Question

Simionie needs S7000 to buy a snowmobile, but only has \(\$ 6000 .\) His bank offers a GIC that pays an annual interest rate of \(3.93 \%,\) compounded annually. How long would Simionie have to invest his money in the GIC to have enough money to buy the snowmobile?

Short answer

Simionie would need to invest his money for about 4 years.

Step-by-step solution

- Write down the compound interest formula

The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:- \(A\) is the amount of money accumulated after n years, including interest.- \(P\) is the principal amount (initial investment).- \(r\) is the annual interest rate (decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time the money is invested or borrowed for in years.

- Identify known values

Given values from the problem:- \(P = 6000\)- \(A = 7000\)- \(r = 3.93\% = 0.0393\)- \(n = 1\) (since the interest is compounded annually)

- Plug in the values

Now plug the given values into the compound interest formula: \[ 7000 = 6000 \left(1 + \frac{0.0393}{1}\right)^{1 \cdot t} \]This simplifies to: \[ 7000 = 6000 \left(1 + 0.0393\right)^{t} \]Further simplifying: \[ 7000 = 6000 \left(1.0393\right)^{t} \]

- Solve for t

First, divide both sides by 6000: \[ \frac{7000}{6000} = \left(1.0393\right)^{t} \]Which simplifies to: \[ 1.1667 \approx (1.0393)^{t} \]Now take the natural logarithm (ln) of both sides to solve for t: \[ \ln(1.1667) = \ln((1.0393)^t) \]Using the logarithmic power rule, this becomes: \[ \ln(1.1667) = t \cdot \ln(1.0393) \]Finally, solve for t: \[ t = \frac{\ln(1.1667)}{\ln(1.0393)} \]Using a calculator, this results in: \[ t \approx 4 \]

Key concepts

Compound Interest Formula

Compound interest is a powerful concept that helps grow your money faster compared to simple interest. The formula for compound interest is \[A = P \times \big(1 + \frac{r}{n}\big)^{nt}\text{\] where:}

• A = total amount of money after time has passed
• P = principal amount (initial amount of money)
• r = annual interest rate (expressed as a decimal)
• n = number of times interest is compounded per year
• t = time the money is invested or borrowed for, in years

Using this formula, we can understand how much money grows when interest is added periodically. If interest is compounded annually, then interest is added once every year. The formula adjusts to capture how multiple smaller interest amounts can add up to a larger one over time. In our example, Simionie wants to know how long to invest his \(6000 to reach \)7000 with a 3.93% annual interest rate compounded yearly.

Solving Exponential Equations

To find the time Simionie needs to invest to have enough money, we need to solve for t in the compound interest formula. After plugging in our values, we end up with the equation: \begintaht{equation} 7000 = 6000(1.0393)^t The next step is to isolate the exponential term: \(\frac7000{6000} =(1.0393)t1.1667=(1.0393)^t\) Solving for t involves finding the exponent in an equation. In general, we can't solve it simply through addition or subtraction. This is where logarithms come into play. Taking the natural logarithm (ln) of both sides gives us a way to manipulate the equation: \(\theta(1.1667)=ln((1.0393)t)\) Using the properties of logarithms, particularly the power rule (ln(a^b) = b*ln(a)), we simplify: \(ln(1.1667) = t . ln(1.0393)\) Finally, we isolate t: t = \frac \theta(1.1667) {ln(1.0393)}$ This helps us understand how many periods it takes for money to grow with exponential growth.

Natural Logarithm

The natural logarithm, denoted as ln, is a vital tool in solving exponential equations. It's the logarithm to the base e, where e is approximately 2.71828. The natural logarithm is used because it naturally arises in continuous growth processes, like compound interest. In our example, we used ln to solve for t when we had an equation with a term in the form of (1.0393)^t.When we apply ln to both sides, the exponent t can move in front due to the logarithmic power rule: ln(a^c) = c*ln(a).This makes the equation manageable and lets us isolate t: ln(1.1667) = t. ln(1.0393). From here, we divide both sides by ln(1.0393) to find t. The natural logarithm turns the multiplying effect of compound growth into manageable addition, enabling us to handle complex exponential equations quickly.In simpler terms, using natural logarithms helps us solve exactly how long it takes for exponential growth to reach a certain point, which is crucial for understanding compound interest scenarios like Simionie's.