Welcome to a lesson on proportions.
Let's begin with some definitions.
First, a ratio is the quotient of two quantities
with the same unit of measure.
For example, a ratio in a recipe might be
six cups of flour to two cups of sugar.
Notice how this is a ratio because we have the quotient
of two quantities with the same units of measure,
in this case cups.
Often we'll see ratios without units
because the units would simplify out.
We can also express this ratio as six to two.
We can also simplify ratios
similar to how we simplify fractions,
that we must leave it as a quotient of two quantities.
So six to two simplifies to three to one.
Next, a rate is the quotient of two quantities
with different units of measure.
For example, let's say you go on a trip
and travel 300 miles in four hours.
The rate would be 300 miles per four hours
and notice this fraction here does simplify.
There's a common factor of four.
We can also express this rate as 75 miles per one hour,
often expressed in this form here as 75 miles her hour.
And then finally, a proportion is a mathematical statement
that two ratios or rates are equal.
So proportion is always in this form here.
An important property about proportions
is that the cross products are always equal,
meaning A times D is always equal to B times C.
As an example, let's consider we know 1/2 is equal to 3/6.
This is in the form of a proportion
and notice if we cross multiply,
one times six does equal two times three.
Notice both products are equal to six.
This is an important property because
if one of the four values are missing,
we can cross multiply and solve the proportion.
Let's look at some examples.
In example one, we're asked to solve for the variable
in each of the following proportions.
So notice in the proportion we have three known values
and one unknown value.
So we can cross multiply, form an equation and solve for T.
It doesn't matter which cross product we find first.
Let's find the cross product with the variable first.
So three times T must equal two times 42.
So again, we have three times T or three T
equals two times 42.
So we have three T equals 84.
Notice here we have a one-step equation.
Three T means three times T.
So to isolate T and solve the equation,
we divide both sides by three.
So in the left side, three divided by three
simplifies to one.
So we have one times T, which equals T.
On the right side, we have 84 divided by three,
which is equal to 28.
This is the solution.
So T equals 28 is our solution.
Looking at the next proportion,
notice how we have one unknown.
We want to find the value of R.
So we'll cross multiply.
R times two must equal three times five.
Well R times two would be equal to two R
equals again three times five.
Well of course three times five is equal to 15.
So we have two R equals 15, divide both sides by two.
Simplifying on the left, two divided by two is equal to one.
One times R is equal to R.
So we have R equals 15/2, which is an improper fraction
though it is simplified.
Let's also convert it to a mixed number.
To convert 15/2 to a mixed number,
we would take 15 and divide by two.
There are seven twos in 15, seven times two is equal to 14.
Subtract, we have a remainder of one,
which means 15/2 is equal to seven and 1/2.
So in doing the homework, if they ask for the value
as a fraction, we can enter 15/2
and if it asks for a mixed number, we'd enter seven and 1/2.
In our third example,
again we'll cross multiply and solve for X.
So seven times X must equal 12 times 35.
Seven times X is equal to seven X
equals again 12 times 35.
So we'd have seven X equals 420.
Solve for X if we divide both sides by seven.
On the left, we have one times X or X.
On the right, we have 720 divided by seven, which equals 60.
Now let's look at an application problem.
The recommended daily allowance or RDA of protein
for active adults 19 years of age and older
is based primarily on body weight.
In general, the RDA of protein for adults is 0.8 grams
for every kilogram or about 2.2 pounds of body weight.
If you weigh 150 pounds, how many grams of protein
should you consume each day?
And we're asked to round the answer
to the nearest 10th if needed.
So we'll solve this problem using a proportion
where we'll set two rates equal to each other.
We'll begin by setting up the known rate
from the given information that the RDA is 0.8 grams
of protein per 2.2 pounds of body weight.
We're using pounds here
because the question involves pounds.
So again, our first rate is 0.8 grams of protein
per 2.2 pounds of body weight.
Now for the second rate, we'll have one unknown value.
Notice how our goal here is to determine
if a person weighs 150 pounds,
how many grams of protein should they consume?
So the unknown is the number of grams of protein.
Let's let X equal the grams of protein.
So we'll have X grams of protein per 150 pounds.
Now that we have our proportion, we'll cross multiply
and solve for X.
So 2.2 times X must equal 0.8 times 150.
We will leave the units off when we cross multiply.
So again, 2.2 times X is 2.2 X
equals 0.8 times 150.
0.8 times 150 equals 120.
So we have 2.2X equals 120.
Now to solve for X, we divide both sides by 2.2.
On the left, this simplifies to one X or X
and now we'll find this quotient
and round to the nearest 10th.
So 120 divided by 2.2
to one decimal place would be approximately 54.5.
Now let's go ahead and put our answer
in a complete sentence.
If you weigh 150 pounds, you should consume
let's say approximately 54.5 grams of protein each day.
I hope you found this helpful.