Question

Sophie bought stock in a company two years ago that was worth xx dollars. During the first year that she owned the stock, it decreased by 21%. During the second year the value of the stock increased by 32%. Write an expression in terms of xx that represents the value of the stock after the two years have passed.

Answer 1

Let's denote the initial value of the stock as \( xx \).

1. After the first year: The stock decreases by 21%. To calculate the value after this decrease, we can subtract 21% of \( xx \) from \( xx \):

   \[ \text{Value after first year} = xx - 0.21 \cdot xx = xx(1 - 0.21) = xx(0.79) \]

2. After the second year: The stock increases by 32%. We will take the value at the end of the first year and increase it by 32%. First, we calculate 32% of the value after the first year:

   \[ \text{Increase} = 0.32 \cdot \left( xx \cdot 0.79 \right) = xx \cdot 0.79 \cdot 0.32 \]

   Therefore, the value after the second year is:

   \[ \text{Value after second year} = \text{Value after first year} + \text{Increase} \] \[ = (xx \cdot 0.79) + (xx \cdot 0.79 \cdot 0.32) \]

   We can factor out \( xx \cdot 0.79 \):

   \[ = xx \cdot 0.79(1 + 0.32) = xx \cdot 0.79(1.32) \]

3. Final expression: Now we can multiply out the constants:

   \[ \text{Final value} = xx \cdot (0.79 \times 1.32) \]

   Calculating \( 0.79 \times 1.32 \):

   \[ 0.79 \times 1.32 = 1.0458 \]

So, the value of the stock after two years is:

\[ \text{Value after two years} = xx \cdot 1.0458 \]

Thus, the final expression representing the value of the stock after two years is:

\[ xx \cdot 1.0458 \]