Let's solve this step by step:
Step 1: Understand the problem.
- John and Mary start at the same point and drive in opposite directions.
- John drives at 55 miles per hour (mi/hr) and Mary at 40 mi/hr.
- Mary's trip takes 15 minutes longer than John's.
- John drives 35 miles more than Mary.
Step 2: Establish variables for the unknowns.
Let x represent the distance Mary travels.
Therefore, x + 35 represents the distance John travels (since he travels 35 miles more).
Step 3: Convert all units to be consistent.
Convert the 15 minutes to hours since our speed is in miles per hour.
15 minutes = 15/60 hours = 0.25 hours.
Step 4: Write down the time equations.
Mary's time = John's time + 0.25 hours
We know that time = distance/speed.
So for Mary, her time is x / 40 hours.
For John, his time is (x + 35) / 55 hours.
Step 5: Set up the time equations based on the information given.
Therefore, x / 40 = (x + 35) / 55 + 0.25
Step 6: Clear the fractions by finding a common denominator or multiplying through by the denominators (in this case, 40 and 55).
Multiplying both sides of the equation by 40 * 55 (the least common multiple of 40 and 55) to eliminate the denominators gives:
55x = 40(x + 35) + 0.25 * 40 * 55
Step 7: Simplify the equation.
55x = 40x + 1400 + 550 (since 0.25 * 40 * 55 = 550)
55x = 40x + 1950
Step 8: Subtract 40x from both sides to get the x terms on one side.
55x - 40x = 1950
15x = 1950
Step 9: Divide both sides by 15 to solve for x.
x = 1950 / 15
x = 130
Step 10: Calculate Mary's and John's distances.
Mary's distance is x, which is 130 miles.
John's distance is x + 35, which is 130 + 35 = 165 miles.
Step 11: Verify the times for Mary and John to check consistency with the original problem.
Mary's time = 130 miles / 40 mi/hr = 3.25 hours.
John's time = 165 miles / 55 mi/hr = 3 hours.
Step 12: Confirm the time difference.
Mary's time - John's time = 3.25 hours - 3 hours = 0.25 hours (which is 15 minutes).
Final Step: Present the solution.
- Mary travels 130 miles at 40 mi/hr, and her trip takes 3.25 hours.
- John travels 165 miles at 55 mi/hr, and his trip takes 3 hours.
- Mary's trip does indeed take 15 minutes longer than John's, confirming the solution fits the problem statement.