Question

Kobe puts $120 every month into an account paying 1.9% for 3 years. He then finds a better interest rate and moves his money into an account paying 4.4% compounded monthly (and continues putting in $120 each month). How long (in years, rounded to one decimal place) will it take for Kobe to have $20,000?

Answer 1

[AI-generated answer — Explain Bot]

To find how long it will take for Kobe to have $20,000, we can use the formula for the future value of an ordinary annuity:

\(FV = P \cdot \left( \dfrac{(1 + r)^n - 1}{r} \right)\)

Where:
\(FV\) = Future Value
\(P\) = Payment amount per period
\(r\) = Interest rate per period
\(n\) = Number of periods

Let's first calculate the future value after 3 years, when Kobe is earning 1.9% interest and contributing $120 each month:

We'll assume the interest is compounded monthly, so the interest rate per period would be \(r = \dfrac{1.9\%}{12} = 0.01583\) (rounded to 5 decimal places).

Using the formula, we have:
\(FV_1 = 120 \cdot \left( \dfrac{(1 + 0.01583)^{3 \cdot 12} - 1}{0.01583} \right)\)

Calculating this gives us:
\(FV_1 = 120 \cdot \left( \dfrac{(1.01583)^{36} - 1}{0.01583} \right) \approx 4604.71\)

Therefore, after 3 years, Kobe would have approximately $4604.71 in his account.

Next, let's calculate the future value starting with $4604.71, but now earning 4.4% interest compounded monthly, until it reaches $20,000:

The interest rate per period would now be \(r = \dfrac{4.4\%}{12} = 0.03667\) (rounded to 5 decimal places).

We are trying to find the number of periods, \(n\), required to reach $20,000, so we can rearrange the formula to solve for \(n\):

\(n = \dfrac{\log\left(\dfrac{FV}{P} \cdot r + 1\right)}{\log(1+r)}\)

Using this formula, we have:
\(n = \dfrac{\log\left(\dfrac{20000}{120} \cdot 0.03667 + 1\right)}{\log(1+0.03667)}\)

Calculating this gives us:
\(n \approx 14.5\)

Therefore, it will take approximately 14.5 years (rounded to one decimal place) for Kobe to have $20,000 in his account.