Trains A and B are 190 miles apart. Train A leaves one hour before

The great thing about these "multiple traveler" questions is that they can be solved in more than one way.
All we need to do is start with a word equation.
Since Train A travels for 1 hour longer than Train B, we can write:

Train A's travel time = (Train B's travel time) + 1
Let d = the distance Train A traveled
This means that 190 - d = the distance Train B traveled

Now let's transform our word equation into an algebraic equation.
Train A's travel time = (Train B's travel time) + 1
Time = distance/speed
We get: d/15 = (190 - d)/10 + 1
Multiply both sides by 30 to get: 2d = 570 - 3d + 30
Simplify: 2d = 600 - 3d
Add 3d to both sides: 5d = 600
Solve: d = 120

Answer:

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In one hour the distance traveled by A is 15 miles..

Thus the effective distance between A and B after 1st Hour is 175 miles (ie, 190 - 15 )

Let the distance traveled by train A be d (When they meet ), so the distance traveled by B is ( 175 - d )

Now, \(\frac{d}{15} = \frac{( 175 - d )}{10}\)

So, 10d = 2625 - 15d

Or, 25d = 2625

or, d = 105

So, Total distance travelled by train A is 120 miles ( i.e 105 + 15 )

Hence answer will be option (E)
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After the first hour, A traveled 15 miles. Starting from this point, B began to travel toward A. Call x as the time for both trains meet. We have: 15x (A) + 10x (B) = 190 -15 => x =7. So, at the time both trains meet, A traveled 15 * (7+1) = 120 miles.

Distance b/w A and B is 190 miles

A takes a head start of 1 hr and covers 15 miles in it.

So after first hour distance b/w them is 190-15 = 175 hrs.

They will meet in time (Relative speed concept) = 175/(15+10) = 7

Total time train A travelled = 1 + 7 = 8 Hrs

Total distance covered = 15*8 = 120 miles.

Option E.

HTH

Official solution from Veritas Prep.

Guiding Principles for Collision Problems

1. Make sure that the two entities are starting at the same time. If not, account for it as shown above, and reset the problem with numbers for when they are starting at the same time.

2. Add the rates of the two trains to get the total speed with which they are covering the distance. If one train is traveling at 15mph and the other at 10mph, then together they are covering 25 miles every hour.

3. Divide the distance between the two trains by their combined speed. The two trains are 175 miles apart when they start, and together they are covering 25 miles per hour, so it takes \(\frac{175}{25}\), or 7 hours, until they collide.

4. Once you have the time to collision, go back to either train to see how far they went. In 7 hours train A traveled 7hrs x 15mph, or 105 miles; train B traveled 7hrs x 10mph, or 70 miles.

5. Lastly, and on this problem most importantly, account for any distances traveled by the train that started first. Here train A traveled 15 miles before they both started and then 105 miles after, so the correct answer choice is E, 120 miles.
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as we can see both the trains meet after traveling for the same time , as a started 1hr. before B then we can calculate that train A has traveled 15miles and now 175miles they both will travel so relative speed equal to 10+15=25 and they will meet after 175/25=7hrs as time is constant then a has traveled 15+7*15=120miles
so answer must be E ,
the better way to tackle time distance problems always find the constant factor such time,distance and rate .

We are given that trains A and B are traveling toward each other, so we have a “converging rate problem” in which we can use the formula:

Distance traveled by train A + Distance traveled by train B = total distance

Since the two trains started 190 apart, the total distance is 190, so we have:

Distance traveled by train A + Distance traveled by train B = 190

We are given that train A travels at a rate of 15 mph and leaves one hour before train B. We are also given that train B travels at a rate of 10 mph.

We can let the time of train B = t and, since train A left one hour earlier and will have traveled for one more hour than train B, at the time they meet, the time of train A = t + 1.

Since rate x time = distance, we can calculate the distance, in terms of t, of both trains A and B.

Distance of train A = 15(t + 1) = 15t + 15

Distance of train B = 10t

Now we can substitute these values into our total distance formula and determine t.

15t + 15 + 10t = 190

25t = 175

t = 7

Thus, when the trains meet, train A has traveled (15 x 7) + 15 = 120 miles.

Answer: E
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Train A would travel 15 miles in 1 hour before Train B starts moving.
So, the distance left between the train = 190 - 15 = 175 miles.
Now, train B has also started moving at 10mph towards train A.
Since, both of them are traveling towards each other, we can add their speed to find the time taken to meet.
Time taken to meet = 175/25 = 7 hr.
Distance traveled by train A = 7 x 15 + 15 = 120 miles.
Hence E.

The problem with your solution starts the how you assigned your variables (above)

We're told that Train A leaves one hour before train B does
This means that Train A's travel time is 1 hour greater than Train B's travel time

If we let A=t and B=t+1, then train B has the longer travel time.

Cheers,
Brent

Responding to a pm: You should use the concept of relative speed here.

Train travels for an hour and covers a distance of 15 miles. So the distance between them now is 190 - 15 = 175 miles.

Now train A and B are travelling toward each other and need to cover 175 miles together.

Time taken = 175/(15 + 10) = 7 hrs

Total distance covered by train A in its total 8 hrs of travel = 15*8 = 120 miles

Answer (E)

Check this: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/0 ... elatively/

Find: distance of A
Given: total D 190, Sb=10, Sa=15, Tb=t, Ta=t+1
one way to solve

A started 1 hour before and covered 15 miles

Remaining distance 190-15 = 175 miles covred at combined speed of (15+10)=25
Total time 175/25 = 7 hours

Distance traveled by A in 7 hours = 15*7= 105
Total distance by A = 105+15 = 120

\(\frac{A}{B} = \frac{15 mph}{10 mph} = \frac{3}{2}\)

When A and B travel toward each other, the 3:2 ratio above implies the following:
Of every 5 miles, A travels 3 miles, while B travels 2 miles, with the result that A travels \(\frac{3}{5}\) of the distance.

Thus:
If A and B were to start traveling toward each other right from the start, A would travel \(\frac{3}{5}\) of the 190 miles:
\(\frac{3}{5}*190 = 114\) miles.
Since A travels for one hour on its own and thus travels for LONGER than B, the distance traveled by A must be MORE THAN 114 MILES.


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Time Speed Distance
A X-1 15 m/h 15(X - 1)
B X 10 m/h 10X

15(x-1) + 10x = 200 - 15
25x = 200
x = 8;
A traveled 15 + 15*7 = 120 miles

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