The simple interest on certain sum at `5%` for 9 month is 7 greater than the simple interest on the same sum @ `3%` for 14 months. What is the sum of interest in both the cases (i.e., total sum of interest)?

Options

A. ₹ 203
B. ₹ 290
C. ₹ 120
D. ₹ 330

Answer

To solve the problem, we need to find the total sum of interest earned at two different rates over specified time periods. Let's break it down step by step.

### Step 1: Define the Variables
Let the principal amount (the sum) be \( P \).

### Step 2: Calculate Simple Interest for the First Case
The formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( R \) is the rate of interest per annum and \( T \) is the time in years.

For the first case:
- Rate \( R_1 = 5\% \)
- Time \( T_1 = 9 \text{ months} = \frac{9}{12} \text{ years} = \frac{3}{4} \text{ years} \)

Substituting these values into the formula:
\[ SI_1 = \frac{P \times 5 \times \frac{3}{4}}{100} = \frac{15P}{400} = \frac{3P}{80} \]

### Step 3: Calculate Simple Interest for the Second Case
For the second case:
- Rate \( R_2 = 3\% \)
- Time \( T_2 = 14 \text{ months} = \frac{14}{12} \text{ years} = \frac{7}{6} \text{ years} \)

Substituting these values into the formula:
\[ SI_2 = \frac{P \times 3 \times \frac{7}{6}}{100} = \frac{21P}{600} = \frac{7P}{200} \]

### Step 4: Set Up the Equation Based on the Problem Statement
According to the problem, the simple interest at 5% for 9 months is 7 greater than the simple interest at 3% for 14 months:
\[ SI_1 = SI_2 + 7 \]
Substituting the expressions we found for \( SI_1 \) and \( SI_2 \):
\[ \frac{3P}{80} = \frac{7P}{200} + 7 \]

### Step 5: Solve for \( P \)
To eliminate the fractions, we can find a common denominator, which is 400:
\[ \frac{3P \times 5}{400} = \frac{7P \times 2}{400} + 7 \]
This simplifies to:
\[ \frac{15P}{400} = \frac{14P}{400} + 7 \]
Multiplying through by 400 to clear the denominator:
\[ 15P = 14P + 2800 \]
Subtracting \( 14P \) from both sides:
\[ P = 2800 \]

### Step 6: Calculate the Total Interest
Now that we have \( P \), we can calculate the total interest earned in both cases.

1. **Interest at 5% for 9 months:**
\[ SI_1 = \frac{3 \times 2800}{80} = \frac{8400}{80} = 105 \]

2. **Interest at 3% for 14 months:**
\[ SI_2 = \frac{7 \times 2800}{200} = \frac{19600}{200} = 98 \]

### Step 7: Total Interest
Now, add both interests:
\[ \text{Total Interest} = SI_1 + SI_2 = 105 + 98 = 203 \]

Thus, the total sum of interest in both cases is **203**.