Question

An investment is made at 7% annual interest compounded daily.
How long does it take to triple the investment?

Answer 1

Answer:

15.7 years

Step-by-step explanation:

To find out how long it takes to triple the investment, we can use the compound interest formula. The formula for compound interest is:

$A = P \times \left(1 + \dfrac{r}{n}\right)^{nt}$

Where:

• $A$ is the amount of money accumulated after $n$ years, including interest.
• $P$ is the principal amount (the initial investment).
• $r$ is the annual nominal interest rate (in decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested for, in years.

In this case, we want to triple the investment, which means $A = 3P$. We're given $r = 0.07$ (7% interest rate) and $n = 365$ since interest is compounded daily.

So, we have:

$3P = P \times \left(1 + \dfrac{0.07}{365}\right)^{365t}$

To solve for $t$, we'll isolate it:

$\left(1 + \dfrac{0.07}{365}\right)^{365t} = 3$

Now, take the natural logarithm of both sides:

$\ln \left( \left(1 + \dfrac{0.07}{365}\right)^{365t} \right) = \ln(3)$

Using the property of logarithms that $\ln(a^b) = b \cdot \ln(a)$, we have:

$365t \cdot \ln \left(1 + \dfrac{0.07}{365}\right) = \ln(3)$

Now, we'll solve for $t$:

$t = \dfrac{\ln(3)}{365 \cdot \ln \left(1 + \dfrac{0.07}{365}\right)}$

Using a calculator, we can find $t$:

$t \approx 15.69596617$

$t \approx 15.7 \textsf{ ( in nearest tenth)}$

So, it takes approximately 15.7 years to triple the investment when compounded daily at 7% annual interest.