Transcript Understanding Ratios and Proportions Power Point Presentation

RATIOS AND PROPORTIONS
Created by Leecy Wise and Caitlyn Reese, © Unlimited Learning, Inc. 2015
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A ratio is a relationship between two numbers in a
defined setting.
Huh?!
Well, by “defined setting,” we
simply mean that each number
represents something, like people
or objects, in a certain situation.
• There are 2 boys for every 3 girls in the class
• There are 2 cups of flour for every 1 cup of sugar in the recipe
• There are 2 nurses for every 20 patients in the nursing home
• There are 6 milligrams of drug in every 50 mL of the medicine
Now there has to be a better way to write out a ratio than to say “there are this
many of these for every that many of those,” right?
In fact, there are actually three different ways to write a ratio!
Let’s use the example of “two boys for every three girls” from the last slide…
1) We can write this relationship as a fraction
2) We can use the “ratio symbol” (which looks like a colon)
3) Or we can simply use the word “to” between the numbers
WOAH there! But how do we know which
number goes on top or is written first?
Ratios should be written in the same order as the words
that express them
For example…
If someone said that there are “5 physician assistants for every 2 doctors” at a
particular clinic, we would write this ratio as:
Write the following numbers in the given situations as a proportion in all three
of the ways you just saw. Fill in your answers in the table given to you on your
Lesson Answer Sheet, then click to check your work.
1)There are 5 trucks for every 7 cars in the parking lot
2)There is only one parent for every 3 kids at the water park
3)There are 8 parks for every 5,000 people in the city
4)There are 68 nurses for every 22 doctors at the hospital
5) There are 96 patients for every 3 providers at the clinic
Now check your work! Cross out any incorrect answers
and write your corrections on your answer sheet.
As a
fraction…
With the ratio
symbol…
Using “to”
1)
5/7
5:7
5 to 7
2)
1/3
1:3
1 to 3
3)
8/5,000
8 : 5,000
8 to 5,000
4)
68/22
68 : 22
68 to 22
5)
96/3
96 : 3
96 to 3
The two numbers in any ratio need to be in reduced form.
This works much like reducing a fraction!
For example:
Let’s say that 5 out of 10 students in a class have an A in the class. Let’s start out by
writing this as a ratio in the fraction form we talked about before:
Now look at
this like a
fraction…
IS this in the most
reduced form?
No!
It’s
not!
What number goes
into both 5 AND 10?
5 goes into
itself once
And 5 goes
into 10
two times
THIS is now in
“reduced form”
Write the following ratios as a fraction, and then put the fraction into reduced
form. Fill in your answers in the table given to you on your Lesson Answer
Sheet, then click to check your work.
1)There are 4 boys for every 6 girls on the team
2)There are 15 players for every 3 referees on the field
3)There are two tables for every 16 people at the party
4)There are 5 doctors for every 100 patients at the clinic
5) There are 750 milligrams of drug in every 5 milliliters of
medicine
Now check your work! Cross out any incorrect answers
and write your corrections on your answer sheet.
As a fraction…
In reduced form..
1)
4/6
2/3
2)
15/3
5/1
3)
2/16
1/8
4)
5/100
1/20
5)
750/5
150/1
Actually, ratios (and the next topic, proportions)
may be the single most important math concept
you will need in your medical career!
Check this out…
Maybe the best example is medications! Of course you’re going to come
across these all the time no matter what medical field you’re in.
And guess what? The dosages of ALL medications, whether in pill,
powder, or liquid form, uses ratios!!!
