A sum of money doubled itself at certain rate of compound interest in 15 years. In how many years will it become four times of itself?

Options

A. 45
B. 30
C. 25
D. 15

Answer

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

### Step 1: Understand the Problem
We know that a sum of money doubles itself in 15 years at a certain rate of compound interest. We need to find out how many years it will take for the same sum of money to become four times itself.

### Step 2: Use the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount (initial sum)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 3: Set Up the Equation for Doubling
Let’s assume the principal amount \( P = 100 \) (for simplicity). When the amount doubles, we have:
\[ A = 200 \]
Using the formula:
\[ 200 = 100 \left(1 + \frac{R}{100}\right)^{15} \]
Dividing both sides by 100:
\[ 2 = \left(1 + \frac{R}{100}\right)^{15} \]

### Step 4: Solve for \( 1 + \frac{R}{100} \)
Taking the 15th root of both sides:
\[ 1 + \frac{R}{100} = 2^{\frac{1}{15}} \]

### Step 5: Set Up the Equation for Quadrupling
Now, we want to find out how long it will take for the amount to become four times the principal:
\[ A = 400 \]
Using the same formula:
\[ 400 = 100 \left(1 + \frac{R}{100}\right)^{T} \]
Dividing both sides by 100:
\[ 4 = \left(1 + \frac{R}{100}\right)^{T} \]

### Step 6: Substitute \( 1 + \frac{R}{100} \) into the Quadrupling Equation
From Step 4, we know:
\[ 1 + \frac{R}{100} = 2^{\frac{1}{15}} \]
Substituting this into the quadrupling equation:
\[ 4 = \left(2^{\frac{1}{15}}\right)^{T} \]

### Step 7: Simplify the Equation
We can express 4 as \( 2^2 \):
\[ 2^2 = \left(2^{\frac{1}{15}}\right)^{T} \]
This implies:
\[ 2 = \frac{T}{15} \]
Thus, we can equate the exponents:
\[ 2 = \frac{T}{15} \]

### Step 8: Solve for \( T \)
Multiplying both sides by 15:
\[ T = 2 \times 15 = 30 \]

### Conclusion
The sum of money will become four times itself in **30 years**.

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