Question

You have decided to invest $9,429. How long would you need to wait to multiply by 5 the initial quantity invested if the average annual growth rate is 8% Your answer should be in years and have two (2) digits after the period. So, if your answer is 5.36 years, answer 5.36.

Answer 1

You can calculate the time it takes to multiply your initial investment by 5 using the formula for compound interest:

`\[A = P(1 + r)^t\]Where:- \(A\) is the final amount (5 times the initial investment)- \(P\) is the initial investment ($9,429)- \(r\) is the annual growth rate (8% or 0.08 as a decimal)- \(t\) is the time in years (which we want to find)`

`In this case, we want to find \(t\), so we'll rearrange the formula:\[5P = P(1 + 0.08)^t\]`

`Now, divide both sides by \(P\):\[5 = (1 + 0.08)^t\]Take the natural logarithm (ln) of both sides to solve for \(t\):\[ln(5) = ln(1 + 0.08)^t\]Use the properties of logarithms to bring \(t\) down:\[ln(5) = t * ln(1 + 0.08)\]Now, solve for \(t\):\[t = ln(5) / ln(1 + 0.08)\]Calculate this using a calculator:\[t ≈ 10.91\]`

So, it would take approximately 10.91 years for your initial investment of $9,429 to multiply by 5 with an average annual growth rate of 8%. Rounding to two decimal places, the answer is approximately 10.91 years.