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### Course: Algebra (all content) > Unit 15

Lesson 3: Word problems with multiple units# Using units to solve problems: Drug dosage

Future doctors and nurses out there, take note. This unit conversion word problem deals with converting drug dosage units, something that is commonly done in hospitals. Give it a try with us. Created by Sal Khan.

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## Video transcript

I just received this drug
calculation problem from a nursing student, and I think
it's essential that the nursing students out there are able to
do this, just in case I'm the patient receiving the drug. So let's do it. And I think it's an interesting
unit conversion problem for pretty much anyone who wants
practice with unit conversion. So the question is that
we have a doctor. The doctor orders drug x. And this is the dosage that
the doctor's requesting. They're saying 5 milligrams per
pound of patient weight-- I'll just write per pound of patient
weight-- every 12 hours. This is what we're
supposed to do. But our supply of the drug--
it isn't just, you know, not just nuggets and milligrams. It's a solution. There's a certain amount of
grams for every milliliter that we have of the solution. It's dissolved in some water. So this is our
supply of drug x. We have 0.9-- I'll
write a 0 in front. My wife, who is a doctor, says
it's essential to write the 0 in front of the decimal. We have 0.9 grams per
milliliter of solution. So if I were to take 1
milliliter out of my solution and give it to someone, I'm
essentially giving them 0.9 grams of this drug. And the final piece of
information we're given is that the patient-- they weigh-- and
maybe we should say they mass, because kilograms is mass,
but we get the idea. The patient is 72.7 kilograms. So there's a couple
interesting things here. We have to figure out the
dosage in terms of milliliters. We have to-- oh, actually,
I didn't even tell you the question. The question is, how many
milliliters of solution do we have to give to
the patient per dose? So milliliters of
solution per dose. That's our question. And there's a couple of things. We have to go from
milligrams to grams. And then convert that
to milliliters. And then they tell us per
pound, but then they gave us the patient's weight, or
their mass, in kilograms. So we have to do some
conversion there. So I definitely can appreciate
how this can be a little daunting and maybe
confusing at times. So let's just do
it step by step. So the first interesting
thing-- and this is just something that you might need
to know, or you might have written down on paper, or you
might have a calculator that does this-- is just how to
convert kilograms to pounds. And it's good to know in
general, if you're converting between the metric and
the English systems. So 1 kilogram is approximately
equal to 2.2 pounds. Not exactly, but that's a
pretty good approximation. And 1 pound-- if you just take
1 over that-- 1 pound is approximately equal
to 0.45 kilograms. So we'll just put
this in a box. This is the only kind of
outside conversion information we'll need to do this problem. Everything else, we'll just
need a calculator, unless we just want to spend a lot of
time doing some arithmetic. So the first thing. Let's figure out our dosage
in terms of per kilogram. This is per pound, and
we really don't need to know every 12 hours. Because they're saying, how
many milliliters of solution do we do per dose? A dose is every 12 hours. So we just really, you know--
the every 12 hours is kind of extra information. So we want to figure out this
5 milligrams per pound. How do we convert that to how
many milligrams per kilogram? So let's do 5-- I'll write
it down here in magenta-- 5 milligrams per pound. And then we want to convert
this to per kilogram. So we can multiply this times
the number of pounds per kilogram-- I'll do it
in yellow-- times this information up here. Times 2.2 pounds per kilogram. And if you ever get confused--
you know, gee, how did Sal know to multiply by 2.2 instead
of dividing by 2.2? Which is the same thing
as multiplying by 0.5. You can pay very close
attention to the units. Notice, I wrote 2.2
pounds per kilogram. 2.2 pounds per 1 kilogram. And you know this'll work out,
because you have a pound in the numerator and you have
a pound in the denominator. It's called
dimensional analysis. If you ever get confused with
these things-- and I think, once you do enough practice,
you'll find that you won't have to pay too much
attention to this. But at first, when you're
getting started, just to make sure you're not multiplying
