Two equal sums of money are deposited in two banks at the rate of 15% per annum for 3.5 years and 5 years respectively. If the difference between their simple interests is Rs 144, find the sum.

Options

A. 460
B. 604
C. 640
D. 406

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI):

\[ \text{SI} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount (the sum of money deposited)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 1: Define the Principal
Let the principal amount deposited in each bank be \( x \).

### Step 2: Calculate Simple Interest for the First Bank
For the first bank, the rate of interest is 15% and the time is 3.5 years.

Using the formula:
\[ \text{SI}_1 = \frac{x \times 15 \times 3.5}{100} \]

### Step 3: Calculate Simple Interest for the Second Bank
For the second bank, the rate of interest is also 15% but the time is 5 years.

Using the formula:
\[ \text{SI}_2 = \frac{x \times 15 \times 5}{100} \]

### Step 4: Find the Difference in Simple Interests
According to the problem, the difference between the two simple interests is Rs 144. Therefore, we can set up the equation:

\[ \text{SI}_2 - \text{SI}_1 = 144 \]

Substituting the values we calculated:
\[ \frac{x \times 15 \times 5}{100} - \frac{x \times 15 \times 3.5}{100} = 144 \]

### Step 5: Simplify the Equation
Factor out \( \frac{x \times 15}{100} \):
\[ \frac{x \times 15}{100} (5 - 3.5) = 144 \]

Calculating \( 5 - 3.5 \):
\[ 5 - 3.5 = 1.5 \]

So the equation becomes:
\[ \frac{x \times 15}{100} \times 1.5 = 144 \]

### Step 6: Solve for \( x \)
Now simplify the equation:
\[ \frac{15 \times 1.5}{100} x = 144 \]

Calculating \( 15 \times 1.5 \):
\[ 15 \times 1.5 = 22.5 \]

Thus, the equation is:
\[ \frac{22.5}{100} x = 144 \]

To isolate \( x \), multiply both sides by \( \frac{100}{22.5} \):
\[ x = 144 \times \frac{100}{22.5} \]

Calculating \( \frac{100}{22.5} \):
\[ \frac{100}{22.5} = \frac{1000}{225} = \frac{40}{9} \]

So:
\[ x = 144 \times \frac{40}{9} \]

Calculating \( 144 \times \frac{40}{9} \):
\[ x = \frac{5760}{9} = 640 \]

### Final Answer
The sum of money deposited in each bank is Rs 640.

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