Episode: 1502
Title: HPR1502: Practical Math - Units - Distances and Area, Part 2
Source: https://hub.hackerpublicradio.org/ccdn.php?filename=/eps/hpr1502/hpr1502.mp3
Transcribed: 2025-10-18 04:20:53

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Hello and welcome to Hacker Public Radio. This is Charles in New Jersey and I'm
back with another series on practical math. Today's show is going to focus on
using and converting between units of distance and area. I'm going to start with an
example. Now for absolute scales like distances all we need is a conversion
factor and a calculator if you need one. I don't think I'll be doing any
calculations that really need a calculator but if you need one get it out
now. Okay. Okay first I guess I'll pose a couple of problems. Now I know that a
foot is 12 inches so how many inches would there be in say 10 feet or with
that the other way? How many feet might there be in 660 inches? Two different
problems I'll warn you in advance so you can't just pick off the number that I've
quoted because they are two different problems but I chose the two problems
because one's going from inches to feet the other one's going from feet to
inches. Now it's clear that going either direction that a factor of 12 should
really be involved because a foot is 12 inches and how do I know when I'm going
to either multiply or divide by 12 in the conversion? Well let's take a look at
it. Now if I multiply 12 inches divided by one foot by any quantity that's in
feet I'm multiplying let's say it's three feet that I'm multiplying by that. Well
the three feet times 12 inches over one foot is equal to well just
rearranged terms and I get 12 inches times three feet over one foot and feet
cancel up and down so I get a unitless value of three so then I can say 12
inches times three is 12 times three inches and I get 36 inches which I know is
three feet because I've used the yardstick I know that three feet is a yard and
it's also 36 inches so that accords with what we already know it accords with
intuition and it uses very simple techniques like multiplying by one in a way
that the unwanted thing cancels out leaving you with the quantity the units
that you want. So let's use this in the problems that we've already talked about
so let's say that I want to convert from feet two inches and I want to go back to
my problem how many inches in 10 feet? Well 10 feet is equal to 10 feet times
one which is equal to 10 feet times 12 inches over one foot which we know to be
one because we've just done that and that is 10 feet divided by one foot times
12 inches. Now feet cancel I get a unitless number 10 so I can say that 12
inches times 10 is the same thing as 12 times 10 inches and that's 120 inches.
It's all very systematic I start with the units I have I multiply by a
conversion factor which has the units I want upstairs the units I want to get
rid of downstairs I get cancellation and I'm left with a multiplication problem
that's all in the unit that I want very cool. So let's solve the second example I
wanted to convert 660 inches to feet so I start with 660 inches which is equal
to the 660 inches times one I multiply that by the fraction one foot divided by
12 inches because I'm cleverly setting myself up to be able to cancel inches
with inches and be left with feet that's why I put feet on top that's the one
I want at the end inches on the bottom that's the one I want to cancel and I
think I'm gonna get the right answer when I just multiply through the numbers
and cancel the units and that's exactly what happens when we rearrange terms
this whole thing on the second step putting in the conversion factor can be
rearranged so they've got the original number 660 inches I was multiplying by
one foot over 12 inches so I can put that 12 inches directly under the 660
inches and then all I'm left with I'm multiplying by one foot oh that looks
good because it's clear when I've got inches over inches that those cancel and I
get a unit list number that's 660 divided by 12 and that ratio that answer 660
by divided by 12 which is 55 is now multiplied by one foot so it's clear that
that 55 times one foot is 55 feet and feel pretty confident that that's probably
the answer if I did the division of those two numbers correctly at least I know
I'm in feet so this gives you a real sense of confidence which you're going to
need because sometimes you don't have direct conversion factors and you have to
actually combine sets of factors may have to take one conversion step using one
identity a second one using a second conversion factor maybe even the third
conversion factor but if you do this step by step by step aiming for conversion
factors that cancel out units you don't want and put in units you do to go to
the next step you will zero in on the right answer in the proper units and if
you carry the units along with it you're much more likely to get the right answer
or at least to have done the right process so that you can go back over your
arithmetic and make sure that it wasn't some silly mistake that you made along
the way you're like calling nine times six fifty six instead of fifty four that
that kind of mistake you can fix because you can see how you did it okay great
so I just have to check some arithmetic that's much easier than determining at
each step do I divide or do I multiply because that gives you two choices to
make on every conversion factor you use if you had to use four of them then
you've got 16 combinations to try you don't want to go down that road make
your life easier carry the units and you'll see that you can convert between
units of distance and we're gonna get into area in just a minute and you can do
this with confidence says great so now we're gonna go to the next segment we're
