Pratibha an amount of Rs. 15,860 in the the names of her three daughters A,B and C in such a way that they get the same interest after 2,3 and 4 years respectively. If the rate of simple interest is 5% p.a. , then the ratio of the amounts invested among A,B and C will be :

Options

A. `5:10:12`
B. `1/(10) : 1/ (15): 1/(20)`
C. `6 :7:8`
D. `6:5:4`

Answer

To solve the problem, we need to find the ratio of the amounts invested by Pratibha for her daughters A, B, and C, such that they receive the same interest after 2, 3, and 4 years, respectively, at a rate of 5% per annum.

### Step-by-Step Solution:

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

2. **Set Up the Interest for Each Daughter**:
- For daughter A (time = 2 years):
\[ \text{SI}_A = \frac{P_A \times 5 \times 2}{100} = \frac{10P_A}{100} = \frac{P_A}{10} \]
- For daughter B (time = 3 years):
\[ \text{SI}_B = \frac{P_B \times 5 \times 3}{100} = \frac{15P_B}{100} = \frac{P_B}{15} \]
- For daughter C (time = 4 years):
\[ \text{SI}_C = \frac{P_C \times 5 \times 4}{100} = \frac{20P_C}{100} = \frac{P_C}{20} \]

3. **Set the Interests Equal**:
Since the interest received by each daughter is the same, we can set the equations equal to each other:
\[ \frac{P_A}{10} = \frac{P_B}{15} = \frac{P_C}{20} = k \quad (\text{where } k \text{ is a constant}) \]

4. **Express Each Principal in Terms of k**:
From the equations, we can express \( P_A \), \( P_B \), and \( P_C \) in terms of \( k \):
- From \( \frac{P_A}{10} = k \), we get:
\[ P_A = 10k \]
- From \( \frac{P_B}{15} = k \), we get:
\[ P_B = 15k \]
- From \( \frac{P_C}{20} = k \), we get:
\[ P_C = 20k \]

5. **Find the Ratio of the Principals**:
Now we can find the ratio of the amounts invested:
\[ P_A : P_B : P_C = 10k : 15k : 20k \]
Dividing each term by \( k \):
\[ = 10 : 15 : 20 \]

6. **Simplify the Ratio**:
To simplify the ratio, we can divide each term by 5:
\[ = 2 : 3 : 4 \]

### Final Answer:
The ratio of the amounts invested among A, B, and C is:
\[ \boxed{2 : 3 : 4} \]