An amount on compound interest is doubled to the original amount in 3 years. In how many yearswill it be eight times its original amount

1. Break Down the Problem

We need to find the time it takes for an investment to grow eight times its original amount, given that it doubles in 3 years.

2. Relevant Concepts

The compound interest formula is given by:

$A = P(1 + r)^n$

Where:

• $A$ = amount of money accumulated after n years, including interest.
• $P$ = principal amount (the initial amount of money).
• $r$ = annual interest rate (decimal).
• $n$ = number of years the money is invested or borrowed.

3. Analysis and Detail

1. Since the amount doubles in 3 years:

   • Set $A = 2P$ and $n = 3$: $2P = P(1 + r)^3 \implies 2 = (1 + r)^3$

2. Taking the cube root of both sides: $1 + r = 2^{1/3}$

3. Therefore, the interest rate $r$ is: $r = 2^{1/3} - 1$

4. Next, we want to find when the amount will be eight times the principal:

   • Set $A = 8P$: $8P = P(1 + r)^n \implies 8 = (1 + r)^n$

5. Substitute $1 + r = 2^{1/3}$: $8 = (2^{1/3})^n \implies 8 = 2^{n/3}$

6. Since $8 = 2^3$: $2^3 = 2^{n/3} \implies 3 = \frac{n}{3} \implies n = 9$

4. Verify and Summarize

After verifying the calculations, the amount will grow eight times its original amount in 9 years.

Final Answer

The investment will be eight times its original amount in 9 years.