If a certain sum at compound interest becomes double in 5 yr, then in how many years, it will be 16 times at the same rate of interest ?

Answer

To solve the problem step by step, we need to find out how many years it will take for a certain sum of money to become 16 times at the same rate of interest, given that it doubles in 5 years.

### Step 1: Understand the given information
We know that a certain sum (let's denote it as \( P \)) becomes double in 5 years. Therefore, the amount after 5 years is:
\[ A = 2P \]

### Step 2: Use the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
where:
- \( A \) is the amount after \( n \) years,
- \( P \) is the principal amount (initial sum),
- \( R \) is the rate of interest,
- \( n \) is the number of years.

### Step 3: Set up the equation for doubling
Since the amount doubles in 5 years, we can set up the equation:
\[ 2P = P \left(1 + \frac{R}{100}\right)^5 \]

### Step 4: Simplify the equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = \left(1 + \frac{R}{100}\right)^5 \]

### Step 5: Solve for \( 1 + \frac{R}{100} \)
Taking the fifth root of both sides:
\[ 1 + \frac{R}{100} = 2^{1/5} \]

### Step 6: Set up the equation for becoming 16 times
Now, we want to find out how many years it will take for the amount to become 16 times the principal:
\[ A = 16P \]
Using the compound interest formula again:
\[ 16P = P \left(1 + \frac{R}{100}\right)^n \]

### Step 7: Simplify this equation
Dividing both sides by \( P \):
\[ 16 = \left(1 + \frac{R}{100}\right)^n \]

### Step 8: Substitute \( 1 + \frac{R}{100} \)
From Step 5, we know:
\[ 1 + \frac{R}{100} = 2^{1/5} \]
Now substituting this into the equation:
\[ 16 = \left(2^{1/5}\right)^n \]

### Step 9: Rewrite 16 as a power of 2
We know that \( 16 = 2^4 \). Therefore, we can write:
\[ 2^4 = \left(2^{1/5}\right)^n \]

### Step 10: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 4 = \frac{n}{5} \]

### Step 11: Solve for \( n \)
Multiplying both sides by 5:
\[ n = 20 \]

### Conclusion
Thus, it will take **20 years** for the sum to become 16 times at the same rate of interest.

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