Math Problem Statement

Two cars are 310 miles apart and are traveling toward each other. One car is traveling 8 mph faster than the other car. The cars meet in 2.5 hours. Find the speed of each car.

Solution

Let's define the speeds of the two cars and use the information given to form equations.

  1. Let the speed of the slower car be xx mph.
  2. Then, the speed of the faster car is x+8x + 8 mph.

Since they are moving toward each other, the total distance they cover when they meet is the sum of the distances each car travels in 2.5 hours. This total distance is given as 310 miles.

Step 1: Set Up the Equation

The distance each car travels is given by its speed multiplied by the time. Since the cars meet in 2.5 hours, the combined distance equation is:

x2.5+(x+8)2.5=310x \cdot 2.5 + (x + 8) \cdot 2.5 = 310

Step 2: Simplify the Equation

Distribute 2.52.5 to both terms:

2.5x+2.5(x+8)=3102.5x + 2.5(x + 8) = 310

Combine like terms:

2.5x+2.5x+20=3102.5x + 2.5x + 20 = 310

5x+20=3105x + 20 = 310

Step 3: Solve for xx

Subtract 20 from both sides:

5x=2905x = 290

Then, divide by 5:

x=58x = 58

Step 4: Find the Speed of Each Car

  • The slower car’s speed is x=58x = 58 mph.
  • The faster car’s speed is x+8=58+8=66x + 8 = 58 + 8 = 66 mph.

Answer

  • The slower car's speed is 58 mph.
  • The faster car's speed is 66 mph.

Would you like further explanation or details on any part?

Here are some additional related questions to explore:

  1. What if the cars had started 500 miles apart? How would that change their speeds?
  2. How long would it take for the cars to meet if the faster car was only 5 mph faster?
  3. What if one car started 30 minutes after the other? How would you set up the problem?
  4. If one car was driving 50 mph, what would be the speed of the other car?
  5. How would the solution change if they were traveling away from each other instead?

Tip: For problems involving relative speeds, remember that when two objects move toward each other, you add their speeds, while if they move away, you subtract.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Distance-Rate-Time Relationships

Formulas

Distance = Speed × Time
Combined speed when moving toward each other = speed1 + speed2

Theorems

Basic properties of linear equations
Distance-rate-time relationship in motion

Suitable Grade Level

Grades 8-10