Trains A & B Are Traveling In The Same Direction On Parallel Tracks. Train A Is Traveling At 100 Mph And Train B Is Traveling At 120 Mph. Train A Passes A Station At 7:10pm. Train B Passes The Same Station At 7:22pm. At What Time Will B Catch Up W/ A?

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Oddman Profile
Oddman answered
Let t be the time in minutes after 7:10 at which train B catches train A.
The distance (in funny units) traveled by train A from the station to the meeting point is given by
d = t*100

The distance traveled by train B from the station to the meeting point is
d = (t-12)120

The two distances are equal (and we don't care what they are), so we can write
t*100 = (t-12)*120
100t = 120t - 1440 (use the distributive property)
0 = 20t - 1440 (subtract 100t from both sides)
0 = t - 72 (divide both sides by 20. 0/20 = 0)
72 = t (add 72 to both sides)

7:10 + 72 minutes = 7:10 + 1:12 = 8:22

Train B will catch train A at 8:22 pm.
The term 100t has units (miles/hour)(minutes). This is distance multiplied by (minutes/hour), or 60 times the distance in miles. Both sides of the equation have these funny units, so no harm is done. Usually, it is a good idea to pay attention to the units of the problem. It helps you make sure you're performing the arithmetic correctly.
If you are not intimidated by numbers in different number bases, you can work with the time directly.
100(t - 7:10) = 120(t - 7:22) (equation where t is actual clock time)
5t - 5(7:10) = 6t - 6(7:22) (divide both sides by 20 to make the numbers manageable)
6(7:22) - 5(7:10) = t (subtract 5t-6*7:22 from both sides)
(5+1)(7:10+0:12) - 5(7:10) = t (rearrange the product on the left. This makes it possible to cancel the 5*7:10 term. It is really an unnecessary step, but it makes the calculation a little more straightforward.)
5(7:10) + 5(0:12) + 1(7:10) + 1(0:12) - 5(7:10) = t (use the distributive law)
5(0:12) + 7:22 = t (collect terms. Recognize that 7:10 + 0:12 is 7:22)
1:00 + 7:22 = t (recognize that 5*0:12 is 0:60, one hour)
8:22 = t (clock time at which the trains meet)
thanked the writer.
Em commented
Oddman, thank you once again for your help with this. You explained it in a way that was very straight forward and made perfect sense to me.

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