For more information, please see full course syllabus of Pre Calculus
For more information, please see full course syllabus of Pre Calculus
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Word Problems
- Word problems form the bedrock foundation of why we should care about math. While arithmetic and algebra are useful tools that we must learn to do well in math, we connect math to the real world through word problems. It is through word problems that we find value in math.
- Learning how to solve word problems teaches us crucial skills such as logic, thoughtfulness, and problem-solving. Even if you have no interest in doing math later in life, these skills are important for virtually anything you might be interested in doing later on. Studying word problems isn't just about doing well in math: it's about preparing yourself for dealing with puzzling situations for the rest of your life.
- There is no one way to solve all word problems because they can take so many shapes. However, there are some general guidelines that will help us work on them. We will use a four step process for approaching word problems. By following this method, you'll have clear, concrete tasks to accomplish at every step. While creativity and thought are still necessary, these guidelines can be used on virtually any problem.
#1 Understand the Problem. Figure out what the problem is asking about. You don't have to solve anything right now, or even figure out exactly what you're looking for: you just want to have some idea of what's going on. This might seem obvious, but remember, you can't solve something before you know what's going on.
#2 What Are You Looking For? Once you understand what's going on, you need to figure out what you are trying to find. In math (especially for the next few years), this will often take the form of setting up variables. Define any variable the problem asks for, along with any others you will need to solve it. Make sure to write down a reminder about what each variable means so you don't forget later.
#3 Set Up Relationships. Use what you know to set up relationships between your unknowns and whatever information the problem gives you. In math (especially for the next few years), this will usually take the the form of making equations. You will set up equations involving your unknown variable(s) and whatever else you know. Sometimes you'll realize you have more unknowns than you originally thought. That's okay: just make some new variables, then figure out equations that will involve these new variables as well.
#4 Solve it! Once you've done the above three steps, you're ready to solve it! Often this is the easiest part. After all, it's like doing any other exercise now: you have some equations to solve. Just roll up your sleeves and work on it like a normal (non-word) problem. [As a rule of thumb, to solve a problem you need as many relationships as you have unknowns. For example, if you need to figure out three variables, you must have three equations relating them to each other.] - This general method of problem-solving is great: it will work in pretty much any situation, no matter what you're working on. But let's simplify it a little and consider the specific approach you'll need in math for the next couple years:
- Understand what the problem is talking about,
- Set up and name any variables you need to know,
- Set up equations connecting those variables to the information in the problem statement,
- Use the equations to solve for the answer.
- Here are some extra tips for working on word problems:
- Tip-Draw Pictures: While it may not be possible for every problem, it can help massively when you can. Basically, if a problem talks about geometry, shapes, or something that is physically happening, you want a picture to look at. If the problem doesn't give you one, draw it yourself! When you're not sure how something works or what to do, sometimes a quick sketch can clear things up.
- Tip-Sum of Parts = Whole; Whole = Sum of Parts: You will often have problems where you can't directly solve for something, but the thing is part of a larger whole or built out of smaller pieces that you can solve for. In that case, figure out how it relates to those other things and solve for those instead, then use that information to get what you want.
- Tip-Try Out Hypothetical Numbers: Sometimes it can be hard to figure out what's going on because we aren't working with numbers. Try plugging in hypothetical numbers to help you understand what is going on. This is a great way to test the equations you set up. It's easy to make a mistake while setting up equations, so check them afterwards with values you understand.
- Tip-Student Logic: The hardest part is often figuring out the relationships the problem gives you. Luckily, you've got a secret weapon: you're a student. This means you can use student logic. You can pretty much be certain the problem is based on whatever you're currently learning. For example, if you're studying logarithms, you know the problem can almost certainly be solved with logarithms.
- Tip-Jump in! Working on word problems can sometimes be paralyzing. You're not sure where to start, you don't know exactly what you're looking for, you don't see how to solve it. It's okay! That happens to everyone sometimes. The important thing is to not freeze up. Instead, just try something. Even if you can't set up the equation right the first time, or you pick the wrong variable, you'll wind up learning from your mistakes. As long as you pay attention to what you're doing and think about what makes sense, you will see where you went wrong. By seeing where you went wrong, you can realize what you have to do for the problem. The fastest way to learn can be by making a mistake!
Word Problems
- First, you need to know what's going on. If the problem doesn't make sense, how can you work it?
- Once you understand the problem, you need a way to talk about what you want to find out. You can represent a number you don't know yet with a variable.
- Now that you have your variable(s) in hand, you need to connect them to the rest of the information in the problem. You can do this by carefully reading the problem and creating equations.
- After you have all your equations in place, solve them for the answer!
2) Variable(s): Set up variables so you can talk about the values you need to know for solving the problem. Always make sure to write a note to yourself about what each variable represents.
