Question

An investment of \(P\) dollars that gains \(r\) percent of its value in one year is worth \(P(1+r)\) at the end of that year. An investment that loses \(r\) percent of its value in one year is worth \(P(1-r)\) at the end of that year. Write a model for the value of an investment \(P\) that loses \(r\) percent one year, then gains \(r\) percent the following year.

Short answer

The model for the value of an investment \(P\) that loses \(r\) percent in one year, then gains \(r\) percent the following year is \[P(1-r)(1+r)\].

Step-by-step solution

Apply Loss in the First Year

Initially, the investment hamounts to \(P\) dollars. After losing \(r\) percent of its value in the first year, the amount left after the first year is given by \(P(1-r)\).

Apply Gain in the Second Year

The gain in the second year is applied to the remaining amount at the end of the first year, \(P(1-r)\). The percent gain is \(r\), therefore at the end of the second year, the amount gained is \(P(1-r)(1+r)\).

Write the Final Model

So, the model for the value of an investment \(P\) that loses \(r\) percent in one year, then gains \(r\) percent the following year is \[P(1-r)(1+r)\].

Key concepts

Percent Increase and Decrease

Understanding percentages is crucial when dealing with investment value calculations. A percent increase indicates that the value of an investment has gone up by a certain proportion, while a percent decrease means the value has gone down. It's a relative measure, compared to the original value.

Let's consider an initial investment of \(P\) dollars. If it gains \(r\) percent of its value, the new amount can be calculated using the expression \(P(1+\frac{r}{100})\). Similarly, if it loses \(r\) percent, the calculation involves the expression \(P(1-\frac{r}{100})\). It's essential to understand that the percent change is applied to the current value of \(P\), not the original value, which is a common misunderstanding.

When an investment undergoes a series of increases and decreases, the changes are compounded. This means the effect of the second change is applied not just to the original amount, but to the already modified amount from the first change. Therefore, if an investment loses \(r\) percent in value and then gains \(r\) percent the following year, the two changes do not cancel each other out due to this compounding effect.

Algebraic Expressions

Algebraic expressions are symbolic representations that include numbers, variables, and arithmetic operations. In the context of investment calculations, we use variables to represent amounts that can change, such as the principal \(P\) or the rate of increase or decrease \(r\).

An algebraic expression that captures the change in investment value needs to accurately reflect the percent increase or decrease. Using algebra, we translate the percent change into an algebraic factor. For example, a \(r\) percent increase translates to the factor \(1 + \frac{r}{100}\), and a decrease translates to \(1 - \frac{r}{100}\).

The beauty of algebraic expressions is that they allow us to model real-life situations abstractly and then apply specific values to predict outcomes. For investments, the algebraic model must reflect the sequential nature of changes; first, the decrease: \(P(1-\frac{r}{100})\), and then the subsequent increase: \(\left[P(1-\frac{r}{100})\right](1+\frac{r}{100})\), or simplified, the expression becomes \(P(1-r)(1+r)\), underscoring that the order of operations and the precise representation of percent changes are central to accurate calculations.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world scenarios to predict and analyze complex behaviors. In finance, mathematical models are used to forecast investment outcomes based on various factors, such as interest rates and market volatility.

In the case of our investment example, the model helps visualize the impact of fluctuating market conditions on an investment's value over time. The initial model for a decrease in value is \(P(1-r)\), and for an increase, it is \(P(1+r)\). By combining these models, we can simulate consecutive years with different market conditions.

The sequential nature of increases and decreases in investment value means that each year's outcome becomes the basis for the next year's calculation. Mathematical modeling not only helps in predicting the end result after a series of such changes but also in understanding the intermediate states of the investment. It's a powerful tool that aids investors and students alike in making informed decisions based on structured, logical progressions.