Related questions with answers

How many years will it take to double your money if you invest in a fund with each of the given interest rates below, compounded continuously? a. 5% b. 6% c. 8% d. 10% e. Financial managers use a quick calculation that does not involve logarithms to approximate how long it will take to double your money given a specific interest rate. They call it the "rule of 72". Determine the relationship between the interest rate, the doubling time and the number 72. Write a function r(x) to approximate the number of years it would take for your money to double given an interest rate of x. f. A savvy financial investor might want to convince their customers that they know how to make their money double faster, by using a more accurate calculation. Determine the best number, rounded to the nearest tenth, to use for a doubling rule. Explain how this number relates to natural logarithms.

If Tanisha has $100 to invest at 8% per annum compounded monthly, how long will it be before she has$150? If the compounding is continuous, how long will it be?

How long will it take $4,000 to grow to$9,000 if it is invested at 7% compounded monthly?

Question

(a) How long does it take for an investment to double in value if it is invested at 8% compounded monthly?

(b) How long does it take if the interest is compounded continuously?

Solution

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Step 1

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(a) If we want to double an investment, the future value $A$ will be equal to $2P$ , where $P$ is the present value. To calculate the time we need to double an investment we will apply formula for future value, which is

A=P\cdot\left(1+\frac{r}{n}\right)^{nt},

where $t$ is time expressed in years, $r$ is per annum interest rate compounded $n$ times per year.We know that $A=2P, r=8\%=0.08$ and $n=12$ because we compound interest monthly, so substituting this into the previous equation we will get

2P=P\left(1+\frac{0.08}{12}\right)^{12\cdot t}.

Dividing the previous equation by $P$ we will get

\begin{align*} 2=\left(\frac{1.08}{12}\right)^{12t}\approx &1.0067^{12t}\\ =&(1.0067^{12})^t\\ \approx &1.083^t \end{align*}

Using the definition of the logarithm we have now that

\begin{align*} 2=(1.083)^t\quad \Longleftrightarrow\quad &t=\log_{1.083}2\\ \Longrightarrow\quad & t=\frac{\ln 2}{\ln 1.083}\\ \Longrightarrow\quad & t\approx 8.69314. \end{align*}

We used the Change-of-Base formula for logarithms in the last step. So, the time needed to double the investment if it is invested at $8\%$ compounded monthly is approximately 8.69 years.