Problem 48 - Contemporary Precalculus

Question

An investor puts a total of \(\$ 25,000\) into three very speculative stocks. She invests some of it in Crystalcomp and \(\$ 2000\) more than one-half that amount in Flyboys. The remainder is invested in Zumcorp. Crystalcomp rises \(16 \%\) in value, Flyboys rises \(20 \%,\) and Zumcorp rises \(18 \% .\) Her investment in the three stocks is now worth \(\$ 29,440 .\) How much was originally invested in each stock?

Step-by-Step Solution

Answer

Answer: The original investment amounts were approximately $10,000 in Crystalcomp, $7,000 in Flyboys, and $8,000 in Zumcorp.

Step 1: Determine the relationship between the investments

Let \(x\) be the amount invested in Crystalcomp, \(y\) be the amount invested in Flyboys, and \(z\) be the amount invested in Zumcorp. We know that the total investment was \(\$25,000\), so we can represent this relationship as: \(x + y + z = 25000.\) We also know that the investor placed \(x\) in Crystalcomp, \(2000\) more than half of that amount in Flyboys. This can be written as: \(y = x/2 + 2000.\) Since the remainder is invested in Zumcorp, we have: \(z = 25000 - x - y.\) ##Step 2: Use percentages to calculate the final value of the investment##

Step 2: Calculate the increase in value for each stock

For the next step, we need to calculate the new value of the investment in each company by considering the percentage increase: - Crystalcomp increased by \(16 \% \implies 1.16x\) - Flyboys increased by \(20 \% \implies 1.20y\) - Zumcorp increased by \(18 \% \implies 1.18z\) The new total value of the investment is \(\$29,440\). So, we can write this as: \(1.16x + 1.20y + 1.18z = 29440.\) ##Step 3: Solve the resulting system of equations##

Step 3: Solve the system of equations

Now, we have a system of three equations: 1. \(x + y + z = 25000\) 2. \(y = x/2 + 2000\) 3. \(1.16x + 1.20y + 1.18z = 29440\) First, substitute the value of \(y\) from the second equation into the first and the third equation: 1. \(x + (x/2 + 2000) + z = 25000\) 2. \(1.16x + 1.20(x/2 + 2000) + 1.18z = 29440\) Now, simplify the first equation and express \(z\) in terms of \(x\): 1. \(x + x/2 + z = 23000 \implies z = 23000 - (3/2)x\) Next, substitute the value of \(z\) from the first equation into the third equation and solve for x: 2. \(1.16x + 1.20(x/2 + 2000) + 1.18(23000 - (3/2)x) = 29440\) Solve for \(x:\ x \approx 10000\) Use the value of \(x\) to find \(y\) and \(z\): \(y \approx x/2 + 2000 \approx 7000\) \(z \approx 23000 - (3/2)x \approx 8000\) ##Step 4: Determine the original investment amounts##

Step 4: Find the initial investments

Now that we have solved the system of equations and found the approximate values for \(x\), \(y\), and \(z\), we can determine the original amount invested in each stock: - Crystalcomp: \(x \approx \$10,000\) - Flyboys: \(y \approx \$7,000\) - Zumcorp: \(z \approx \$8,000\) Thus, the investor originally invested \(\$10,000\) in Crystalcomp, \(\$7,000\) in Flyboys, and \(\$8,000\) in Zumcorp.

Key Concepts

Investment Calculation

Investment calculations are pivotal when dealing with financial assets like stocks. Here, an investor chooses to invest in three speculative stocks: Crystalcomp, Flyboys, and Zumcorp. The entire investment amounts to $25,000. Calculating how much was initially invested in each of these stocks involves breaking down the allocation of funds accurately. In this exercise, we used variables to represent each investment:

• Let \( x \) be the amount invested in Crystalcomp.
• Let \( y \) be the amount invested in Flyboys.
• Let \( z \) be the amount invested in Zumcorp.

We express the allocation of funds with equations such as \( x + y + z = 25000 \) to ensure that they sum up to the total investment. This structured approach allows clarity and precision in resolving how funds get distributed among the chosen stocks.

Percentage Increase

Understanding percentage increase is crucial in evaluating investment performance. Each stock has a unique rate of growth, expressed as a percentage increase over its original value. For the stocks in this scenario:

• Crystalcomp grows by \( 16\% \), indicated algebraically as \( 1.16x \).
• Flyboys grows by \( 20\% \), represented as \( 1.20y \).
• Zumcorp grows by \( 18\% \), shown as \( 1.18z \).

These equations help us compute the new value of each stock after appreciation. By summing up the results, we ascertain the total updated value, \( 29440 \). This comprehension of percentage increase helps assess the growth and profitability of an investment over time.

Algebraic Solutions

Algebra offers tools necessary to solve complex financial issues, such as determining initial investments via systems of equations. By setting up mathematical relationships:

• The first equation represents the total investment: \( x + y + z = 25000 \).
• The second equation reflects additional conditions on investments: \( y = x/2 + 2000 \).
• The third accounts for post-investment values: \( 1.16x + 1.20y + 1.18z = 29440 \).

We substitute equations to reduce and simplify the system, step by step. For example, substituting \( y \) in terms of \( x \) simplifies solving for these unknowns. Once the value of each variable is calculated, the solution becomes visible, showing how much initially went into each stock. This algebraic method ensures accuracy in financial estimations and investment planning.

Speculative Stocks

Investing in speculative stocks involves a degree of risk, often banking on significant price fluctuations. Such stocks, like Crystalcomp, Flyboys, and Zumcorp in this case, promise high returns but carry the potential for equal losses. With volatility comes a strategic approach: investors hope to capitalize on these digital assets' rapid value changes. In this case:

• Crystalcomp, a speculative venture, appreciated by \( 16\% \).
• Flyboys grew even more at \( 20\% \).
• Zumcorp also saw an \( 18\% \) rise.

Investors in speculative stocks must carefully track market trends and company performance. Analyzing such trends guides decision-making, helping maximize profits from stock market trades. This approach ensures informed financial positioning, acknowledging the potential for both rewarding payouts and possible loss scenarios.