Question

Investment of ₹ 4,000 at \(R\%\) per annum compounded annually becomes 16,000 in 8 years. If 2,000 is invested at \(R\%\) per annum compounded annually, then in how many years will the investment become 16,000?

• A. 8
• B. 10
• C. 12
• D. 14

Answer 1

The Correct Option is C

Compound Interest Calculation 

The given problem involves compound interest, where the formula is:

\[ A = P \left(1 + \frac{R}{100}\right)^t \]

where:

• A is the final amount,
• P is the principal amount,
• R is the annual interest rate,
• t is the number of years.

Step 1: Determine the Rate of Interest

Given that an investment of ₹4000 becomes ₹16000 in 8 years:

\[ 16000 = 4000 \left(1 + \frac{R}{100}\right)^8 \]

\[ \frac{16000}{4000} = \left(1 + \frac{R}{100}\right)^8 \]

\[ 4 = \left(1 + \frac{R}{100}\right)^8 \]

Taking the 8th root on both sides:

\[ 1 + \frac{R}{100} = \sqrt[8]{4} \]

Step 2: Find the Time for ₹2000 to Become ₹16000

Now, using the same rate \( R \), we solve for time \( t \) when ₹2000 grows to ₹16000:

\[ 16000 = 2000 \left(1 + \frac{R}{100}\right)^t \]

\[ \frac{16000}{2000} = \left(1 + \frac{R}{100}\right)^t \]

\[ 8 = \left(1 + \frac{R}{100}\right)^t \]

From Step 1, we know:

\[ 4 = \left(1 + \frac{R}{100}\right)^8 \]

Squaring both sides:

\[ 8 = \left(1 + \frac{R}{100}\right)^{12} \]

Thus, \( t = 12 \) years.

Final Answer:

Option (C) 12 years