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(male narrator) In this video,
we will look at solving distance problems
where individuals are moving in opposite directions.
To set up distance problems, we're gonna organize
the information of the problem in a table
using the equation that the rate, times the time,
will tell you the distance traveled.
If people are traveling in opposite directions--
one going to the left and one going to the right--
while the distances may not be the same...
the first distance and the second distance
will give us a total distance.
We will add the distances together.
So let's take a look at an example
where we do just that.
In this problem,
Brian and Jennifer both leave at the same time,
traveling in opposite directions.
Let's organize what we have in a table:
rate times time, equals distance.
For Brian, we're told that he drove 35 miles per hour.
This is his rate.
For Jennifer, we're told she drove 50 miles per hour.
We wanna know after how much time--
we don't know, but it's the same time for both--
will the total distance between them be 340 miles.
We calculate distance by multiplying rate by time:
35t and 50t.
In other words, Brian's distance is 35t,
and Jennifer's distance is 50t.
Adding those distances together--35t plus 50t--
will give us the total distance traveled of 340 miles.
We can solve this equation by combining like terms
to get 85t, equals the 340 miles.
Dividing both sides by 85 to solve for t,
we find the amount of time needed is 4 hours...
for them to be 340 miles apart.
Let's try another example
where the two are traveling in opposite directions.
In this problem, Maria and Tristan...Tristan...
are 126 miles apart...
coming towards each other.
Setting up our table,
knowing that rate times time, equals a distance,
Maria is biking 6 miles per hour faster than Tristan.
We don't know how fast Tristan is biking.
We just know Maria is 6 miles per hour faster,
which we show with r plus 6.
We know the time they are traveling is 3 hours,
and so, we fill in our table.
To get the distance, we multiply these together:
3 times r, plus 6; and 3 times r.
This means Maria's distance is 3 times r, plus 6;
and Tristan's distance is 3r.
Traveling in opposite directions,
we know we can take the two distances
and add them together to equal the total distance of 126.
We can solve this equation
just as we've solved all our equations before.
First distributing through the parentheses,
3r plus 18 is... plus 3r equals 126.
Combining like terms gives us: 6r plus 18, equals 126.
Subtracting 18 from both sides will give us: 6r equals 108.
Dividing both sides by 6
gives us our final answer for r, which is 18.
r, we said, was Tristan's rate,
so Tristan is traveling 18 miles per hour.
Maria--traveling 6 miles faster--
is 24 miles per hour.