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Let's do some more applications.
And the problems you'll see in these lectures
are going to be where you have to read the words
and formulate a mathematical equation so you
can solve and get the answer.
In order to do that, you should read the problem
and understand what is being asked.
Draw a rough sketch, because that
might help you understand the dynamics amongst all
the components.
Assign variables for all the unknown quantities.
Here are some guidelines on how you
decide what quantities get variable.
When you read a problem if it says "how much" or "how far,"
"how long," "find," "when," these
are some key words you should look for.
And that will tell you what your variables should stand for.
In assigning variables use letters that make sense.
So for example, if it says "How much time is required?"
Then you would say t for time.
Sometimes a problem tells you what variables to use
and then you have to use those variables.
In which case pay careful attention
to whether they are small letters or capital letters,
because that can make a big difference.
When using known formulas to solve a problem
try to organize information in a chart
form because that will help.
And then sometimes you have to translate words
into mathematical expressions as we saw in Module 1.
Let's review some of that here.
So "is" translates mathematically
to equal to sign, "sum" translates to addition,
"product" as multiplication, "square" means
to the power of 2, "twice" means two times.
Substitute any known quantities into the formula,
and then solve for the missing variable.
When working with equations and inequalities,
solve for the unknown variable.
And always check your answer to see if it makes sense
or if it satisfies [? your end of ?] the equation.
Also see if the solution that you found
makes sense to the problem that you're solving.
Keep track of units throughout your work
so that your final answer has appropriate units.
And again I cannot emphasize enough,
please check your answer.
And you should have some estimates in your head
so that you know whether the answer you got is reasonable
or not.
Write your final answer always in a sentence
that fits the question being asked with appropriate units.
All right, so let's take some examples then.
So at times, before you could do a problem,
you might have to make some assumptions that the two
quantities are related linearly.
And most times they'll tell you whether they
are related linearly.
Assign variables to represent quantities that change or move.
Come up with one or more equations that
relate to variables and then use the algebra tools
we have developed so far.
Or you can solve a problem visually
so that you understand what's going on.
So let's start with this problem.
Recipe calls for 1 and 3/4 cups of flour, 1/2 a cup of sugar,
1 stick of butter.
It's critical that these proportions
are met so that your cookies turn out well.
Paul was careless as he started and put 2 cups of flour instead
of 1 and 3/4 cup and had added the sugar before realising his
error.
Determine how much additional sugar and how much total butter
he should use so that his cookies will turn out well.
So what do you think we should do?
I'm going to ask you to pause the video here and give
this problem a try and then we'll go over it.
So go ahead try it.
Let's see what you come up with.
Assuming you have paused and come back, let's take a look.
Let's draw a picture to represent the original recipe.
So we are asked to have 1 cup.
Here's 1 cup and here is 3/4 cup.
So 1 and 3/4 cups of flour, 1/2 a cup of sugar and 1
stick of butter.
That's our original recipe.
Now if you look at the proportions of flour
to butter to sugar we have 1, 2, 3, 4, 5, 6, 7 pieces of flour.
So let's take our sugar and break it into 7 parts,
so you know that this 1/7 of a 1/2
is the amount that will go over this 1 strip of flour.
You can do the same here and this 1/7 of the butter
will go over to 1 strip of flour.
That helps us understand what the proportions are
flour to sugar to butter.
All right, let's take a look at what Paul did.
Paul had 2 cups of flour so instead of 1 and 3/4 cups,
he has this 1 extra strip of flour.
The sugar he already had added from the original recipe
and the butter he originally had it would be that much.
So what do you think we should do
for the excess strip of flour?
This excess strip of flour corresponds do 1 strip of sugar
so we need 1 extra strip of sugar,
and 1 extra strip of butter.
And so that would be our solution then.
So this is how much butter and sugar
we need in addition to the original amount
and so our answer is going to be what then?
We want to add 1/7 of 1/2, which is 1/14 cup of sugar,
and 1 and 1/7 stick of butter.
That will make the recipe come out well.
Remember we did not use any algebra,
it was just a visual solution.
In fact, when you're cooking, most often
this is probably the way you might solve the problem.
You're not going to get a paper and pencil
and solve an algebra problem.
All right, let's take another example.
The tank on Karl's truck went from 1/3 full to 1/2 full
when he added 4 gallons of gas.
Use the information to determine how many gallons the tank holds
when full.
All right so let's draw the truck.
We have 1/3 and 1/2 represented in the same tank
so that's why we have 1, 2, 3, 4, 5, 6 pieces in the tank.
