Question

A company publishes to its customers that at a certain compound interest rate, a sum of money deposited by anyone will become 8 times in 3 years. If the same amount is deposited at the same compound rate of interest, then in how many years will it become 16 times?

• A. 4 years
• B. 4.5 years
• C. 3 years
• D. 3.5 years

Answer 1

The Correct Option is A

To solve the given problem, we first need to understand the concept of compound interest. The question states that a sum of money becomes 8 times in 3 years at a certain compound interest rate. We are asked to find the number of years it will take for the same sum of money to become 16 times at the same rate. 

In compound interest, the relation between the principal amount (initial money), the final amount, the time period, and the interest rate can be described using the formula:

\(A = P \left(1 + \frac{r}{100}\right)^n\)

where:

• \(A\) is the amount after time \(n\) years.
• \(P\) is the principal amount.
• \(r\) is the annual interest rate (in percentage).
• \(n\) is the number of years.

According to the problem, the amount becomes 8 times in 3 years. Thus, we have:

\(8P = P \left(1 + \frac{r}{100}\right)^3\)

By simplifying, we get:

\(8 = \left(1 + \frac{r}{100}\right)^3\)

We need to find \(n\) such that the money becomes 16 times. So:

\(16P = P \left(1 + \frac{r}{100}\right)^n\)

Dividing both sides by \(P\) simplifies to:

\(16 = \left(1 + \frac{r}{100}\right)^n\)

We know from above that \(\left(1 + \frac{r}{100}\right)^3 = 8\). Therefore, we can express:

\(16 = 8^{\frac{n}{3}}\)

Since \(8 = 2^3\) and \(16 = 2^4\), we use these identities:

\(2^4 = \left(2^3\right)^{\frac{n}{3}}\)

This implies that:

\(\left(8\right)^{\frac{n}{3}} = 16 \rightarrow \left(2^3\right)^{\frac{n}{3}} = 2^4 \rightarrow 2^n = 2^4\)

Thus, \(n = 4\).

Conclusion: The amount will become 16 times in \(4\) years. Therefore, the correct option is 4 years.