Problem 71 - Introductory and Intermediate Algebra for College Students 4th

Question

Suppose that you put \(\$ 10,000\) in a rather risky investment recommended by your financial advisor. During the first year, your investment decreases by \(30 \%\) of its original value. During the second year, your investment increases by \(40 \%\) of its first-year value. Your advisor tells you that there must have been a \(10 \%\) overall increase of your original \(\$ 10,000\) investment. Is your financial advisor using percentages properly? If not, what is the actual percent gain or loss on your original \(\$ 10,000\) investment?

Step-by-Step Solution

Answer

Your advisor's assertion that there has been a 10% increase in the investment is incorrect. In reality, there was a 2% loss.

Step 1: Calculating Value After First Year

The problem states that during the first year, the investment decreased by 30% of its original value. In order to find out how much money is left after the first year, calculate 30% of $10,000, which is \(0.30 * 10000 = 3000\). Then, subtract this amount from the initial investment: \(10000 - 3000 = 7000\). Thus, the investment is worth $7,000 after the first year.

Step 2: Calculating Value After Second Year

During the second year, the investment increases by 40% of its first-year value. To know how much money is added during the second year, calculate 40% of $7,000, which is \(0.40 * 7000 = 2800\). Next, add this amount to the $7,000 value after the first year: \(7000 + 2800 = 9800\). So, the investment is worth $9,800 after the second year.

Step 3: Finding the % Loss or Gain

To check if the financial advisor's calculations are correct, compare the final value from the original one. Since the final value is $9,800 and the initial amount was $10,000, some loss occurred. To find the percentage loss, subtract the final amount from the original, divide by the original one and multiply by 100. Let's calculate, \((10000 - 9800) / 10000 * 100 = 2 \% \) loss.

Key Concepts

Understanding Percentage Decrease

When an investment or value "decreases by a certain percentage," it means the value is reduced by that specific fraction of its original amount. For instance, if an investment of \(10,000 decreases by 30%, this decrease is calculated as a portion of the original amount. This is done by multiplying the original value by the percentage (expressed as a decimal). For example:

• The original investment is \)10,000.
• Percentage of decrease: 30% = 0.30 (in decimal form).
• Amount of decrease: \(0.30 \times 10000 = 3000\).
• Value after decrease: \(10000 - 3000 = 7000\).

After reducing by 30%, the investment is now worth $7,000. Understanding this concept is crucial as it allows for the calculation of the remaining value after the decrease. This also highlights that percentages refer to proportional parts of the specific amount they are taken from, not the entire original value again after each adjustment.

The Concept of Percentage Increase

A percentage increase indicates that an amount is now worth more than its previous level by a given percentage of that former value. Consider an investment worth \(7,000 (after a previous decrease). If this investment then increases by 40%, it means the increase is a percentage of the \)7,000, not the original \(10,000.

• Original value after first change: \)7,000.
• Percentage increase: 40% = 0.40 (in decimal).
• Amount of increase: \(0.40 \times 7000 = 2800\).
• Value after increase: \(7000 + 2800 = 9800\).

This results in a new value of \(9,800. It's important to distinguish that the percentage increase is applied to the value at the time of increase, not the initial \)10,000. Therefore, each successive percentage change applies to the previously adjusted amount.

Evaluating Financial Analysis with Percentage Changes

Financial analysis often involves assessing investment performance, which frequently requires understanding percentage changes over time. In the example, it was crucial to account for the separate effects of a 30% decrease and a 40% increase, both on different base values. To find out the actual effect on an investment over multiple periods:

• Calculate each percentage change sequentially, using the adjusted value from the previous computation as the new base.
• Compare the final amount to the original investment to understand overall performance.
• In our case, beginning with $10,000 and concluding with $9,800 implies an overall 2% decrease (not a 10% increase).

Therefore, financial analysis sometimes reveals that intuitive assumptions (like expecting two consecutive 10% changes to sum to a 20% change) can be misleading. Proper understanding and calculation of sequential percentage changes are essential. This ensures decisions are based on accurate financial insights rather than potentially flawed logical assumptions.