Leah here from leah4sci.com and in this video
we’re going to continue our discussion on dimensional analysis unit conversions specifically
looking at problems that involve more than one conversion. You can find part 1 along
with the unit conversion practice quiz and SI units to memorize for the MCAT cheat sheets
on my website leah4sci.com/conversions. In part 1 we looked at the setup of given
times ratio to help you take the units from where you are to the conversion of where you
need to go. But we kept the conversion simple, from one unit to another. For example, say
we’re given 24 meters and we’re ask to convert it to kilometers, we set it up with a given,
in this case 24 meters, times a ratio that has the unit for the given in the opposite
numerator or denominator so that we can cancel it out. Here we would put it in the denominator
and then the unit we’re trying to get to, we put in the numerator. Assuming there is
a direct conversion between them. How do we convert from kilometers to meters? We know
that 1000 meters is equal to 1 kilometer so we set it up as 1 over 1000. This allows us
to cancel out the units leaving us with the math of 24 times 1 over a thousand. You should
know how to do this using the move the decimal trick that I teach in the MCAT Math Series,
leah4sci.com/MCATMath. For every zero in the denominator we kick the decimal and move it
back one space so that gives us one two three with an answer of 0.024 kilometers which you
can also represent in scientific notation as 2.4 x 10 to the minus two. How do we get
this answer? If we move this back one, two spaces we get a whole number decimal the rest
of the unit, 2 spaces back is 10 to the minus 2. Another way to do this is to recognize
that a thousand ten to the third and that means we have 24 divided by ten to the third.
If we write that as 24 times 1 times 10 to the minus 3 because 1 over ten to the third
is one times ten to the minus three. Then we simply do the move the decimal trick, again
taught in the Math series. In this case we move the decimal to one smaller so we make
this one, 1 bigger, goes from 24 times 10 to the minus three to 2.4 times ten to the
minus 2 that we have here. Again, this was just review. If you were uncomfortable with
a setup or the Math, make sure to review the appropriate videos that are all linked below
so that you can understand what we do in the next step when we start adding multiple steps
to each conversion. For example, say you’re told that a vehicle
is moving at a velocity of 47 meters per second but you wanna convert this into kilometers
per hour so you setup your equation as given times ratio but then what do we fill in? The
given in this case is 47 meters per second but now we need a ratio that allows us to
have meters to cancel out or seconds to cancel out, which one? And the answer is we want
both but we can do it one at a time. Remember that with multiplication, if you have a unit
times another unit times another unit, the order ultimately doesn’t matter because the
answer will still be the same no matter what order you use to multiply. So when you have
multiple conversions just treat it as one conversion at a time. Going for 47 meters
per second, we’re really trying to convert meters to kilometers and we’re trying to convert
seconds to hours and so we just take the steps to go from one to the next to the next as
a given times a ratio times another ratio times another ratio. Let’s move it over so
we have room starting out with 47 meters per second, let’s choose meters to convert first,
we wanna go from meters to kilometers. We have two options, we can say that 1000 meters
is equal to 1 kilometer or that 1 meter is equal to 1 times 10 to the minus 3 kilometers.
I know a lot of students get confused here, so here is how I think about it. I have many
small equal one big, right coz that one big is so big it has many smaller pieces in it
but if I’m looking at just one small,it’s gonna be a tiny tiny portion of that giant
unit, so 1 meter which is small compared to a kilometer is a tiny tiny portion 1 times
10 to the minus 3 of that giant thing. Which one do I prefer? I like to avoid scientific
notation as much as possible, I like to avoid decimals and fractions, I like my numbers
to be clean so this is my preferred method. If you’re asked to do this in scientific notation
you can also look at this as ten to the third meter is equal to one kilometer and the reason
I point is out is if we have to use scientific notation I don’t like the negatives. How do
we get the positives we just look at the opposite direction. A lot of small is equal to one
big and it’s all positive values. Both are correct it’s really just about what’s simpler
and faster and most importantly, less confusing. The less confused you are the more confident
you will be going through the problem. We’re doing a conversion of meter so we’re
gonna put meters on the bottom, let’s go with 1000 meters is equal to 1 kilometer, I like
to cancel as I go. When you’re just learning this, write it, cancel so you’ll know you’re
on track and we have a unit that we want. So this one we’ll leave alone, next we go
to seconds. For seconds we don’t have direct conversion to hours. We know that 60 seconds
is equal to 1 minute, so let’s set it up with seconds on top allowing us to cancel that
means we have 1 minute on the bottom, 60 seconds on top. Seconds cancelled out, minute is not
the unit we want so we need one more step for this conversion and that conversion is
going to be 60 minutes is equal to 1 hour which goes on top which goes on bottom, we
need minutes to cancel. So minutes go on top allowing hour on the bottom, 60 minutes, one
hour, minutes cancel. The only units remaining are kilometers divided by hours which is perfect
because we’re trying to find the answer in kilometers per hour and then we have the Math.
