Speed, distance and time question

Question

R and A travel with uniform speed from A to B and B to A respectively via the same route. R starts at 5:00 am while A starts fours hours later. They meet on the way at 1:10 PM. At what time do they reach their destinations if both of them reach at the same time?

Answer 1

Answer: 7:00 pm.

Solution:

The key to solving this problem is three observations:

1. The first train travels the last part of its route to City B in the same amount of time as the second train travels the last part of its route to City A.
2. At the time of intersection, both trains are equidistant from City A.
3. At the time of intersection, both trains are equidistant from City B.

From (1), let h represent the number of hours past 1:10 pm that the trains travel to their destinations. We will use 1⅙ to represent 1:10 and 13⅙ to represent 1:10 pm.

Both trains arrive at their destinations at the same time. Since the second train leaves 4 hours (at 9:00 am) after the first train, then it must be traveling faster that the first train.

Let v = the constant speed of the slower train -- the one headed towards City B.

Let V = the constant speed of the faster train -- the one headed towards City A.

At 1:10 pm, the first train has been traveling since 5:00 am at speed v. The distance it has traveled is then v*(8 ⅙) = 49 v / 6.

At 1:10 pm, the second train needs h more hours at speed V to reach City A. The distance it will travel is then h * V.

From (2) above, these two quantities must be equal:

49 v / 6 = h V

or

49 v = 6 h V

At 1:10 pm, the first train will travel h more hours at speed v to reach City B. The distance it will travel will then be v*h.

At 1:10 pm, the second train consumed 4 ⅙ hours at speed V to reach the intersection point. The distance it traveled was V*(4 ⅙) = 25 V / 6.

From (c) above, these two distances must be equal:

v h = 25 V / 6

or

6 v h = 25 V

These two equations, when combined, will yield h. To get the answer, we simply add 13 ⅙.

Cross multiply the equations as follows:

(49 v) * (25 V) = (6 h V) * (6 v h)

49 * 25 = (6 h)²

h = 7 * 5 / 6 = 35 / 6 = 5 ⅚ hours

Answer: 13⅙ + 5 ⅚ = 19, or 19:00, or 7:00 pm