Do you see the ratio on this label?
250 milligrams (mg) per 5 milliliters (mL) is
definitely a ratio!
Now if we put this in fraction form we have…
So what does that mean?
Remember
to reduce…
(5 goes into both numbers)
This means that in
every 1 mL of the
liquid, there is 50 mg
of medicine
Another great example of ratios in medicine is Intravenous
(in-truh-VEEN-us) fluids, usually called IV fluids for short.
IV fluids are not simply water! The typical fluid that is given
through an IV in the hospital is called normal saline, which is
salt and water, mixed in a specific ratio.
IV fluids always have some kind of substance dissolved in
them, and the amount of that substance is usually written as
a percent. But guess what? Percent is a ratio too!!!
For example…
Let’s say we have a
bag of 2% normal
saline. 2% saline just
means that there
are 2 parts salt to
100 parts water,
which is a ratio…
2 parts
100 parts
Again,
remember to
reduce…
(2 goes into both
numbers)
1
50
So in every bag of normal saline, there is 1 part salt for every 50 parts of water
Make sense? How about…
Answer the following questions about medication dosages
(Hint: Turn them into a ratio and then convert to most reduced form.)
Write your answers on your Lesson Answer Sheet.
1) A full dose of Ibuprofen is 800 mg in four pills.
How many mg are there in each pill?
2) There are 5 mg of medicine in 50 mL of cough syrup.
How many mL contain just 1 mg of medicine?
3) A bag of IV fluid has 5% potassium.
How many parts potassium are there versus parts of water?
4) A bottle of nasal spray has 500 mg of medicine and contains
100 spray doses. How many mg of medicine are in each spray?
Now check your work! Cross out any incorrect answers
and write your corrections on your answer sheet.
As a fraction…
In reduced form..
1)
800 mg/4 pills
200 mg/1 pill
2)
5 mg/50 mL
1 mg/10 mL
3)
5 parts potassium/
100 parts water
1 part potassium/
20 parts water
4)
500 mg/100 sprays
5 mg/1 spray
Let’s say that a physician orders you to
give 300,000 units of penicillin IM
(intramuscular) to a patient. You find a
supply of penicillin in the storage area.
However, it states 400,000 units per mL…
Now, you know that the amount
prescribed is 300,000 units.
But how will you figure out how many mL to
give the patient when the bottle tells you that
there are 400,000 units per 1 mL?
Ratios show the relationship between two numbers,
as you have learned.
Well a proportion is basically a set of two ratios. The
numbers in each ratio are different, but…
The relationship between the numbers
in each set is the same.
Basically what this means is that
size doesn’t matter in proportions, only relationships matter.
This man is a
certain height,
let’s say
6 feet
He also has a certain
width, let’s say
It does NOT mean that
they are the same
height or width, as
you can see…
2 feet
It only means that the
relationship between his
height and width is the same
as the first man.
Let’s look at this further…
As we said before, the first man is
6 feet tall and 2 feet wide:
Okay, to find the relationship between his
measurements, let’s write a ratio:
6 feet
Now reduce,
just like
before…
(2 goes into both
numbers)
2 feet
Okay, now what about our smaller man?
Remember, he is proportionate to the larger man, meaning the relationship
between each man’s height and width is the same
Knowing that, what if we now told you
that the smaller man is exactly 3 feet tall?
3 feet
What does this tell us about his width?
1 foot
On the other hand, these two men are NOT
proportionate!
They have
the same
height…
But VERY
different
widths…
In other words, the relationships between their heights and
widths are far from the same
As you can see, a proportion is a set of two
ratios where the numbers in each one may be
