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Shen deposited $5000 into an account with a 6.4% annual interest rate, compounded quarterly. Assuming that no withdrawals are made, how long will it take for the investment to grow to $6145 ?

Do not round any intermedlate computations, and round your answer to the nearest hundredth.

User Snorex
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1 Answer

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Answer:

Time ≈ 3.25 years

Explanation:

Formula for compound interst:

The formula for compound interest is given by:

A(t) = P(1 + r/n)^(nt), where

  • A(t) is the amount in the account after t years,
  • r is the interest rate (the percentage is converted to a decimal when using the formula),
  • and n is the number of compounding periods.

Determining n (the number of compounding periods):

  • Note that when money is compounded quarterly, there are 4 compounding periods.
  • This means the money is compounded once every 3 months and there are four of these 3-month periods in 1 year (i.e., 12 months).

Using the compound interest formula:

Now we can solve for t (the amount of time in years) by substituting 6145 for A, 5000 for P, 0.064 for r, and 4 for n:

(6145 = 5000(1 + 0.064/4)^(4t)) / 5000

log (1.229) = log ((1.016)^4t)

(log (1.229) = 4t * log(1.016)) / log (1.016)

(log (1.229) / log (1.016) = 4t) * 1/4

1/4 * (log (1.229) / log (1.016)) = t

3.247594892 = t

3.25 = t

Thus, it would take about 3.25 years for the $5000 investment to grow to $6145, given that there's a 6.4% annual interest rate and that the money is compounded quarterly.

User RamHS
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