Pratibha invests an amount of ₹ 15,860 in 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 step by step, we will determine the amounts invested for A, B, and C based on the simple interest they receive after their respective investment periods.

### Step 1: Understand the Problem
Pratibha invests a total amount of ₹15,860 among her three daughters A, B, and C. They receive the same interest after 2, 3, and 4 years respectively at a rate of 5% per annum.

### Step 2: Set Up the Formula for Simple Interest
The formula for Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (the amount invested)
- \( R \) = Rate of interest (5% in this case)
- \( T \) = Time (in years)

### Step 3: Calculate the Interest for Each Daughter
Let the amounts invested for A, B, and C be \( A \), \( B \), and \( C \) respectively.

- For daughter A (2 years):
\[ \text{SI}_A = \frac{A \times 5 \times 2}{100} = \frac{10A}{100} = \frac{A}{10} \]

- For daughter B (3 years):
\[ \text{SI}_B = \frac{B \times 5 \times 3}{100} = \frac{15B}{100} = \frac{3B}{20} \]

- For daughter C (4 years):
\[ \text{SI}_C = \frac{C \times 5 \times 4}{100} = \frac{20C}{100} = \frac{C}{5} \]

### Step 4: Set the Interests Equal
Since all three daughters receive the same interest:
\[ \frac{A}{10} = \frac{3B}{20} = \frac{C}{5} \]

### Step 5: Express A, B, and C in Terms of a Common Variable
Let’s denote the common interest amount as \( k \):
1. From \( \frac{A}{10} = k \) ⇒ \( A = 10k \)
2. From \( \frac{3B}{20} = k \) ⇒ \( B = \frac{20k}{3} \)
3. From \( \frac{C}{5} = k \) ⇒ \( C = 5k \)

### Step 6: Find the Total Amount Invested
Now, we can sum up the amounts invested:
\[ A + B + C = 10k + \frac{20k}{3} + 5k \]

To combine these, we need a common denominator:
- The common denominator for \( 1 \), \( \frac{20}{3} \), and \( 1 \) is \( 3 \):
\[ A + B + C = \frac{30k}{3} + \frac{20k}{3} + \frac{15k}{3} = \frac{65k}{3} \]

### Step 7: Set the Total Equal to ₹15,860
Now, we set this equal to the total investment:
\[ \frac{65k}{3} = 15860 \]

### Step 8: Solve for k
Multiplying both sides by 3:
\[ 65k = 15860 \times 3 \]
\[ 65k = 47580 \]

Now, divide by 65:
\[ k = \frac{47580}{65} = 732 \]

### Step 9: Calculate A, B, and C
Now substitute \( k \) back to find A, B, and C:
- \( A = 10k = 10 \times 732 = 7320 \)
- \( B = \frac{20k}{3} = \frac{20 \times 732}{3} = 4880 \)
- \( C = 5k = 5 \times 732 = 3660 \)

### Step 10: Find the Ratio
Now we have:
- \( A : B : C = 7320 : 4880 : 3660 \)

To simplify the ratio, we can divide each term by the greatest common divisor (GCD):
- The GCD of 7320, 4880, and 3660 is 2440.

Thus, the simplified ratio is:
\[ \frac{7320}{2440} : \frac{4880}{2440} : \frac{3660}{2440} = 3 : 2 : 1.5 \]

### Step 11: Convert to Whole Numbers
To express this in whole numbers, multiply through by 2:
\[ 6 : 4 : 3 \]

### Conclusion
The ratio of the amounts invested among A, B, and C is:
**6 : 4 : 3**