A person borrows Rs. 5000 for 2 years at 4% per annum simple interest . He immediately lends it to another person at `6 (1)/(4)`% per annum simple interest for 2 years. Find his gain in this transaction.

Answer

To solve the problem step by step, we will calculate the simple interest for both the borrowing and lending transactions and then find the gain from the transaction.

### Step 1: Calculate the Simple Interest on the Borrowed Amount
The formula for Simple Interest (SI) is given by:

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

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

Here, the person borrows Rs. 5000 for 2 years at a rate of 4% per annum.

- \(P = 5000\)
- \(R = 4\%\)
- \(T = 2\)

Substituting the values into the formula:

\[ SI = \frac{5000 \times 4 \times 2}{100} \]

Calculating this:

\[ SI = \frac{5000 \times 8}{100} = \frac{40000}{100} = 400 \]

So, the simple interest he has to pay after 2 years is Rs. 400.

### Step 2: Calculate the Simple Interest on the Lent Amount
Now, the person lends the same amount (Rs. 5000) at a rate of \(6 \frac{1}{4}\%\) per annum for 2 years.

First, convert \(6 \frac{1}{4}\%\) into an improper fraction:

\[ 6 \frac{1}{4} = \frac{25}{4}\% \]

Now, using the same formula for Simple Interest:

- \(P = 5000\)
- \(R = \frac{25}{4}\%\)
- \(T = 2\)

Substituting these values into the formula:

\[ SI' = \frac{5000 \times \frac{25}{4} \times 2}{100} \]

Calculating this:

\[ SI' = \frac{5000 \times 25 \times 2}{4 \times 100} = \frac{250000}{400} = 625 \]

So, the simple interest he earns after 2 years is Rs. 625.

### Step 3: Calculate the Gain from the Transaction
Now, to find the gain from this transaction, we subtract the interest he has to pay from the interest he earns:

\[ \text{Gain} = SI' - SI \]

Substituting the values:

\[ \text{Gain} = 625 - 400 = 225 \]

Thus, the gain in this transaction is Rs. 225.

### Final Answer
The gain in this transaction is Rs. 225.

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