1/3 will be two of them, so this is 1/3 full.
Now, when you added 4 more gallons it became 1/2 full
so 1/2 full would be this much more then.
So that represents 4 gallons and so our full tank
would be 4 times 6, or 24 gallons.
So that's our visual solution.
Let's see how we would do it algebraically.
Since we don't know how many gallons the tank holds
when full, that's called x, the full capacity of the tank.
Then going from 1/3 x to 1/2 x requires 4 gallons.
So that means what?
1/3 x plus 4 is equal to 1/2 x, and that's
this picture right here, and then solve for x.
So x is 4 times 6, or 24, so full tank's capacity
is 24 gallons.
Look at the similarity between this process
and this visual process.
For some people one method is better than other.
You decide which you prefer, and it does not
matter as long as you can solve the problem.
[? If you write ?] correct mathematically
if you're writing it this way.
All right let's take the next problem Why don't you
pause the video here and see what you can do here.
Again how many meals?
So let's take that as our variable.
Let's say n is the number of meals that Jenn purchased.
So since the debit card dropping from 800 down 4.50 for each
meal, and we have n meals, we're going to have the following.
800 minus-- minus means you're taking away--
4.50 for every meal.
4.50 times n.
And that is $71.
That's the balance on the card left.
And then solve for n.
You know how to do that now go ahead pause the video
and continue.
All right so you should end up with 162 for n.
And so that means Jenn purchased 162 meals.
Now how do you know if it's reasonable or not?
Two months, so that's like 60 days.
Let's say you eat 2 meals a day, or 3 meals a day,
depending on how many meals a day you eat.
So 60 times 3 is about 180, and 60 times 2 is about 120.
So that is a reasonable amount of meals in the 2 months,
even though it looks like a really big number.
All right, try this on your own.
Pause the video please.
All right so draw a picture if you can
and let the unknown length of the property
be determined by l.
So we know that the width is 1/4 mile
and let's say this length is l miles.
An area of a rectangle is length times width.
And the perimeter is going to be 2 lengths plus 2 widths.
Perimeter is given to you as 1.5 miles, so let's solve for l.
And so the length is 1/2 a mile.
And so the area of the property is going to be 1/2 times
1/4, or 1/8, square miles.
You should always write your answer in words in a sentence.
All right try that one on your own.
So you have a situation where you
need to mix yellow, blue, and red paints in the ratio 3
to 2 to 4, to up obtain a certain shade of paint.
You've decided to make a batch with 1/2 of the numbers above.
So 1/2 of 3 would be 1.5, 1/2 of that would be 1, 1/2 of that
would be 2.
So 1 and 1/2 cups to 1 cup to 2 cups of yellow, blue, and red
paint accordingly.
All right.
Now by accident you started by putting
in 2 and 1/2 cups of yellow and did not realize this
until the 1 cup of blue and 2 cups of red had been mixed in.
Should you start over, or just add appropriate amount
of blue and red so that you have the correct shade?
So how much more blue and red should we add?
That's the question.
So go ahead and do that on your own.
Pause the video.
I'm going to show you a visual solution.
So let's say, look here's our original paint combination.
We have 3 to 2 to 4.
3 yellow cups, 2 blue cups, and 4 red cups.
That's what's giving us the paint, the shade of the paint
that we need.
If you wanted to use 1/2 of that,
then you would have to take 1 and 1/2 cups of yellow,
1 cup of blue, and 2 cups of red.
That's what you planned, you planned
on doing half the batch.
Instead, what did you have?
You did 2 and 1/2 cups of yellow.
So before we can proceed further,
let's just do what we did with the flour.
Since I have 2 cups of blue and 4 cups of red,
let's break everything into thirds.
So you can see that for every cup of yellow
you are going to need 1, 2, 2/3 of a cup of blue.
And here we're going to have 1, 2, 3, 4.
4/3 cups of red.
So for 1 cup yellow paint, you need 2/3 cups of blue and 1, 2,
3, 4 thirds cups of red.
Can you see that?
So it is breaking everything into 3
so that you can see what 1 cup of yellow corresponds to.
All right, so we originally planned for 1/2
a batch, which was this much amount.
But then we have instead of 1 and 1/2,
we have 2 and 1/2 cups of yellow.
So 2 and 1/2 cups of yellow.
So for this excess cup of yellow we
need to add 2/3 cup off blue and 4/3 cup of red.
That's what we have to add extra.
So you can see how visually this will be the answer then.
So add 2/3 cup of blue and 1 and 1/3 cup of red
which is the same as 4/3.
That will preserve the shade and you don't have to start over.