A sum of money at simple interest doubles itself in 5 years 6 months . In how much time will it triple itself at the same rate ?

Options

A. 11 years
B. 12 years
C. 16 years
D. None of these

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the given information
We know that a sum of money doubles itself in 5 years and 6 months. This means that the simple interest earned in this time is equal to the principal amount.

### Step 2: Convert the time into years
5 years and 6 months can be converted into years:
- 6 months = 0.5 years
- Therefore, 5 years 6 months = 5 + 0.5 = 5.5 years or \( \frac{11}{2} \) years.

### Step 3: Set up the equation for simple interest
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

Since the amount doubles, we have:
\[ SI = P \]
Thus, we can write:
\[ P = \frac{P \times R \times \frac{11}{2}}{100} \]

### Step 4: Simplify the equation
Cancelling \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 1 = \frac{R \times \frac{11}{2}}{100} \]
Multiplying both sides by 100:
\[ 100 = R \times \frac{11}{2} \]
Now, multiplying both sides by 2:
\[ 200 = R \times 11 \]
Finally, divide by 11 to find \( R \):
\[ R = \frac{200}{11} \]

### Step 5: Set up the equation for tripling the amount
Now we need to find the time taken to triple the amount. When the amount triples, the simple interest earned will be:
\[ SI = 2P \]
Using the same formula:
\[ 2P = \frac{P \times R \times T}{100} \]
Cancelling \( P \):
\[ 2 = \frac{R \times T}{100} \]
Substituting \( R = \frac{200}{11} \):
\[ 2 = \frac{\frac{200}{11} \times T}{100} \]
Multiplying both sides by 100:
\[ 200 = \frac{200}{11} \times T \]
Now, multiplying both sides by 11:
\[ 2200 = 200 \times T \]
Dividing by 200:
\[ T = \frac{2200}{200} = 11 \]

### Final Answer
The time taken for the sum of money to triple itself at the same rate is **11 years**.

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