gonna do conversions using compound conversion factors and that's just I
don't have a factor that goes directly from A to B so I have to go or A to Z
so I go A to B B to Z C to D and so on until I get to the unit I want
great or you can just run the units program and bash and be done with it okay
so suppose I want to find the number of inches in a furlong which is probably
quite a lot or the number of acres or hectares in a square mile hectare is the
pronunciation I learned for hectare which is 100 errors or 10,000 square
meters in the metric system but I'm getting ahead of myself so since my
almanac is poor and doesn't carry direct conversion factors for either one of
these I'm gonna start with what I have which is a bunch of these separate identities
and I'm gonna work my way through it now I know by definition that a furlong is
10 chains equal to 10 chains I've actually introduced that up above you can
rewind or look at the show notes to see that I've actually given you that
identity before now a chain is four rods so if I want to go stepwise from
chains to what inches and I have number of rods in a chain then if I multiply 10
chains by the quantity or the ratio of quantities I should say four rods
divided by one chain and that's equal to one then I'll now have an answer that's
not in chains but it'll be in rods okay but I want inches I don't have rods to
inches but I have rods to feet I know that a rod is 16 and a half feet but
multiplying by that's going to give me something in square rods per square
rods per foot not helpful but if I multiply the answer I've got so far that's
in rods by the ratio 16 and a half feet per rod then the rods will cancel and
I'll add up and feet excellent so now I'm in feet and I just need to convert
feet to inches so if I've got feet and I'm multiplied by one the ratio of 12
inches per one foot then feet upstairs and downstairs cancel I have a unit
less one down below and I multiply my quantity by 12 inches and I am good that
gives me my answer so if you look at the show notes you can see this drawn out
in full detail so a furlong is ten chains times four rods per chain well
chains cancel so that's 40 rods okay 40 rods times 16 and a half feet per
rod rods cancel so I get 40 times 16 and a half feet well that's same thing as
20 times 33 feet or 660 feet okay but that's still in feet I want inches all
right well 660 feet times 12 inches per foot feet cancel that 660 times 12
which is 660 times 10 plus 660 times two I had 660 times 10 which is 6600
and 660 times two is 1320 so I've got 7,920 inches and if I look at my
show notes peaking at the answer that's exactly what I have very cool and if
you unwind all that by dividing by 12 or very yet multiplying by one foot
over 12 inches and one rod over 16 and a half feet and one chain over four rods
you'll get back to 10 chains or one furlong excellent it all works so now let's
go for a leap we're going to do an area conversion very intimidating let's say
we're looking for the number of acres in a square mile which is equal to one
mile squared or one mile times one mile in this kind of equation it's as if one
mile is included as a factor twice so don't be too intimidated by the square
mile thing now area means that we're dealing in two dimensions so if we ever
end up with a unit of distance that's by itself when we started with an area
we are probably wrong keep an eye out for that so one square mile is how many
acres one square mile equals the square mile times one if you remember
correctly an acre is equal to 10 square chains or 10 chains squared or 10
chain times chain so if I'm going to multiply by anything I'm probably going to
aim for something that gives me acres to chain by chain but acres and chains
don't work with miles hmm so I can't go directly well what do I have that
relates chains and miles hmm well a furlong is an eighth of a mile so I want to
have a conversion factor that converts miles to furlongs and gets rid of the
miles on top in the numerator of the fraction and puts furlongs up above
since I have square miles I have to do this twice so before we get to involved
let's go from miles to square miles to square furlongs and I promise you
we'll get to the answer so one square mile is a square mile times eight fur
longs per one mile times eight furlongs times one mile that cancels both of the
one mile quantities in the one square mile that I started with gives me fur
longs squared and I promise this is going somewhere so one square mile times
eight furlongs per mile times eight furlongs per mile is eight times eight times
furlong times furlong or 64 square furlongs okay what do I do now well a fur
long is ten chains it's a definition or a conversion factor so I can
multiply 64 square furlongs by a conversion factor of what 64 square
furlongs times ten chains per furlong times ten chains per furlong gives me
square chains as my unit and furlong and furlong in the denominator cancel the
furlong and furlong in the numerator so I end up with 64 hundred 64 hundred
square chains okay but now I'm in square chains well fortunately in the
definition of an acre I did an extra little step that told me that one acre is
equal to ten square chains so I can multiply 64 hundred square chains times one
acre oh that's what I want divided by ten square chains the square
chains cancel and I end up with 64 hundred divided by ten or 640 acres equals
one square mile if you can handle those two calculations you're pretty good
with compound conversion factors once you practice this enough how to keep all
the straight and use your units in the calculation don't ever let go of
labeling your units because if you get into electrical units where you start
equating voltage with current if you make a mistake you'll want to know that
you're doing that and if you intended to do that you're wrong if you didn't
