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## Algebra (all content)

### 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.

## Want to join the conversation?

- is there any way that you can post more problems like this im a pharmacy student and I am having a difficult time with conversions for some reason. I get confused on setting up the problems. Can you also cover other methods of conversions such as this example problem that I have in my book.

You receive a prescription for amoxicillin 75mg 4times a day (QID) for ten days. How many mL of amoxicillin 250mg/5ml do you need to fill this prescription to last a full 10 days.(35 votes)- SuSu,
**The secret to conversions is that words can cancel out each other just like numbers**.

You have a ratio of 250mg to 5ml. The ratio can be used either as

250mg/5ml or

5ml/250 mg.

If you know the mg and you want your answer is ml, you want to multiply

mg * ml/mg = ml

Notice that the mg/mg on the left cancel out leaving you an answer in ml.

If you know ml and want to find mg you use

ml * mg/ml = mg

Notice that the ml/ml on the left cancel out leaving you an answer in mg.

75 mg/dose * 4 dose/day * 10days = 3000 mg

Again notice that the words mg*dose*day are in you numerator and

dose*day are in you demoniator. The dose/dose and day/day cancel out leaving your answer in mg.

So your complete formula is

75mg/dose * 4doses/day*10days*5ml/250mg.

Your numerator is 75*4*10*5mg*doses*days*ml

Your denominator is 250*dose*day*mg

The words dose*day*mg in both the numerator and denominator cancel out giving the answer in ml

and the numbers 75*4*10*5/250 = 15000/250=60

So the answer is 60ml

I hope that helps make it click for you.

It is important for a pharmasist to never mess up on conversions.

Good luck on your studies.(56 votes)

- Why is it essential to include the zero when writing decimals?(15 votes)
- If you are dealing with medications you ALWAYS include the zero in front of the decimal point. This not so much a math rule as it is a rule for patient care. In math, it is acceptable to write .4 mg. If you are dealing with medications you must write 0.4 mg. This prevents accidental dosing errors which could hurt or kill the patient. Without the zero in front of the decimal, someone may read .4 mg as 4 mg and give 10 times the intended dose.(40 votes)

- Is there a list of basic useful conversions every one should know and have memorized?(13 votes)
- I don't think so, unless I do not know the unwritten rule too.. but I've noticed that older people know 1yd=3ft, 3tsp =1 tb (got that one from who wants to be a millionaire), 12 inches in 1 ft, 1 mile= 5280 ft, quart = 4 cups, pint= 2 cups, 2.2 lbs= kg, 2.54 cm to inch..(21 votes)

- I don't really get how to convert decimals of inches into big thins like miles. All the numbers and equations get jumbled up in my head.(13 votes)
- so as not to get confused in converting units, i use the unitary method. first i try to find the value for 1unit and then multiply with the said number of units(7 votes)

- I thought you couldn't make pounds into kilograms. Is it really 0.45 KG.= 1 lb.?(4 votes)
- Yes. Any units of measurement that measure the same thing (weight, mass, volume, distance etc.) can be converted into and from one-another.

I can make up a completely new unit of measurement and if I give it some parameters, you could convert any other unit into it and vice-versa.

A pound is a measurement of weight, a kilogram is a measurement of weight. It is also completely reasonable that both have fixed values based on something, which means there will also be a RATIO of pounds to kilograms that will not change as long as their values remain fixed. In this case 2.2 : 1

However in the case of pounds to kilograms, the actual number has too many decimal places, so usually, the conversion rate is approximated to some degree. For everyday use, this is completely fine.(9 votes)

- Why doesn't the whole world just adopt the metric system?(8 votes)
- The English system allows for the use of fractions, something which is difficult to do with the metric system. A majority of the world's recipes are written using the Imperial System. So changing to metric will be hard. It also will be hard to convince everyone and answer the question: why?(0 votes)

- Instead of multiplying 2.2 lb/1kg, I did 1lb/.45kg and got a slightly different answer at the end: .8975ml. Is that just because the .45 and 2.2 figures Sal wrote at the top were estimates or is what I did not a valid way to do the conversion?(4 votes)
- At7:45, Sal writes 1 mL / 0.9 grams. I thought it was forbidden (at the very least not recommended) to write fractions with decimals in either the numerator or denominator. Wouldn't the best practice be to re-write the expression as 10 ml / 9 gram?(4 votes)
- It's just a matter of notation. It's definitely not forbidden, however, it is less convenient, so may be not recommended for younger students ;)(1 vote)

- This is getting a bit confusing.(3 votes)
- maybe you can watch the metric system units of weight or more basic ones to get the idea and then work towards harder ones like these. Don't worry like it said this is advanced. Good Luck!(2 votes)

- how do i figure out all of these conversions? Sal seems to know them all by heart.(2 votes)
- The conversions were preset that way, or discovered to be that way after equivalent amounts of different units were connected. Sal knows many by heart because he has been using them often and memorized many by now.

Just so you know, different units and ways of measuring were designed by different nations/countries (and at first knowing little to none about what other places used). Now that the world is largely international, we have "shared" the measuring units and developed appropriate ways to convert, if using one unit is more efficient than another, while able to let others understand the same thing in another way.(2 votes)

## 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.