3) Equation(s): Use the information in the problem to set up equations using the variables you just created.
4) Solve: Now that you have all your equations, use them to find the answer. This will normally be a matter of solving for what one of the variables is.
- Begin by making sense of the problem. They're picking oranges, and they have each picked some portion of the total share.
- Set the thing we want to know as a variable:
[We could use any symbol, but x is fine and pretty common.]x=number of oranges Ray picked. - Using the variable, set up an equation. If we add x to the number Lola picked, we get the total:
x+17 = 25. - Solve the equation for x.
- Begin by making sense of the problem. There are three libraries with some number of books amongst them. Each of the libraries contributes to that total number.
- Set the thing we want to know as the variable:
[We could use any symbol, but x is fine and pretty common.]x = number of books in Hecate Library. - Using the variable, set up an equation. If we add x to the number of books in each of the other two libraries, we will have the total number of books:
127 + x + 390 = 892. - Solve the equation for x.
- Begin by making sense of the problem. Because the problem has to do with a geometric shape, it would help to draw a picture.
- We want to know the width, so we set it as a variable:
[We already wound up doing this over the course of drawing the picture, so it would probably be fine to skip this step if we wanted.]w = width of the rectangle. - The area of a rectangle is equal to the product of the length and width. [If we had forgotten this fact, we know that we need to somehow connect the width and length to the area, so we could start looking up formulas for the area of a rectangle.]
160 = 20 ·w. - Solve the equation for w. Don't forget that it needs to have a unit on it because it's a measurement.
- Begin by making sense of the problem. To understand this, we need to know what `consecutive integers' are. If we don't know it, we can look it up. Consecutive integers are two numbers that are "next to each other": 1 and 2, 7 and 8, 512 and 513 are all examples.
- Set up the variables. We want to know both, so we could set up variables for both:
[We could actually write this with just one variable as well, as we will see in the next step.]x = the first integer y = the second integer - We know that the two numbers add up together to 109:
However, that's not enough information to solve it. So we have to go back to the question ands see what other equations we can create. We know that the numbers are consecutive, so y must be 1 greater than x:x+y = 109.
[Alternatively, we could have realized this from the beginning, and never written out y when we were setting up the variables. We could have just set the two "numbers" as x and x+1 since we knew they were consecutive from the start.]x+1 = y. - Using this information, we have
x + (x+1) = 109. - Solve the equation for x. Don't forget that the question asks for both numbers, not just x. This means we have to give the value of x and x+1.
- Begin by making sense of the problem. Two people are picking strawberries. Each of them has picked some number, and together they have picked a total of 210.
- Set up the variables. We want to know how many each of them have picked:
R = Ray′s number of strawberries L = Lola′s number - From the problem, we know they have picked 210 total:
However, that's not enough information to solve it. So we go back to the problem and see what other equations we can make. We have a relationship between how many Ray has picked and how many Lola has picked-"Lola has picked 120 fewer strawberries than five times the amount that Ray has picked":R + L = 210.
[Note: Turning a sentence like that into math can be tough. If you try to make an equation, but you're not sure you set it up right, test it out with some hypothetical numbers. See if it does the same thing as the sentence said.]L = 5R − 120. - Substituting one equation into the other, we get
R + (5R − 120) = 210. - Solve the equation for R, then use that to find L as well.
Hecate Library has three times as many books as Porter Library. Siward Library has 120 more books than half the number of books in Hecate. How many books are in each of the three libraries?
- Begin by making sense of the problem. There are three libraries with some number of books amongst them. Each of the libraries contributes to that total number.
- We don't know the number of books in any of the libraries, so let's make a variable for each one:
P = number of books in Porter H = Hecate S = Siward - From the problem, we know that there are 2397 books in total:
However, that's not enough information to solve it. So we go back to the problem and see what other equations we can make. We see we are given two other relationships amongst the libraries. "Hecate Library has three times as many books as Porter Library":P + H + S = 2397.
"Siward Library has 120 more books than half the number of books in Hecate":H = 3 P. S = 1 2H + 120. - We can put each of the libraries in terms of P:
We can now plug these in to our equation involving all three libraries:H = 3 P S = 1 2(3P) + 120 P + 3P + 1 2(3P) + 120 = 2397. - Solve the equation for P, then use that information to find out H and S as well.
- Begin by making sense of the problem. Since we have a picture already, we just need to understand the picture. We can think of it as if the square has had a circular portion removed from its area.
- In this case, we could get away without creating any variables because we won't need to do any complicated algebra with the unknown. Still, let's set one up anyway:
A = area of shaded portion. - To find out what the shaded area is, we need to break this into pieces. We can think of the square as being a combination of the shaded area and the circle.
