Question

You have been following your stock investment over the past 4 months. Your friend tells you that his stock's value has doubled in the past 4 months. You know that your stock did better than that. Without telling your friend how much your original investment was, you would like to do a little boasting. The following table shows your stock's activity each month for 4 months. (TABLE CAN NOT COPY) a. Represent your original investment by \(x\) dollars and write an algebraic expression to represent the value of your stock after the first month. b. Use the result from part a to determine an expression for the value at the end of the second month. Simplify the expression if possible. Continue this procedure until you determine the value of your stock at the end of the fourth month. c. Explain to your friend what has happened to the value of your stock over 4 months. d. Instead of simplifying after each step, write a single algebraic expression that represents the changes over the 4 -month period. e. Simplify the expression in part d. How does the simplified algebraic expression compare with the result in part b? f. Your stock's value was 4678.75 dollar four months ago. What is your stock's value today?

Short answer

Answer: The final stock value after four months is 1.41372x, or 1.41372 times the initial investment.

Step-by-step solution

a. Stock Value After First Month

For the first month, let's assume that the original investment is \(x\) dollars. If the stock's value increased by 10%, the new value will be \(x + 0.10x\). This can be represented as \(x(1 + 0.10)\) or \(1.10x\).

b. Stock Value for Second, Third, and Fourth Month

To find the stock value for each month, we will multiply the previous month's value by the respective growth rate (1 + percent increase). Second Month: After the first month, the value is \(1.10x\). In the second month, the value increases by 5%. So, the new value will be \(1.10x (1+.05) = 1.10x\times 1.05 = 1.155x\). Third Month: After the second month, the value is \(1.155x\). In the third month, it increases by 2%. Hence, the new value will be \(1.155x\times(1+0.02) = 1.155x\times1.02 = 1.1781x\). Fourth Month: After the third month, the value is \(1.1781x\). In the fourth month, it has a 20% increase. So, the new value will be \(1.1781x\times(1+0.20) = 1.1781x \times 1.20 = 1.41372x\).

c. Explain the Value Change to Your Friend

During the four-month period, the stock's value increased as follows: In the first month, it increased by 10%, then 5% in the second month, 2% in the third month, and finally 20% in the fourth month. Overall, the stock's value increased from the original investment to 1.41372 times the initial amount, showing a greater growth than your friend's stock, which only doubled in value.

d. Single Expression for 4-Month Period

Instead of simplifying after each step, you can use a single algebraic expression to calculate the stock's value after the four-month period like this: \((1 + 0.10)(1 + 0.05)(1 + 0.02)(1 + 0.20)x\)

e. Simplify the Expression in Part d

Simplifying the expression from part d, we have: \((1.10)(1.05)(1.02)(1.20)x\) Multiplying the values, we get \(1.41372x\). This simplified expression matches the result from part b, showing the same increase in the stock value.

f. Stock's Value Today

The value of the stock four months ago was 4678.75 dollars, and we need to find its value today. We can use the simplified expression from part e to calculate the stock's current value: 1.41372x = 1.41372 * 4678.75 Current Value = 6613.31 dollars So, the stock's value today is 6613.31 dollars.