different, but when you reduce each ratio, they
are actually equal! We call this “equivalent”
Let’s look at a couple more examples of
proportions, so we can see how in each one,
the ratios are equivalent…
2 4
=
5 10
3 15
=
5 25
Can you tell that these
two ratios (or fractions)
are equal?
2 goes
into
both
numbers
If we reduce the right
side (3 goes into both
top and bottom), we
end up with the same
thing again!
Now it’s
exactly
equal to
the other
side!!!
Look at the following sets of ratios, and decide if they are proportions or not.
You’ll find each of the following sets of ratios on your answer sheet. Use the free
space to reduce your fractions, and then write “yes” if they are proportions
(they are equivalent), or “no” if they are not proportions (not equivalent).
When you’re finished, click to check your work.
1)
2)
3)
4)
Now check your work! Cross out any incorrect answers
and write your corrections on your answer sheet.
2)
1)
Reduce
by 6
Can’t be
reduced
Reduce
by 5
Reduce
by 3
Yes!
NO!
4)
3)
Can’t be
reduced
NO!
Reduce
by 3
Can’t be
reduced
Reduce
by 9
Yes!
A good understanding of proportions will
help A LOT in your health career.
So far we’ve learned what they are and
how they work…
What if we know two things are
proportional, but we don’t know one of the
numbers in one of the two ratios?
These two men are proportional.
The man on the left is 6 feet
tall and weighs 142 lbs.
6 ft
142lbs
The man on the right is 4
feet tall… but how much
does he weigh?
4 ft
??lbs
In every proportion problem, the key is to remember that …
Why???
Because the fact that the two ratios are equivalent is what
makes it a proportion in the first place!!!
6 ft
142lbs
=
4 ft
??lbs
To solve a proportion with an unknown
number in it, use what is called the
“Cross-Multiplication Property.”
It works every time!
We’ll call it the X rule.
In any true proportion (made up of two equivalent ratios) you
can cross-multiply, which means multiply the numbers
diagonally across the equal sign, and the products you get on
either side will be equal… every time!
And this is a super easy way to check if any two ratios are equivalent!
Multiply across…
2 4
=
2 x 10 = 20
5 10
2 4
=
5 10
Now the other way…
5 x 4 = 20
So YES, according to the X rule, this is definitely a
true proportion
Determine whether or not the following ratios are true proportions using the
X rule. Do your work on your Lesson Answer Sheet. If they are true
proportions, write “yes,” and if not write “no.” When you are done, click to
check your work.
1)
7
14
and
8
16
2)
3
9
and
5
10
3)
12
36
and
17
51
4)
2
10
and
3
15
Now check your work! Cross out any incorrect answers
and write your corrections on your answer sheet.
7
14
and
8
16
7 X 16 = 112 and 8 x 12 = 112. Yes, this
is a true proportion.
3
9
and
5
10
3 X 10 = 30 and 5 x 9 = 45. No, this is a
NOT a true proportion.
12
36
and
17
51
12 X 51 = 612 and 17 x 36 = 612. Yes,
this is a true proportion.
2
10
and
3
15
2 X 15 = 30 and 3 x 10 = 30. Yes, this is a
true proportion.
Now that you know it works, you can solve problems with
unknown numbers in them by applying the X rule.
A cough syrup contains 3 grams (g) of
medication in every 15 milliliters (mL).
How many grams would be in 125
mL of this cough syrup?
Step 1: Set up the KNOWN ratio as a fraction
We know that there are 3 grams (top) in every 15 mL (bottom)
Step 2: Set up the UNKNOWN ratio, using an
X to represent the unknown amount
The question asks how many grams are in 125 mL.
So the number of the grams (top) is our unknown number…
3g
15mL
Xg
125mL
Step 3: Now just throw an equal sign in
the middle to create a proportion!
(We know they are going to be equal, right? It’s the same
medicine! All that is different is the amount of liquid…)
3g
Xg

