A certain sum of money becomes 4 times of itself in 30 years at a rate of simple interest. In how many years it will become double of itself at the same rate of simple interest?

Options

A. 20
B. 15
C. 10
D. 12

Answer

To solve the problem step by step, we need to find out how long it will take for a certain sum of money to double itself at the same rate of simple interest, given that it becomes four times itself in 30 years.

### Step 1: Understand the relationship between Principal, Amount, and Simple Interest
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Set up the equation for the given condition
We know that the amount becomes four times itself in 30 years. If we let the principal amount be \( P \), then:
\[ \text{Amount} = 4P \]
The Simple Interest earned in this case is:
\[ \text{SI} = \text{Amount} - \text{Principal} = 4P - P = 3P \]

### Step 3: Substitute into the Simple Interest formula
Using the Simple Interest formula:
\[ 3P = \frac{P \times R \times 30}{100} \]

### Step 4: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 3 = \frac{R \times 30}{100} \]
Multiplying both sides by 100:
\[ 300 = R \times 30 \]
Dividing both sides by 30:
\[ R = 10\% \]

### Step 5: Find the time to double the principal
Now we need to find out how long it will take for the principal to double. If the principal is \( P \), then the amount when it doubles is:
\[ \text{Amount} = 2P \]
The Simple Interest earned in this case will be:
\[ \text{SI} = 2P - P = P \]

### Step 6: Set up the equation for doubling
Using the Simple Interest formula again:
\[ P = \frac{P \times R \times T}{100} \]

### Step 7: Simplify the equation for doubling
Cancel \( P \) from both sides:
\[ 1 = \frac{R \times T}{100} \]
Substituting \( R = 10 \):
\[ 1 = \frac{10 \times T}{100} \]
Multiplying both sides by 100:
\[ 100 = 10T \]
Dividing both sides by 10:
\[ T = 10 \text{ years} \]

### Final Answer
It will take **10 years** for the sum of money to become double at the same rate of simple interest.

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