Sunita divide a sum of money in 3 parts and deposit these in 3 different bank. The ratio the rate of interest is 4:6:3 of the banks respectively. If he receive the same interest from these banks after 9 months. Find the ratio of deposited sum.

Options

A. a)`3:2:4`
B. b)`4:5:6`
C. c)`6: 3:8`
D. d)`5 : 4 :6`

Answer

To solve the problem step by step, we need to find the ratio of the deposited sums in three different banks based on the given rates of interest and the fact that the interest received from each bank is the same after 9 months.

### Step 1: Understand the problem
Sunita divides a sum of money into three parts and deposits them in three banks with interest rates in the ratio of 4:6:3. We need to find the ratio of the deposited sums.

### Step 2: Assign variables
Let the amounts deposited in the three banks be \( A \), \( B \), and \( C \). The rates of interest for these banks are given as:
- Bank 1: \( R_1 = 4\% \)
- Bank 2: \( R_2 = 6\% \)
- Bank 3: \( R_3 = 3\% \)

### Step 3: Use the formula for Simple Interest
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

### Step 4: Calculate the interest for each bank
Since the time is 9 months, we convert this to years:
\[ T = \frac{9}{12} = \frac{3}{4} \text{ years} \]

Now, we can express the interest from each bank:
1. Interest from Bank 1:
\[ SI_1 = \frac{A \times 4 \times \frac{3}{4}}{100} = \frac{3A}{100} \]
2. Interest from Bank 2:
\[ SI_2 = \frac{B \times 6 \times \frac{3}{4}}{100} = \frac{9B}{200} \]
3. Interest from Bank 3:
\[ SI_3 = \frac{C \times 3 \times \frac{3}{4}}{100} = \frac{9C}{400} \]

### Step 5: Set the interests equal
Since the interest received from all three banks is the same, we set the equations equal to each other:
\[ \frac{3A}{100} = \frac{9B}{200} = \frac{9C}{400} \]

### Step 6: Solve for ratios
1. From \( \frac{3A}{100} = \frac{9B}{200} \):
\[ 3A \times 200 = 9B \times 100 \implies 600A = 900B \implies \frac{A}{B} = \frac{900}{600} = \frac{3}{2} \]
So, \( A = \frac{3}{2}B \).

2. From \( \frac{3A}{100} = \frac{9C}{400} \):
\[ 3A \times 400 = 9C \times 100 \implies 1200A = 900C \implies \frac{A}{C} = \frac{900}{1200} = \frac{3}{4} \]
So, \( A = \frac{3}{4}C \).

### Step 7: Express B and C in terms of A
From \( A = \frac{3}{2}B \):
\[ B = \frac{2}{3}A \]
From \( A = \frac{3}{4}C \):
\[ C = \frac{4}{3}A \]

### Step 8: Find the ratio A:B:C
Now we can express the ratio \( A:B:C \):
\[ A : B : C = A : \frac{2}{3}A : \frac{4}{3}A \]
This simplifies to:
\[ 1 : \frac{2}{3} : \frac{4}{3} \]
To eliminate the fractions, we multiply through by 3:
\[ 3 : 2 : 4 \]

### Final Answer
Thus, the ratio of the deposited sums \( A : B : C \) is \( 3 : 2 : 4 \).

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