In a certain time, a sum becomes 3 times at the rate of 5% per annum. At what rate of interest, the same sum becomes 6 times in same duration?

Answer

To solve the problem step by step, we will use the concept of simple interest and the relationship between the principal amount, rate of interest, time, and the final amount.

### Step 1: Understand the given information
We know that a sum of money becomes 3 times itself at a rate of 5% per annum in a certain time period.

Let the principal amount be \( P \).
Then, the final amount \( A \) after the time period is:
\[ A = 3P \]

### Step 2: Use the formula for Simple Interest
The formula for the final amount in terms of principal, rate, and time is:
\[ A = P + SI \]
where \( SI \) (Simple Interest) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Here, \( R \) is the rate of interest and \( T \) is the time in years.

### Step 3: Set up the equation for the first scenario
From the information given:
\[ 3P = P + \frac{P \times 5 \times T}{100} \]

### Step 4: Simplify the equation
Subtract \( P \) from both sides:
\[ 3P - P = \frac{P \times 5 \times T}{100} \]
\[ 2P = \frac{P \times 5 \times T}{100} \]

### Step 5: Cancel \( P \) from both sides
Assuming \( P \neq 0 \):
\[ 2 = \frac{5T}{100} \]

### Step 6: Solve for \( T \)
Multiply both sides by 100:
\[ 200 = 5T \]
Now divide by 5:
\[ T = 40 \text{ years} \]

### Step 7: Set up the equation for the second scenario
Now we need to find the rate \( R_2 \) at which the same sum becomes 6 times itself in the same time period \( T \):
\[ A = 6P \]
Using the same formula:
\[ 6P = P + \frac{P \times R_2 \times T}{100} \]

### Step 8: Simplify the equation for the second scenario
Subtract \( P \) from both sides:
\[ 6P - P = \frac{P \times R_2 \times T}{100} \]
\[ 5P = \frac{P \times R_2 \times T}{100} \]

### Step 9: Cancel \( P \) from both sides
Assuming \( P \neq 0 \):
\[ 5 = \frac{R_2 \times T}{100} \]

### Step 10: Substitute \( T \) into the equation
Now substitute \( T = 40 \):
\[ 5 = \frac{R_2 \times 40}{100} \]

### Step 11: Solve for \( R_2 \)
Multiply both sides by 100:
\[ 500 = R_2 \times 40 \]
Now divide by 40:
\[ R_2 = \frac{500}{40} = 12.5\% \]

### Conclusion
The rate of interest at which the same sum becomes 6 times in the same duration is \( 12.5\% \).

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