A car X left town A for town B at the same time that another car Y left town B for town A, each travelling at constant speed. They met 80 minutes later and car X arrived at town B 36 minutes after car Y reached town A. How long did it take car X to reach town B?
Let x = speed of car X
y = speed of car Y
L = distance of AB
"meet 80minutes later"
L = 80x + 80y (distance = velocity * time)
The time for car X to reach town B is:
tx = ty + 36
But, tx = L/x and ty = L/y
Substitute,
tx = ty + 36
[80(x+y)]/x = [80(x+y)]/y + 36
Multiplying both sides by xy
80(x+y)y = 80(x+y)x + 36xy ~ 20(x+y)y - 20(x+y)x = 9xy
Expand:
20y^2 - 20x^2 - 9xy = 0
Divide both sides by x^2, and let k=y/x
20k^2 - 9k - 20 = 0
k = 5/4, so x = 4 and y= 5
Substitute values to the equation tx
tx = 180mins.
Is this correct? Thank you.
So, did you actually check your answer?
tx=180 means ty=144
So, total time for X is 180+80=260
total time for Y is 144+80 = 224
distance from A of X to meeting place is L/260*80
distance from B of Y to meeting place is L/224*80
Add those up and you do not get L.
Your mistake is here:
But, tx = L/x and ty = L/y
tx is not L/x. tx is the time X took after the initial 80 minutes. Not sure how you can fit that into your equations, but I did it like this:
If y = kx, then in the first 80 minutes, L=80x+80y = 80x(1+k)
(80x(1+k)-80x)/x = (80x(1+k)-80kx)/kx + 36
80(1+k)-80 = (80(1+k)-80k)/k + 36
80k^2 = 80+36k
20k^2-9k-20 = 0
k = 5/4
So, Y travels 5/4 as fast as X.
So, in the 1st 80 minutes, X went 4/9 of the way to B and Y went 5/9 of the way to A.
So, X took 5/4 as long to make the rest of the trip, or another 100 minutes.
Not sure whether the question is X's total time (180 min) or remaining time (100 min)
Check:
Y took 4/5 as long to get to A as it did to get to the meeting place. That's 64 minutes. Y's total time = 144 minutes.
180 = 144+36
From what I understood in the problem, the required is the total travel time of car X from town A to town B.
In the 180minutes, I think that the "80minutes later" is already included so no need to add it up.
To solve this problem, we need to determine the relationship between the speeds of the two cars and the time it took for car X to reach town B.
Let's assume that the speed of car X is denoted as Vx (in units of distance per time) and the speed of car Y is denoted as Vy.
We know that the total time taken for the two cars to meet is 80 minutes. This means that car X traveled for 80 minutes, and car Y also traveled for 80 minutes.
Now, let's break down the journey of each car:
1. Car X: Car X traveled from town A to the meeting point, which took 80 minutes. Then, it continued to town B and took an additional t minutes to reach there.
2. Car Y: Car Y traveled from town B to the meeting point, which took 80 minutes. Then, it reversed direction and traveled back to town A, taking an additional 36 minutes to reach there.
We need to find the value of t, which represents the time taken by car X to reach town B.
Given that the speed of a vehicle is equal to the distance traveled divided by the time taken, we can use the formula:
Distance = Speed × Time
Since car X and car Y met at the midpoint of their respective journeys, the distances traveled by both cars are equal.
Using this information, we can set up the following equation:
Vx × 80 minutes = Vy × 80 minutes + Vy × 36 minutes
Here, we are equating the distance covered by car X in 80 minutes to the distance covered by car Y in 80 minutes, plus the distance covered by car Y in an additional 36 minutes.
Simplifying the equation:
Vx × 80 = Vy × (80 + 36)
80Vx = 116Vy
Now, let's use this equation to find the relationship between Vx and Vy.
Once we have the relationship between the speeds of the two cars, we can find the time (t) it takes for car X to reach town B.
Since we are given that the speeds are constant, we can assume that Vx and Vy are constant throughout their journeys.
By rearranging the equation, we get:
Vx/Vy = 116/80
Simplifying further,
Vx/Vy = 29/20
This means that the ratio of the speeds of car X to car Y is 29/20.
Now, let's find the time (t) taken by car X to reach town B.
Since we know that the time taken by car Y to reach town A is 36 minutes, car Y traveled for a total of 80 + 36 = 116 minutes.
Using this information and the relationship between the speeds, we can find the value of t.
We can set up the following equation:
Vx × t = Vy × (t + 116 minutes)
Substituting Vx/Vy = 29/20,
(29/20) × t = t + 116
29t = 20t + (20 × 116)
9t = 2320
t = 2320 / 9
Therefore, car X took t = 257.8 minutes to reach town B.
So, it took car X approximately 257.8 minutes to reach town B.