Proportions with unit conversion

Now dimensional analysis
is also really handy when we have a problem
like this one that’s gonna involve
unit conversion. So a bicycle is traveling
15 miles per hour. How many feet will it cover
in 20 seconds? So the problem here,
of course, is that, uh…this measurement
is in hours, while this measurement
is in seconds. This measurement
is in miles, while this measurement
is in feet. And so we’re gonna need
to do some conversion. Uh…so let’s start
with our quantity–20 seconds. So to convert this
into some other units, um…I’m gonna multiply
by an equivalence. And in this case,
I wanna get rid of seconds, uh…and introduce
something else to replace it. And the immediate conversion
that comes to mind is I know that 1 minute
is 60 seconds. Uh…and minutes
are better than seconds, but I really want hours. So I’m gonna do
the same thing again. I’m gonna
eliminate minutes. Notice that this is
in the numerator. This is in the denominator,
so they will cancel…or reduce. Uh…and,
uh…let’s see, I know that 1 hour
is 60 minutes. Uh…and so now,
the seconds are gonna reduce. The minutes
are gonna reduce. Let’s see,
multiplying, I’m gonna end up
with 20 over 3600 is 1 over 180…
1 over 180. And this is now
inhoursas the units. So the time we’re talking about
is 1 over 180 hours. So I’ve got 1 over 180–
and again that’s hours. And…but I’m really interested
in distance. And so I’ve got to multiply this
by another equivalence, and the equivalence
I’m gonna use is this one– in this case,
coming from the speed that the bike
is traveling. So I want to eliminate hours,
introduce miles, uh…and this tells me that
the bicycle is traveling 50 miles in one hour… Hours are gonna get
reduced. We’re gonna be left
with miles. We’re gonna be left
with 15 over 180, uh…is…one-twelfth
of a mile. So in those 20 seconds, the bicycle travels
one-twelfth of a mile. And now all we need to do
is convert into feet. So to do that, we’re going to eliminate miles,
introduce feet. Uh…and looking on my…
looking it up, um… it looks like 1 mile
is 5,280 feet. And so…we end up
with 5,280, uh…feet divided by 12 is…
let’s see, is…440 feet. So in those 20 seconds, the bicycle
will travel 440 feet. Uh…and now this conversion
works just fine. Um…and notice
that we sort of did it in three different steps here. We converted the time. Then we converted it
to a distance, and then we converted
the distance into feet. Um…if we
were really wanting to do this sort of…
as quickly as possible, we actually could’ve done
this a little bit differently by, uh…just stringing
all these conversions… into one. So notice that this conversion
now is here. This conversion
now is here. And if we multiply all
of our numerators and all of our denominators
and divide them, we would end up
with exactly the same 440 feet that we came up with doing it
in three different steps.

One thought on “Proportions with unit conversion

  1. In a testing environment, how would you work out the miles to feet conversion without looking it up?

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