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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

Question

An amount on compound interest is doubled to the original amount in 3 years.

In how many years will it be eight times its original amount?

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Solution

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 A = P(1 + r)^n

Where:

  • A A = amount of money accumulated after n years, including interest.
  • P P = principal amount (the initial amount of money).
  • r r = annual interest rate (decimal).
  • n 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 A = 2P and n=3 n = 3 : 2P=P(1+r)3    2=(1+r)3 2P = P(1 + r)^3 \implies 2 = (1 + r)^3
  2. Taking the cube root of both sides: 1+r=21/3 1 + r = 2^{1/3}

  3. Therefore, the interest rate r r is: r=21/31 r = 2^{1/3} - 1

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

    • Set A=8P A = 8P : 8P=P(1+r)n    8=(1+r)n 8P = P(1 + r)^n \implies 8 = (1 + r)^n
  5. Substitute 1+r=21/3 1 + r = 2^{1/3} : 8=(21/3)n    8=2n/3 8 = (2^{1/3})^n \implies 8 = 2^{n/3}

  6. Since 8=23 8 = 2^3 : 23=2n/3    3=n3    n=9 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.

This problem has been solved

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