Welcome to a video that will
go over several application problems
that can be solved using proportions.
Let's take a look at the first example.
The directions on the label of an over the counter
fever reducer states to take 10 milligrams of liquid
per 40 pounds of body weight every six hours.
If you weigh 140 pounds, how much medicine should you take?
So we're going to write two rates
that compare the amount of medicine to body weight.
The first rate will be 10 milligrams
per 40 pounds of weight.
10 milligrams to 40 pounds must equal the rate
of milligrams to 140 pounds,
so we'll say x milligrams to 140 pounds.
These two must be equal and therefore form a proportion,
therefore the cross products must be equal,
so we'll leave off the units for right now.
40 times x must equal 10 times 140,
and now we just need to solve this equation.
So we have 40x equals 1,400.
Divide both sides by 40,
and 1,400 divided by 40 equals 35, so x equals 35.
You should take 35 milligrams of medicine
if you weigh 140 pounds.
Let's take a look at another example.
A team of forest rangers need to estimate
the population of deer on government land.
They catch 100 deer and tag and release them.
Later, they catch 40 deer and 8 of them have tags.
Estimate the deer population.
So we'll form two ratios that compare tagged deers
to the total number of deer.
So if they catch and tag 100 deer, the ratio of tagged deers
to total deer would be 100 to x,
because we don't know the total number of deer.
Then, when they catch 40 and eight of them are tagged,
40 would be the total number of deer,
and eight would be the number tagged,
and notice these ratios are consistent when comparing
tagged deer to total deer here,
and tagged deer to total deer here,
and that's important because we can't change
the order of the ratios and still form a proportion.
Now we can cross multiply eight times x must equal
100 times 40, that would be 4,000.
Dividing both sides by eight, we have x equals,
well 4,000 divided by eight would be 500.
So using this tagging technique, the rangers would estimate
there are a total of 500 deer on the government land.
On this problem, you plan to drive from Phoenix, Arizona
to Houston, Texas which is 1,175 miles.
If you travel 442 miles in two days,
how many total days will this drive take?
We're going to compare miles to days for our two rates.
The first rate will be 1,175 miles
to the total number of days, which we don't know,
so we'll say x days.
Must equal the rate of 442 miles
per 2 days, and of course this problem does assume
that we would travel at a constant rate per day.
Now we'll cross multiply.
442 times x, or 442x must equal 1,175 times 2.
So we have 442x equals 2,350.
Dividing both sides by 442,
x will be approximately equal to the number of days
the total trip will take.
Let's go to the calculator.
2,350 divided by 442.
Looks like it will take approximately 5.3 days.
Here we have a geometry example.
These two triangles are similar,
meaning they're the same shape but different sizes,
and similar triangles are proportional,
which means the ratio of corresponding sides are equal.
So to determine the length of this missing side here,
we'll set up two ratios comparing
the larger triangle to the smaller triangle.
So we can say that 16 is to five,
as 12 is to x.
Again, it's important that we compare the larger triangle
to the smaller triangle consistently as we did here.
So now we'd have 16 times x, or 16x must equal 5 times 12.
That'd be 60.
Dividing both sides by 16 will give us
the length of this missing side.
60 divided by 16 is equal to 3.75.
So this side has length 3.75 centimeters.
I think we have time for one more example.
One of my favorites.
In this problem, we have a yard stick
that casts a shadow two feet long.
If the shadow of a flagpole is 25 feet long,
how tall is the flagpole?
So on a sunny day, we can determine the height of a flagpole
without ever having to measure the actual flagpole.
So if this is our yardstick, this would be three feet,
and the shadow would be two feet,
and then it also tells us that the shadow
of the flagpole is 25 feet long measured here,
and the height of the flagpole is actually the unknown.
So if we were to form a triangle here
these would be similar triangles.
So we can set up a proportion to find the unknown,
and we'll compare the heights of the poles
to the lengths of the shadows, set up our proportion.
So we would have three is to x
as two is to 25,
and, as always, we'll find our cross products.
Two times x must equal three times 25, that'd be 75.
Dividing both sides by two,
so x equals 37.5 feet, which would be
the height of that flagpole.
I hope you found these sample problems helpful.