Problem Solving

ktrout2020 wrote:Trains A and B are 190 miles apart. Train A leaves one hour before train B does, traveling at 15 miles per hour directly toward train B.Train B travels at 10 miles per hour directly toward train A. When the trains meet, how many miles has train A traveled?

a) 70
b) 85
c) 95
d) 105
e) 120

Source: Veritas

When Train B starts, Train A had covered 15*1 = 15 miles; thus, both trains together cover 190 - 15 = 175 miles

Since the trains are moving towards each other, their relative speed would be the sum of their speed = 15 + 10 = 25 mph

Time taken to cover 175 miles = 175/25 = 7 hours

Thus, Train A traveled for 1 + 7 = 8 hours. And Train A traveled for 8*15 = 120 miles

The correct answer: E

Hope this helps!

-Jay
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ktrout2020 wrote:Trains A and B are 190 miles apart. Train A leaves one hour before train B does, traveling at 15 miles per hour directly toward train B.Train B travels at 10 miles per hour directly toward train A. When the trains meet, how many miles has train A traveled?

a) 70
b) 85
c) 95
d) 105
e) 120

Source: Veritas

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.
In Scott's solution, he starts with the following word equation: Distance traveled by train A + Distance traveled by train B = total distance

Let's try a different 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: E

Cheers,
Brent

ktrout2020 wrote:Trains A and B are 190 miles apart. Train A leaves one hour before train B does, traveling at 15 miles per hour directly toward train B.Train B travels at 10 miles per hour directly toward train A. When the trains meet, how many miles has train A traveled?

a) 70
b) 85
c) 95
d) 105
e) 120

Source: Veritas

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 miles 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 thus 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