Nitin’s money becomes double in 4 years at compound interest. In how many years will it become sixteen times at compound interest ?

Options

A. a) 20
B. b) 24
C. c) 16
D. d) 28

Answer

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

### Step 1: Understand the Problem
Nitin's money doubles in 4 years at compound interest. We need to find out how many years it will take for his money to become sixteen times.

**Hint:** Identify the relationship between the doubling time and the time to reach 16 times.

### 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:
- \( A \) is the amount after time \( t \),
- \( P \) is the principal amount,
- \( r \) is the rate of interest,
- \( t \) is the time in years.

### Step 3: Set Up the Equation for Doubling
Since Nitin's money doubles in 4 years:
\[ 2P = P \left(1 + \frac{r}{100}\right)^4 \]

**Hint:** Cancel \( P \) from both sides since \( P \) is not zero.

### Step 4: Simplify the Doubling Equation
Cancelling \( P \) gives:
\[ 2 = \left(1 + \frac{r}{100}\right)^4 \]

### Step 5: Set Up the Equation for Becoming Sixteen Times
Now, we need to find when the amount becomes 16 times:
\[ 16P = P \left(1 + \frac{r}{100}\right)^x \]

**Hint:** Again, cancel \( P \) from both sides.

### Step 6: Simplify the Sixteen Times Equation
Cancelling \( P \) gives:
\[ 16 = \left(1 + \frac{r}{100}\right)^x \]

### Step 7: Relate the Two Equations
From the doubling equation, we know:
\[ 2 = \left(1 + \frac{r}{100}\right)^4 \]
We can express 16 as:
\[ 16 = 2^4 \]

### Step 8: Substitute and Equate the Powers
Now we can write:
\[ 16 = \left(1 + \frac{r}{100}\right)^{4x/4} \]
This gives us:
\[ 2^4 = \left(1 + \frac{r}{100}\right)^x \]

### Step 9: Set the Exponents Equal
Since both sides have the same base (2), we can equate the exponents:
\[ 4 = \frac{x}{4} \]

### Step 10: Solve for \( x \)
Multiplying both sides by 4 gives:
\[ x = 16 \]

### Conclusion
Thus, it will take 16 years for Nitin's money to become sixteen times at compound interest.

**Final Answer:** 16 years.

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