intend to do that you've made a mistake that you need to correct and actually if
you've done everything correctly on the arithmetic you may actually be able to
unwind a conversion factor or two and get back to the right answer and that
will be something that you can thank me for later so practice carrying units
through your unit conversions so start with your units on the left hand side
thing that you want to convert multiply by by conversion factors to take away
the units you don't want and give you a step toward the units you want and keep
working until you get into the right units and if you've done the arithmetic
and cancellations properly you'll have the right answer it helps to write it down
now it's time for segment three hey you ready to try metric the U.S. listeners
are going to be at disadvantage now because the metric system except in
certain circles never really caught on here maybe in medicine or in chemistry
class and that's true even though most of the English speaking world has
already adopted it even for things like speed limits and temperatures now if
you're thinking of units that are entirely within the metric system the units
conversions pretty easy because everything's in powers of 10 but even so it's
still a good idea to carry along the units when you're doing calculations
because even though you just have to move the decimal point really you have to
move it in the right direction so you might as well carry along the units even
when you're doing calculations in metric because it'll tell you exactly what
to do next you'll know whether you're going to multiply or divide now before I
start playing around with unit conversions let's get a sense of the metric
system and it's basic units and some relationships with the with the English
system and I'm not going to go with the the teeny tiny measurements I'm just
going to go with the ones we can kind of see with the naked eye and and relate to
in our daily lives I guess for the basics I'm going to start small and say that
I'm going to use the oh centimeter as the basic unit of distance I could have
chosen the meter it really doesn't matter distances are arbitrary as long as
you always use the system to convert from one unit to another it really doesn't
matter which one you call your basic unit as long as you've defined your
conversions correctly you'll always stay in good shape okay now the English
and metric systems have moved toward each other so that conversions are a little
bit easier than they used to be I won't get into who moved but now the
conversion from one inch to a certain number of centimeters is exact and that
conversion factor is one inch equals 2.54 centimeters exactly I don't know why
2.54 to do anything any different like making it 2.5 would probably be too much
of a move given that there are space programs using both types of units and
to make too much of a change would probably create too much disruption because
you're talking about vast distances and even small errors can mount up when you
scale them by really really really large numbers okay well that's the
conversion between English and and metric now let's go to other metric units
because that those conversions are easier a meter is 100 centimeters actually a
centimeter is probably more often defined as one one hundredth of a meter but in
in my world the centimeter is central because it's my little pet measurement
that I keep in a box and 100 of those make a meter now if I want to know what a
meter is in inches I take my previous equation it's in the show notes I say
1 meter is 100 centimeters that's equal to 100 centimeters times one which is
100 centimeters times one inch divided by 2.54 centimeters that puts
centimeters in the denominator to potentially cancel with my original unit so
I'll end up in inches so now my centimeters have canceled and I end up with
100 unitless over 2.54 unit less times one inch and that's about 39.37
inches which is I think what they taught you in school a kilometer or a
kilometer it's a same Canada is just a thousand meters pretty easy you can
convert those to inches pretty easily if you already know these two numbers
that I've given you so I won't go any further with that now go into area now I'm
not sure how these are pronounced because I've only read them and I've heard
the names of them pronounced in in a couple of different ways I'm going to say
that an air ARE is 100 square meters that's an area of a square that's 10
meters on the side the unit of area that's used in measuring lands and in
reports of forest fires or amount of arable land in a country or something like
that land under cultivation is called a hectare or a hectare in my
woods and that's a hundred errors a hundred errors is a hundred errors times one
which is a hundred errors times a hundred square meters divided by one air from my
previous identity I know that's equal to one so then heirs cancel I end up with
100 unitless times 100 square meters which is 10,000 square meters all very
cool now here's where we get practical even people in the US as provincial as
some people think we are like to watch sporting events like the Olympics or
drive cars or something like that for short distances not too short like
microns or something and not too long like space travel but let's say the the
middle distances that people run in races in the Olympics we can probably get
along with an approximation when we're converting from English to metric I'm
gonna let you in on a little secret when distances are really short you can't
really do this because there's a higher relative error from rounding off too
soon so if you're measuring a piece of wood for a project that's not a large
scale now you probably want to be close so you would probably do your conversion