15mL 125mL
Now what do you think Step 4 will be,
based on what you just learned?
If you said, “Cross multiply,” you
would be absolutely correct!
3g
Xg

15mL 125mL
PSST! When we are doing problems with X as the
unknown number, a better way to write out
multiplication is to just put both numbers in
parentheses to avoid confusion!
(
)(
)
is the same as
×
3g
Xg

15mL 125mL
(3) (125) = (15)(X)
Okay, now do the
multiplication…
(3)(125) = 375 and (15)(X) = 15X
Now remember, according to the X rule, the two products are the same, right?
So put an equal sign between them, and we have…
375 = 15X
Now we just need to solve by getting X by itself…
375 = 15X
Division!
Remember, this is 15 times X
To get X alone, we need to
reverse what is being done to it.
Okay, so let’s divide by 15…
The two sides of the problem are equal, right? So to make sure it stays equal,
anything we do needs to be done to both sides…
375 15x
=
15
15
375 ÷ 15 is and
25
=
15X ÷ 15 is
And remember, they’re equal!
1X
(or just X)
25 = X
So what does that mean for our original problem? (Yes, that was a while ago )
Here is the original problem:
A cough syrup contains 3 grams (g) of medication in every 15 milliliters (mL).
How many grams would be in 125 mL
of this cough syrup?
We were able to solve and figure out that X
= 25
There are 25 grams of medicine in 125 mL of the cough syrup
Now that you understand the steps, let’s go through
one more example…
Step 1: Set up the KNOWN ratio
2 grams
4.5 liters
Step 2: Set up the UNKNOWN ratio, using an X grams
X to represent the unknown amount
13.5 liters
2
x
Step 3: Set them equal to create a proportion
=
3 9
Step 4: Cross multiply
(2)(13.5) = (3)(X)
2
x

4.5 13.5
27 = 4.5X
Which is…
Step 5: Divide on both sides to get X by itself
27 4.5 X

4.5
4.5
6=X
So there are 6 grams
of salt in 13.5 liters of
the IV solution!
Now it’s your turn. Find the missing (X) value in the following
proportions. Do your work on your Lesson Answer Sheet. When
you are done, click to check your answers.
15 5

63 X
0.4 mg 6 mg

1 tablet
X
0.01 X

5
2.5
7 mg 1.5 mg

1 mL
X
1.5 mg
4 mg

3 capsules
X
350 mg
X mg

2 tablets 4 tablets
(Round to
two decimal
places)
Note: It doesn’t matter where the X is placed in the proportion.
The process is the same.
Now check your work! If you missed any, figure out the
mistake was and correct it on your Answer Sheet.
(0.01)(2.5)  (5)( X )
0.025  5 X
0.025 5 X

5
5
(15)( X )  (63)(5)
15 X  315
15 X 315

15
15
0.005  X
X  21
(1.5)( X )  (3)(4)
1.5 X  12
1.5 X 12

1.5 1.5
X 8
Click again to see
the answers to 4-6
Now check your work! If you missed any, figure out the
mistake was and correct it on your Answer Sheet.
(0.4)( X )  (1)(6)
0.4 X  6
0.4 X
6

0.4
0.4
(7)( X )  (1)(1.5)
7 X  1 .5
7 X 1 .5

7
7
X  15
X  0.21
(350)(4)  (2)( X )
1400  2 X
1400 2 X

2
2
700  X
A physician ordered you to give 300,000 units of penicillin
IM (intramuscular) to a patient. You find a supply of
penicillin in the storage area. However, it states 400,000
units per mL. How many units would you give the patient
using the supply you found?
Do your work on your Lesson Answer Sheet, and then
click to check your answer!
First, set up your known ratio .
The problem told us that there are
400,000 units in 1 mL of the medicine.
400,000
1 mL
Then, set up your unknown ratio .
We need to know how many milliliters (X)
we will need to get 300,000 units.
=
(Set them equal to
make a proportion)
300,000
X mL
(400,000)(X) = (1)(300,000)
Now divide on both
sides to get X by itself
400,000X = 300,000
400,000
400,000
X = 0.75 mL
Review and Practice
Turn in your Lesson Answer Sheet to your coach. Then
complete the following worksheets (you should have a
print out of them):


Writing and Reducing Ratios
Using Proportions
Once you complete each worksheet, ask your coach for the
answer key and correct your work. Don’t worry, you’ll only be
graded for completion on this part.
Finally, click here to review some flash cards with all of the
terms you learned in this lesson (you can also play games with
the terms!):
https://quizlet.com/128869519
CONGRATULATIONS!
You now have a great
introduction to ratios and
proportions!
Ready to take the quiz?