Manish borrowed a sum of 1150 from Anil at the simple rate of 6% per annum for 3 years. He then added some more money to the borrowed sum of lent it to Sunil for the same time at 9% per annum at simple interest. If Manish gains 274.95 by way of interest on the borrowed sum as well as his own amount from the whole transaction, then what is the sum lent by him to Sunil?

Options

A. 1290
B. 1785
C. 1285
D. 1200

Answer

To solve the problem step by step, we need to find out how much money Manish lent to Sunil after borrowing a sum from Anil. Let's break it down:

### Step 1: Calculate the interest Manish has to pay Anil.
Manish borrowed a sum of 1150 at a simple interest rate of 6% per annum for 3 years.

The formula for simple interest is:
\[ \text{Interest} = \frac{P \times R \times T}{100} \]
where:
- \( P \) = principal amount (1150)
- \( R \) = rate of interest (6)
- \( T \) = time in years (3)

Substituting the values:
\[ \text{Interest} = \frac{1150 \times 6 \times 3}{100} = \frac{20700}{100} = 207 \]

### Step 2: Calculate the total amount Manish has to pay back to Anil.
The total amount to be paid back to Anil after 3 years is:
\[ \text{Total Amount} = \text{Principal} + \text{Interest} = 1150 + 207 = 1357 \]

### Step 3: Set up the equation for the total gain.
Manish gains a total of 274.95 from the entire transaction. This gain comes from the interest he earns from lending money to Sunil minus the interest he has to pay Anil.

Let \( x \) be the additional amount Manish added to the borrowed sum to lend to Sunil. Therefore, the total amount lent to Sunil is \( 1150 + x \).

The interest earned from lending to Sunil at 9% per annum for 3 years is:
\[ \text{Interest from Sunil} = \frac{(1150 + x) \times 9 \times 3}{100} = \frac{(1150 + x) \times 27}{100} \]

### Step 4: Set up the equation for the gain.
According to the problem, Manish's gain can be expressed as:
\[ \text{Interest from Sunil} - \text{Interest to Anil} = 274.95 \]
Substituting the values we calculated:
\[ \frac{(1150 + x) \times 27}{100} - 207 = 274.95 \]

### Step 5: Solve the equation.
Rearranging the equation gives:
\[ \frac{(1150 + x) \times 27}{100} = 274.95 + 207 \]
\[ \frac{(1150 + x) \times 27}{100} = 481.95 \]
Multiplying both sides by 100:
\[ (1150 + x) \times 27 = 48195 \]
Dividing both sides by 27:
\[ 1150 + x = \frac{48195}{27} = 1785 \]
Now, solving for \( x \):
\[ x = 1785 - 1150 = 635 \]

### Step 6: Calculate the total sum lent to Sunil.
The total amount lent to Sunil is:
\[ 1150 + x = 1150 + 635 = 1785 \]

### Final Answer:
The sum lent by Manish to Sunil is **1785**.

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