or dividing by the wrong thing, just make sure the
dimensions cancel out. Pounds in the numerator,
pounds in the denominator. So let's do that. Pounds in the numerator, pounds
in the denominator cancel out. And you multiply 5 times 2.2. This is equal to-- let's see. 5 times 2 is 10. 5 times 0.2 is 1. So this is equal to 11. And then in our numerator,
we have milligrams. 11 milligrams per kilogram. So we just converted our
dosage information to a pure metric system. It was actually a mix
between the metric and the English system before. Now let's see what we can do. Well, let's see if we can get
it in terms of how many milliliters we have to
deliver per pound. So once again, we want this--
well, actually, let's go to grams first. Because we have
milligrams here. We have grams up here. So let's see if we can
convert this thing to grams. So just like we did before,
we want a milligrams in the denominator. I'll do it in orange. We want a milligrams in the
denominator and we want a gram in the numerator. Why did I say that? Because I want this
and this to cancel. And I want a grams
in the numerator. So how many grams are
there per milligram? You can just think it through. There's 1 gram per
1,000 milligrams. Or 1,000 milligrams per gram. And you just multiply it out. So the milligrams cancels with
the milligrams, and then we get-- this is equal to
11/1,000 grams per kilogram. So now we have everything in
terms of grams, but we want it in terms of milliliters. The question is, how
many milliliters of solution per dose? So let me go down here on
this line right here. So we had this result. We have 11/1,000-- I won't do
the division just yet-- grams of drug x per kilogram. This is really just a re--
we've just rewritten this dosage information
in different units. And let's see how much solution
we need per kilogram. So I want to cancel out
the grams here and have a milliliters there. So to cancel out that grams,
I'm going to have to have a gram in the denominator and a
milliliter in the numerator. So in our solution, how many
grams are there per milliliter? Well, they told us. There are 0.9 grams
per milliliter. Or for every 1 milliliter,
there are 0.9 grams. Notice, I just took
the inverse of that. Because we want a milliliter
in the numerator, grams in the denominator, so that
these two cancel out. And let's do this
multiplication now. So our grams cancel out. We have milliliters
per kilogram. And then we multiply it out. 11/1,000 times 1 over 0.9. So I'll just keep-- let me
just write it like this. So there's going to be 11/1,000
times 0.9 milliliters of our solution per kilogram. So we've gotten this far. So this is per kilogram
of patient body weight. And then finally, they tell
us how many kilograms our patient weighs. So let's do that last
multiplication, and then we can actually get our calculator out
and do it all at once. So let's multiply this times--
we want to know how many milliliters per patient. We want the kilograms
to cancel out. So we want kilograms
per patient. Now we're talking about
this particular patient. Not every patient is
going to be the same number of kilograms. But if we do this,
kilograms will cancel out. We'll have milliliters per
patient-- milliliters of solution per patient-- which
is exactly what we want. We want milliliters of solution
per dose per patient. But everything we've assumed
so far has been per dose. So how many kilograms
does the patient weigh? Well, there's 72.7
kilograms per patient. That's how much the
patient weighs. So we just do this final
multiplication and we'll be done. So our answer-- and as these
two things are going to cancel out-- so our final answer is
going to be 11 times 72.7 divided by 100 times--
actually, 100 times 0.9 is pretty easy to figure out. That's 900. Divided by 900
milliliters per patient. Or you can just say
milliliters per dose. However you want to say it. Per dose per patient. Let's get our calculator
out and do this. So we have 11 times 72.7 is
equal to 799 divided by 900. Is equal to 0.88-- well,
we could round up. 0.889. Hopefully the
doctor won't mind. So that is equal to-- I'll
write it in a nice, vibrant color-- 0.889 milliliters
of solution per dose. So this is what we're going
to give every 12 hours. If they ask, how many total
milliliters over the course of 2 days? We would have to say,
oh, there's 48 hours. We'd multiply it by 4. But that 12 hours was extra
information in this problem. But anyway, hopefully this is
useful, and it'll ensure that any nurses serving me in the
future are giving me my proper dosage. And hopefully, the doctor even
got the right dosage to begin with, because otherwise
it's all for naught. Anyway.