let's say if it's one foot especially while that's one foot is 12 inches times
one we've just looked at conversion factor to go from inches to centimeters so
that would be 12 inches a foot is 12 inches times 2.54 centimeters per inch inches
cancel so I have a unitless 12 times 2.54 centimeters which is 30.48 centimeters
cut carefully now for larger but not too large moderate distances like the
distances covered in track and field events or football you can live with
approximation because it gives you a chance to get a mental picture or
intuition for the comparison between units that you know and a newer set of
units that aren't as familiar to you okay so we've already seen that a meter
somewhere around 39.37 inches let's suppose just among friends that I call
that about oh 1.1 yards just to make a benchmark so that a meter is now in my
approximate world 39.6 inches it's off a little yeah but my meter in this
reckoning is maybe what a quarter of an inch too long it's 0.23 inches that's
pretty close now if I'm planning an interplanetary space mission I'd probably
be in trouble with that kind of crude approximation but how bad really could it
be forgetting like a feel for the distances covered by athletes in the Olympic
games you know if I look at a hundred meter race and say it's 110 yards I'm
probably I'm about 23 inches over that's like two feet over 110 yards that's
not bad gives me a pretty good idea of how far it is and saves me a lot of fuss
in doing more precise multiplications and divisions and cancellations and
all that so at 200 meters it's four feet off at a kilometer it's 20 feet over I
mean unless you're trying to solve a ballistics problem of shooting a cannonball
a kilometer and trying to hit a small target 20 feet over kilometers pretty
decent you could apply this to how many kilometers there are in a mile how
would I do that well I could say hmm well if a kilometer is a thousand meters
all right and a meter is 1.1 yards a mile 1760 yards which is wow it's 1.1
times 1600 so I could call a metric mile about 1600 meters and not be
terribly far off in fact I think my error would be about 32 feet that's close
enough for me over approximately a mile distance 30 feet over over a mile is
pretty good so I'm I'm taking that one more thing before I before I close
suppose I hear about a forest fire in Quebec and they give the acreage not see
acreage an English centric they give the amount of area land area in hectares
well how do I know what that is in acres or square miles hmm that's a hard
problem right as it turns out it's not as terribly hard as you thought because I
know for example that a hectare is 10,000 square meters I know that there are
640 acres in a square mile an acre is a chain by a furlong all right does that
does that help me in any way well furlong is 1 that's 220 yards oh but that in my
approximation that's 200 meters so 200 meters by 20 okay by one chain which is 20
meters in my approximation okay so 20 times 200 that's 4,000 square meters in an
acre so a hectare would be about 2 and a half acres pretty cool I like it so if I
hectare's 2 and a half acres that means there's about 256 hectares in square
mile and if I make that 250 that's not too far off that's a pretty easy
conversion hmm 250 hectares to the square mile wow so if there's a thousand
hectares that's four square miles no yes 1024 among friends but it's nice easy
conversion unlike in it so what if there's a million hectares of land under
cultivation in a country how many square miles would that be now I can't remember
what the number was let's say it was a million a million hectares times well 1
over 250 square miles per hectare so I get hectares cancel 1 million over 250 is
unitless times hectimes square miles oh well a million is a thousand times a
thousand that's 4,000 square miles million hectares wow that's not so hard to
convert anymore I think we've done it Watson so if you work with approximations
and you're not too concerned about being exact because I'm sure that people
talking about 20,000 hectares being destroyed by or at risk in a forest fire I'm
betting that number's not 20,401 it's around 20,000 it could be 24,000
probably or 16,000 so they amount you're throwing away or throwing into the
pot you're going over by making these little approximations is probably not
hurting you all that much because you're coming close enough because what you
want is an idea of what they're saying so that you're not watching the TV and
every minute going to Google saying what is this many hectares in square miles I
mean you can do that but if you'd like to pay attention and understand where
people are saying without having your phone in your face or your computer your
tablet then get yourself some rough and ready rules with approximations and
this one's not bad at all if you like multiplying by powers of 2,256 is
preferable to 250 for the number of hectares in a mile but it really does help you
think about it when you can make quick and close enough approximations of what
one set of units means in this other system that you do know so you can get a
much faster mental mapping of what each one what the new system is and
eventually you can maybe even leave the old system behind as a relic from the
past and use what the international community is using while you're talking
with them it might make the world a more peaceful place if we can learn to
talk to each other and familiar terms and we learn how to communicate then to
translate between not only languages but units of distance I think it'll make
the world a smaller and more comfortable place where we can live together okay
that's it for today's show be back with more practical map and more units of
different kinds here on Hacker Public Radio thanks for listening bye
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