It is 74.4 miles from Dwane and Patti's house to Martha's house. Patti leaves home at 10:00 and drives toward Martha's house. Dwane and Martha set off half an hour later from their own houses and drive toward the other's house. The average speed of the three friends is 54 mph and they all meet at 11:06. How fast does each drive?
This is quite confusing problem and the people setting the problem should keep common sense handy when designing a problem.
The confusion is created because Dwane and Patti live in the same house but leave at different times to go to meet the same person Martha at the same time and point on the road. More confusing is Martha also leaves her house same time as Dwane. Considering this is car driving (the distance is 74.4 miles) it doesn't make much sense and then all-three meet in-between. That it just not a close to real life situation.
In any case, the once this is understood the problem is simple: Speeds P(Patti)+D(Dwane)+M(Martha)=3x54, 1.1P=0.6D (As Patti and Dwane travel the same distance in 1.1 hr and 0.6 hr), and 0.6D+0.6M=74.4 (Dwane and Martha's distance added to the total distance).
Solving these three equations:
P+D+M=162
1.1P=0.6D
0.6D+0.6M=74.4
That gives:
P=38 mph
D=69 2/3 or 69.67 mph
and
M=54 1/3 or 54.33 mph
To determine how fast each person drives, we first need to determine the time it takes for each person to meet. Let's break down the timeline:
Patti leaves home at 10:00.
Dwane and Martha leave half an hour later, which would be at 10:30.
They all meet at 11:06.
To find the time it takes for Patti to reach the meeting point, we need to subtract the departure time from the meeting time:
11:06 - 10:00 = 1 hour and 6 minutes.
Now, let's convert this time into decimal hours to make it easier for calculations:
1 hour + (6 minutes ÷ 60) hours = 1.1 hours.
Since Patti drives at an average speed of 54 mph, we can use the formula: distance = speed × time to find the distance Patti traveled.
Distance = 54 mph × 1.1 hours = 59.4 miles.
Since Dwane and Martha started their journey at 10:30 and met Patti at 11:06, their travel time is 36 minutes.
36 minutes ÷ 60 = 0.6 hours.
Since they meet at the same point, the total distance covered by all three is 74.4 miles.
Now, let's calculate Dwane's and Martha's speeds. Let's assume Dwane drives at speed x mph and Martha drives at speed y mph.
The equation for Dwane's distance is: Dwane's speed × Dwane's time = 74.4 miles - 59.4 miles (Patti's distance).
x mph × 0.6 hours = 15 miles
0.6x = 15
x = 15 ÷ 0.6
x = 25 mph.
Similarly, the equation for Martha's distance is: Martha's speed × Martha's time = 15 miles.
y mph × 0.6 hours = 15 miles
0.6y = 15
y = 15 ÷ 0.6
y = 25 mph.
Therefore, both Dwane and Martha drive at a speed of 25 mph, while Patti drives at a speed of 54 mph.