A sum of money become three times at the simple interest rate of 4% by at what percent rate will becomes it 6 time?

Options

A. `9%`
B. `10%`
C. `11%`
D. `12%`

Answer

To solve the problem step by step, we need to find out the rate at which a sum of money becomes six times the principal amount, given that it becomes three times at a simple interest rate of 4% per annum.

### Step 1: Understand the problem
We know that a certain principal amount (P) becomes three times itself (3P) at a simple interest rate of 4%. This means the simple interest earned is 2P (since 3P - P = 2P).

**Hint:** Identify the relationship between the principal, the total amount, and the simple interest.

### Step 2: Use the Simple Interest formula
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) is the simple interest
- \(P\) is the principal amount
- \(R\) is the rate of interest per annum
- \(T\) is the time in years

From the information given, we have:
\[ 2P = \frac{P \times 4 \times T}{100} \]

**Hint:** Substitute the known values into the simple interest formula.

### Step 3: Simplify the equation
We can simplify the equation:
\[ 2P = \frac{4PT}{100} \]
Dividing both sides by \(P\) (assuming \(P \neq 0\)):
\[ 2 = \frac{4T}{100} \]
Now, multiply both sides by 100:
\[ 200 = 4T \]
Dividing both sides by 4 gives:
\[ T = 50 \text{ years} \]

**Hint:** Isolate the variable to find the time period.

### Step 4: Find the new simple interest for six times
Now we need to find the rate at which the principal becomes six times itself (6P). The simple interest in this case will be:
\[ SI = 6P - P = 5P \]

**Hint:** Calculate the new simple interest based on the new total amount.

### Step 5: Set up the equation for the new scenario
Using the simple interest formula again:
\[ 5P = \frac{P \times R \times 50}{100} \]
Dividing both sides by \(P\):
\[ 5 = \frac{R \times 50}{100} \]
Multiplying both sides by 100 gives:
\[ 500 = 50R \]
Dividing both sides by 50 gives:
\[ R = 10\% \]

**Hint:** Solve for the rate by isolating \(R\).

### Conclusion
The rate at which the sum of money will become six times the principal is **10% per annum**.

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