest at the end of these periods. The ratio between these parts is....

Options

A. `2:3:4`
B. `4:3:2`
C. `6:4:3`
D. `3:4:6`

Answer

To solve the problem, we need to find the ratio of three parts of a sum of Rs. 65,000, such that when these parts are lent out for 2, 3, and 4 years at 8% simple interest, they yield the same interest. Let's denote the three parts as A, B, and C.

### Step-by-Step Solution:

1. **Understand the Simple Interest Formula**:
The formula for simple interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

2. **Set Up the Interest Equations**:
For the three parts A, B, and C, we can write the interest equations for each part:
- Interest from A for 2 years:
\[ \text{SI}_A = \frac{A \times 8 \times 2}{100} = \frac{16A}{100} = 0.16A \]
- Interest from B for 3 years:
\[ \text{SI}_B = \frac{B \times 8 \times 3}{100} = \frac{24B}{100} = 0.24B \]
- Interest from C for 4 years:
\[ \text{SI}_C = \frac{C \times 8 \times 4}{100} = \frac{32C}{100} = 0.32C \]

3. **Set the Interests Equal**:
Since the interests are the same, we can set up the following equations:
\[ 0.16A = 0.24B = 0.32C \]

4. **Express A, B, and C in Terms of a Common Variable**:
Let’s express A, B, and C in terms of a common variable \( k \):
- From \( 0.16A = 0.24B \):
\[ A = \frac{0.24}{0.16}B = \frac{24}{16}B = \frac{3}{2}B \quad \text{(1)} \]
- From \( 0.16A = 0.32C \):
\[ A = \frac{0.32}{0.16}C = \frac{32}{16}C = 2C \quad \text{(2)} \]

5. **Substituting Values**:
From equation (1), we can express B in terms of A:
\[ B = \frac{2}{3}A \]
From equation (2), we can express C in terms of A:
\[ C = \frac{1}{2}A \]

6. **Substituting into the Total Sum**:
Now, we know that:
\[ A + B + C = 65000 \]
Substituting the values of B and C in terms of A:
\[ A + \frac{2}{3}A + \frac{1}{2}A = 65000 \]
To combine these, we need a common denominator, which is 6:
\[ A + \frac{4}{6}A + \frac{3}{6}A = 65000 \]
This simplifies to:
\[ \frac{6A + 4A + 3A}{6} = 65000 \]
\[ \frac{13A}{6} = 65000 \]
Multiplying both sides by 6:
\[ 13A = 390000 \]
Dividing by 13:
\[ A = 30000 \]

7. **Finding B and C**:
Now, substituting back to find B and C:
\[ B = \frac{2}{3}A = \frac{2}{3} \times 30000 = 20000 \]
\[ C = \frac{1}{2}A = \frac{1}{2} \times 30000 = 15000 \]

8. **Finding the Ratio**:
The parts A, B, and C are:
- A = 30000
- B = 20000
- C = 15000
The ratio of A : B : C is:
\[ 30000 : 20000 : 15000 = 6 : 4 : 3 \]

### Final Answer:
The ratio between the parts A, B, and C is **6 : 4 : 3**.