The price of Stock A at 9 A.M. was ​$15.13. Since​ then, the price has been increasing at the rate of ​$0.09 each hour. At noon the price of Stock B was ​$15.63. It begins to decrease at the rate of ​$0.13 each hour. If the two rates​ continue, in how many hours will the prices of the two stocks be the​ same?

Let's set up the problem by defining the variables and formulating equations for the prices of Stock A and Stock B over time.

### Step 1: Define the variables
- Let $ t $ be the number of hours after 9 A.M.
- The price of Stock A at time $ t $ will be denoted as $ P_A(t) $.
- The price of Stock B at time $ t $ will be denoted as $ P_B(t) $.

### Step 2: Price of Stock A
- At 9 A.M. (when $ t = 0 $), the price of Stock A was $15.13.
- The price increases by $0.09 per hour.
- Thus, the price of Stock A after $ t $ hours is:
  \[
  P_A(t) = 15.13 + 0.09t
  \]

### Step 3: Price of Stock B
- At noon (which is 3 hours after 9 A.M.), the price of Stock B was $15.63.
- It decreases by $0.13 per hour.
- The price of Stock B at noon corresponds to $ t = 3 $, so for time $ t \geq 3 $, the price of Stock B after $ t $ hours is:
  \[
  P_B(t) = 15.63 - 0.13(t - 3)
  \]
  Simplifying the equation:
  \[
  P_B(t) = 15.63 - 0.13t + 0.39
  \]
  \[
  P_B(t) = 16.02 - 0.13t
  \]

### Step 4: Set the prices equal
To find when the prices of the two stocks are the same, set $ P_A(t) = P_B(t) $:
\[
15.13 + 0.09t = 16.02 - 0.13t
\]

### Step 5: Solve for $ t $
- Move all terms involving $ t $ to one side:
  \[
  0.09t + 0.13t = 16.02 - 15.13
  \]
  \[
  0.22t = 0.89
  \]
- Solve for $ t $:
  \[
  t = \frac{0.89}{0.22} = 4.045 \text{ hours}
  \]

### Conclusion:
The prices of the two stocks will be the same approximately 4.045 hours after 9 A.M. This means they will be equal at around 1:02 P.M.

Would you like more details on any part of the solution?

Here are 5 related questions for practice:
1. How would the solution change if the rates of change for the stocks were different?
2. What if the price of Stock A started decreasing instead of increasing?
3. How would you solve if Stock B’s price was given at a different time, like 11 A.M.?
4. How would the equation change if Stock B decreased at a non-constant rate?
5. What would happen if the price of Stock A had a time delay before starting to increase?

**Tip**: When dealing with rate problems, always set up clear equations based on the initial condition and rate of change.

Discover how to calculate when two stocks, Stock A and Stock B, will have the same price based on given rates of increase and decrease. This problem explores key algebraic concepts and linear equations. Suitable for students in Grades 8-10.

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