The price of Stock A at 9 AM was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at... The price of Stock A at 9 AM was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at the rate of $0.14 per hour. If the stocks continue to increase and decrease at the same rates, in how many hours will the prices of the stocks be the same?
Understand the Problem
The question is asking how long it will take for the prices of Stock A and Stock B to be equal, given their rates of increase and decrease over time. Stock A increases by $0.06 per hour starting from $12.73, while Stock B decreases by $0.14 per hour starting at $13.48. We will set up an equation to find the time when the two stock prices are the same.
Answer
The prices will be the same in approximately $2.85$ hours.
Answer for screen readers
The prices of the stocks will be the same in approximately ( 2.85 ) hours after 12 PM.
Steps to Solve
- Define the Variables for Stock Prices Let ( t ) represent the number of hours after 12 PM when Stock B starts decreasing.
- The price of Stock A at 12 PM, after 3 hours of increase (from 9 AM to 12 PM), is calculated as follows: $$ \text{Price of Stock A} = 12.73 + 3 \times 0.06 = 12.73 + 0.18 = 12.91 $$
- Write Equations for Stock Prices Next, express the prices of both stocks as functions of ( t ).
-
Price of Stock A after ( t ) hours since 12 PM: $$ \text{Price of Stock A} = 12.91 + 0.06t $$
-
Price of Stock B after ( t ) hours since 12 PM: $$ \text{Price of Stock B} = 13.48 - 0.14t $$
-
Set the Prices Equal and Solve for ( t ) Now, set the two equations equal to each other to find when the stock prices are the same: $$ 12.91 + 0.06t = 13.48 - 0.14t $$
-
Combine Like Terms Rearranging the equation gives: $$ 0.06t + 0.14t = 13.48 - 12.91 $$ $$ 0.20t = 0.57 $$
-
Solve for ( t ) Divide both sides by 0.20 to isolate ( t ): $$ t = \frac{0.57}{0.20} = 2.85 $$
-
Convert to Hours Since ( t ) is in hours and may need to be expressed in a clearer format:
- 2.85 hours equals 2 hours and 51 minutes (0.85 hours × 60 minutes/hour).
The prices of the stocks will be the same in approximately ( 2.85 ) hours after 12 PM.
More Information
This solution shows how to model real-world situations mathematically by using linear equations. Stock price movements can be predicted using rates of changes.
Tips
- Forgetting to account for the time difference between when Stock A starts increasing and Stock B starts decreasing.
- Incorrectly rearranging the terms when setting the equations equal to each other.
AI-generated content may contain errors. Please verify critical information