Again that’s 47 times 60 times 60 divided by 1000. On MCAT you can’t use the calculator
but on the MCAT , close enough is good enough so let’s see how we can simplify it. First
thing I’m gonna do is knock out my zeros, one from the top and bottom, another from
the top and bottom, this gives us 47 times 6 time 6 divided 10. Let’s do a quick rounding,
47 rounds up to 50 which gives me another zero to knock out, that means I have 5 times
6 times 6 divided by 1. 5 times 6 is 30 times another 6, I don’t know but I know that 6
times 3 is 18, add that zero back, the answer is 180 or approximately 180. The calculator
gives me an answer of 169.2 which is on the MCAT is close enough. Now here is one trick you could have applied.
Given that 60 seconds is 1 minute and 60 minutes is 1 hour, if you see that conversion over
and over and over, you may as well memorize that 36 hundred seconds is equal to 1 hour.
Another confusing type of problem is when you are forced to use exponents but you have
been going in multiple directions meaning you have multiple steps with exponents. For
example, if we look at a Bacterial Cell, which is about a 10th of the size of an animal cell
and yes you need to know this for the MCAT. Say you’re given a specific bacterial cell
that is approximately point two microns in length and you are asked to convert this into
kilometers. We setup our equation as Given times Ratio. Knowing that our given is 0.2
microns. What the heck is a micron? It’s just a fancy short way of saying micrometer which
you can write like this, which is u m or you can just write it as u m if you’re trying
to type it out. So we have point two micrometers and we wanna convert that to kilometers. I
don’t know about you but I don’t know the conversion between micrometers to kilometers
but I do know how to go towards meters because you should have memorized according to the
cheat sheet link below how to go from meters to everything else. Meters to micrometers,
to millimeters, to kilometers, and so on. So let’s set this up as given times multiple
ratios. We have .2 micrometers and the first thing we wanna convert to is meters so we’re
gonna go from micrometers to meters keeping in mind the following conversion; 1 times
10 to the sixth micrometers is equal to 1 meter or 1 micrometer is equal to 1 times
10 to the minus six meters. Again a whole lot of tiny is equal to 1 big or 1 tiny is
a tiny, tiny portion of that big. This is where a lot of students get confused, make
sure that you understand that it can’t be the other way around. You can’t have 1 micrometers
equals to 1 times 10 to the 6th meter coz you’r saying one tiny is equal to a whole
lot of huge and that just doesn’t work. Which one do I wanna use? I don’t like negative
exponents if I can avoid them so I’m gonna take the first one and say that 1 meter is
equal to 10 to the 6th micrometers, where that 1 is self understood so you don’t have
to include it. This allows us to cancel out the micrometers taking us to the next conversion
from meters to kilometers which we already absolutely know. In this case we can use the
exponents or use a thousand given that we already have one exponents and they’re fairly
easy to add, let’s go ahead and put exponents on the bottom for the other one. 1 kilometer
is equal to the 10 to the third meters, this allows us to cancel meters leaving us with
our desired unit for the answer kilometers and now we just have to carry out the math. The Math itself is 0.2 times 1 times 1 which
doesn’t change, divided by 10 to the 6th times 10 to the 3rd. First combine the unit on the
bottom 10 to the 6th and 10 to the 3rd is equal to 10 to the 9th because when you multiply
all you do is add the exponents and I’m gonna write this out but I want you to do this in
your head. Point two is really 2 times 10 to the minus 1 and a number divided by 10th
to the 9th is really 1 times 10 to the minus 9 so we can think of this as 0.2 times 10
to the minus 9 but for proper scientific notation you need this number to be one whole number
and then the decimal. So we have to multiply this by 10. We’re using that times 10 divided
by 10 trick again. We multiply it by 10 to give us 2, that means we have to divide this
one by 10 because if we make one bigger we have to make the other one smaller giving
us to the minus ten for a final answer of 2 times 10 to the minus 10 kilometers. Again,
if the Math at any point was overwhelming especially the tricks make sure you go back
and study the MCAT Math without a calculator tutorial series. If you were able to follow
the Math and the concepts but want more practice, make sure you try the dimensional analysis
unit conversion quiz which I have linked below. You can also find it on my website leah4sci.com/conversions.