



You drive
down the road at 55 miles per hour for 2 hours. 





How far
did you go? 





Some
people can do this one in their heads. 





The
answer is 110 miles. 





The
actual formula for doing problems like this one is ... 











So for
the problem we started with ... 





Speed you
were going = 55 mph 


Time you
were traveling = 2 hours 


Distance
traveled = ? (what we need to find out) 





So: 


Distance
Traveled = 55 × 2 = 110






Remember
the page on percents? 


We said
that percent was the combination of per 


(meaning
divided by) 


and cent
(meaning 100). 





Here,
when we say 55 miles per hour 


the per
means the same thing, divided by. 


So 55
miles per hour actually means 


"55
miles divided by one hour." 











In the
problem we multiplied this by 2 hours. 


What we
were actually doing was: 











Hey, this
one looks like one of those 


unit
conversion things! 





Well,
that's part of what's going on here. 


We cancel
the hours in the numerator 


with the
hours in the denominator. 











We don't
need to write in a denominator of 1 


so we can
lose it ... 











The moral
of he story is this. 


If the
problem said 


"You
were going 55 miles per hour for 20 minutes," 


you would
have to convert the 20 minutes 


to
whatever that is in hours before you could do the problem. 





Here's a
picky point. 


Generally
in these problems, they don't use the word speed. 


Instead,
they use the word rate. 


Just
remember that rate means speed. 


It's a
thesaurus thing I guess. 





Example: 





You drive
down the highway at 65 miles per hour for 20 minutes. 


How far
IN FEET did you go? 





OK, OK,
they want the answer in feet. 


That's no
big deal. 


Somewhere
along the line in the problem, 


we need
to convert our measures to feet. 





We could
work the whole problem in miles if we want 


and then
convert to feet at the end. 





We could
also change 65 miles per hour 


to
whatever that is in feet per hour to begin with 


and then
work the problem with that. 


Do
whatever is easier for you. 


It
doesn't matter. 


If you do
the math right, you'll get the right answer. 





Distance =
(What you want to find out) 


Rate (means
speed) = 65 miles per hour 


Time = 20
minutes 











So we've
got hours and minutes in the same problem. 


One of
them needs to be changed 


to be the
same units as the other. 


The
easiest way is to change 20 minutes 


to some
amount of hours. 





Remember
how we do it? 











So now we
can do the problem ... 











The
little line over the 66 means that this number 


(the 6's)
repeat forever. 





So now we
just need to change miles to feet. 











Let's do
another one ... 





Example: 





We drive
for 4 hours and go 190 miles. 


What was
our average speed? 





We might
not have been going the same speed 


for the
whole 4 hours, 


so we
want the average speed for this time ... 





Distance
Traveled = Rate ×
Time






Distance
= 190 miles 


Rate =
(We don't know) 


Time = 4
hours 





So ... 


190
miles = Rate ×
4 hours






Let's use
R to stand for rate and solve the problem for it. 


The math
works just the same with an R 


as it
does with an X ... 





190
miles = Rate x 4 hours






Divide
both sides by 4 hours and simplify ... 











The rate
(speed) was 47.5 miles per hour. 





OK, one
last one ... 





Example: 





Two
trains start at the same point. 


One train
travels east at 40 miles per hour for at least 4 hours.. 


The other
train travels west at an unknown speed. 


4 hours
later, the trains are 360 miles apart. 


What was
the average speed of the second train. 








OK, now
we have 2 moving objects. 


The
distance we are given is the total distance they went together 


(Since
they were traveling in opposite directions). 





We have: 





360 = Distance
traveled by train 1 + Distance traveled by train 2 





We know
that: 


Distance
Traveled = Rate ×
Time






So; 


360 = Rate of
train 1 × Time + Rate of train 2 × Time 





And we
know: 





Rate of train
1 = 40 miles per hour 


Rate of train
2 = we don't know; call it R 


Time (for
both) = 4 hours 





So we
have: 








The rate
of the second train is 50 miles per hour. 





copyright 2005 Bruce Kirkpatrick 
