At what rate per cent compound interest does a sum of money become 16 times in 4 years?

Answer

To find the rate of compound interest at which a sum of money becomes 16 times in 4 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the rate of interest (R) where the amount becomes 16 times the principal (P) in 4 years.

2. **Use the Compound Interest Formula**: The formula for the amount (A) in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where:
- \(A\) = final amount
- \(P\) = principal amount
- \(R\) = rate of interest per annum
- \(T\) = time in years

3. **Set Up the Equation**: Since the amount becomes 16 times the principal, we can write:
\[ A = 16P \]
Substituting this into the compound interest formula gives:
\[ 16P = P \left(1 + \frac{R}{100}\right)^4 \]

4. **Simplify the Equation**: We can cancel \(P\) from both sides (assuming \(P \neq 0\)):
\[ 16 = \left(1 + \frac{R}{100}\right)^4 \]

5. **Take the Fourth Root**: To solve for \(1 + \frac{R}{100}\), we take the fourth root of both sides:
\[ 1 + \frac{R}{100} = 16^{1/4} \]

6. **Calculate \(16^{1/4}\)**: We know that:
\[ 16 = 2^4 \implies 16^{1/4} = 2 \]
Thus, we have:
\[ 1 + \frac{R}{100} = 2 \]

7. **Solve for \(R\)**: Rearranging gives:
\[ \frac{R}{100} = 2 - 1 = 1 \]
Therefore:
\[ R = 100 \text{ percent} \]

### Final Answer:
The rate of compound interest is **100% per annum**.

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