Question

You are saving money for a trip to Europe that costs \(\$ 3600 .\) You make an investment of \(\$ 1000\) two years before the trip and an investment of \(\$ 1800\) one year before the trip. Use \((1+r)\) as a growth factor to write an equation that can be solved to find the growth rate \(r\) that you need in order to have \(\$ 3600\) at the time of the trip to Europe.

Short answer

The equation to be solved is \(1000 (1 + r)^2 + 1800 (1 + r) = 3600\). The valid solution for the growth rate \(r\) is the one given by the quadratic formula for this equation. The solution \(r\) must be positive to be meaningful.

Step-by-step solution

Understand the Compound Interest Concept

Compound interest refers to the idea that you earn interest on both your original money and on the interest you have already received. In this case, it is being used to calculate how much an investment grows over a certain period of time. The compound interest formula is \(A = P(1 + r/n)^(nt)\), where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount or the initial amount that was invested, \(r\) is the annual interest rate, \(n\) is the number of times that interest is compounded per unit \(t\), or the number of times an interest is applied, and \(t\) is time the money is invested for in years. However, as the interest is compounded annually, that is once per year, the formula can be simplified to \(A = P(1 + r)^t\). Since we have multiple investments we have an series of compound investments instead of just one.

Set Up the Equation

Given, the initial investment amount \(\$ 1000\) after two years and \(\$ 1800\) after one year should sum up to \(\$ 3600\). We can write these two investments as compound investments using the formula \(A = P(1 + r)^t\), where \(A\) is the final amount, \(P\) is the principal amount or initial investment, \(r\) is annual growth rate, and \(t\) is time in years. That gives us: \(1000 (1 + r)^2 + 1800 (1 + r) = 3600\). The growth rate \(r\) is what we need to solve this equation.

Solve the Equation for \(r\)

This is a quadratic equation in \(r\). To solve it, you must either factor it or use the quadratic formula. If factoring is not possible, using the quadratic formula is the only method. In this case, the equation is already simplified, so factoring or further simplification is not required before using the quadratic formula. The solutions for \(r\) will give us the necessary growth rates. If there are two solutions, the meaningful rate is the one that is positive, as a negative growth rate wouldn't make sense in this context.

Key concepts

Growth Factor

The concept of a growth factor plays a critical role in understanding compound interest. It represents the multiplier applied to the principal amount to calculate the amount of money that will be present after a certain period, considering the compound interest effect. In the exercise, the growth factor is denoted by \(1 + r\), where \(r\) is the annual growth rate. For each period, or year, the investment grows by a factor of \(1 + r\).

A single investment of \$1000 growing for two years would therefore be represented as \$1000 \times (1 + r)^2\. The power of 2 represents the compound interest being applied for two compounding periods (years in this case). Similarly, a subsequent investment of \$1800 growing for one year is expressed as \$1800 \times (1 + r)\. Understanding this, you realize that each initial amount, \(P\), when multiplied by its respective growth factor over the given time, contributes to the final total of \$3600 needed for the trip.

Investment Growth Rate

When planning an investment, determining the investment growth rate is key to achieving financial goals. In our exercise, we are interested in finding the annual growth rate \(r\) that will allow the initial investments to grow to a specified amount, \$3600, in the given time frame. The growth rate represents the percentage increase of the investment per compounding period - annually in this situation.

To calculate the needed growth rate, we explore different values of \(r\) that satisfy the equation from the compound interest formula. It is crucial to understand that the growth rate influences how quickly an investment will grow over time, and selecting a realistic and attainable growth rate is essential for accurate financial planning. Whether you are saving for a trip, as in the example, or another financial target, mastering the concept of the investment growth rate is fundamental to investment success.

Solving Quadratic Equations

A quadratic equation typically has the structure \(ax^2 + bx + c = 0\), where \(^2\) indicates that the variable is squared, making it a second-degree polynomial equation. To solve a quadratic equation, such as the one in our compound interest problem, one might use simplification, factoring, or the quadratic formula, depending on the specific equation's characteristics.

The quadratic formula, \((-b \pm \sqrt{b^2 - 4ac})/(2a)\), provides a method to find solutions for \((x\) when factoring is not straightforward. As we apply this to find the growth rate \(r\), remember to consider only the meaningful solution - in this context, a positive growth rate - as a negative rate would not be logical for a growth scenario. Solving quadratic equations is a fundamental skill in algebra, often applied in financial calculations, physics, engineering, and other sciences.