This means we need formulas for the area of a square and the area of a circle. If we don't know them already, we can look them up and find out:A_{square} = A+ A_{circle} ⇒ A = A_{square} − A_{circle} A_{square} = s^{2} A_{circle} = πr^{2} - We already know that the radius of the circle is r=4 from the problem, but what is the length of the side of the square? We can look at our picture again to figure this out. Since the radius spans from the center to the edge of the circle, we can use two radii to create a side as in this picture:
- Putting this all together, we get
A = 8^{2} − π·4^{2} .
- Begin by making sense of the problem. He has various types of coins and they all add up to a certain amount of money.
- We want to know how many quarters he has so we will definitely need a variable for that. It seems likely that we will need to know the number of pennies to work out the problem, so we'll make a variable for that as well:
q = number of quarters p = number of pennies - We know the total value of all his coins, so we can create an equation:
We will need another equation to solve since we have two unknowns. Going back to the problem, we also know the total number of coins:q ·0.25 + 3 ·0.10 + p ·0.01 = 2.09. q + 3 + p = 14. - We can solve the second equation for p in terms of q:
Then we can substitute that in to the first equation to find an equation that only uses q:q + 3 + p = 14 ⇒ p = 11 − q q ·0.25 + 3 ·0.10 + (11−q) ·0.01 = 2.09. - Solve the equation for q.
- Begin by making sense of the problem. This is a difficult problem to understand, but we can make it a lot easier by representing it with a picture: Our goal is to find out what the total length of all the fence is (the black lines).
- To figure out the total amount of fence, we will need to know the width of the pens: w = width of each pen.
- We know the total area is 54 m^{2}, so if we multiply the area of one pen (3·w) by three, that would be equal to the total area:
3 ·(3w) = 54. - We can solve this to find that the width of each pen is w = 6. Now we need to find what the total length of fencing is.
- The width (w=6) is used a total of six times because it makes the top and bottom of each pen. The length (3) is used only four times though because it is shared by some pens. This means the total amount of fencing is
6 ·6 + 4 ·3.
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Word Problems
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Introduction
- What is a Word Problem?
- Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
- Requires Us to Think
- Why Are They So Hard?
- Reason 1: No Simple Formula to Solve Them
- Reason 2: Harder to Teach Word Problems
- You Can Learn How to Do Them!
- Grades
- 'But I'm Never Going to Use This In Real Life'
- Solving Word Problems
- First: Understand the Problem
- Second: What Are You Looking For?
- Third: Set Up Relationships
- Fourth: Solve It!
- Summary of Method
- Examples on Things Other Than Math
- Math-Specific Method: What You Need Now
- Understand What the Problem is Talking About
- Set Up and Name Any Variables You Need to Know
- Set Up Equations Connecting Those Variables to the Information in the Problem Statement
- Use the Equations to Solve for an Answer
- Tip
- Example 1
- Example 2
- Example 3
- Example 4
- Intro 0:00
- Introduction 0:05
- What is a Word Problem? 0:45
- Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
- Requires Us to Think
- Why Are They So Hard? 2:11
- Reason 1: No Simple Formula to Solve Them
- Reason 2: Harder to Teach Word Problems
- You Can Learn How to Do Them!
- Grades
- 'But I'm Never Going to Use This In Real Life'
- Solving Word Problems 12:58
- First: Understand the Problem
- Second: What Are You Looking For?
- Third: Set Up Relationships
- Fourth: Solve It!
- Summary of Method 19:04
- Examples on Things Other Than Math 20:21
- Math-Specific Method: What You Need Now 25:30
- Understand What the Problem is Talking About
- Set Up and Name Any Variables You Need to Know
- Set Up Equations Connecting Those Variables to the Information in the Problem Statement
- Use the Equations to Solve for an Answer
- Tip 26:58
- Draw Pictures
- Breaking Into Pieces
- Try Out Hypothetical Numbers
- Student Logic
- Jump In!
- Example 1 34:03
- Example 2 39:15
- Example 3 44:22
- Example 4 50:24
Precalculus with Limits Online Course
Transcription: Word Problems
Hi--welcome back to Educator.com.0000
Today, we are going to talk about word problems.0002
This lesson is going to tackle solving those dreaded word problems.0005
Hopefully, when this lesson is over, you are going to dread them a little less.0008
We are going to start by talking about why we should care about them, and essentially why we should care about learning in general.0011
Then, we are going to go over a general structure for approaching and solving word problems, and really any form of problem.0017
We will see how it applies to what we are working on now.0022
And then, once we understand that method, we will see a variety of different tips and strategies to help us get the most out of it.0025
So, we will go over some specific tips that apply to that general strategy that we talked about.0031
And then finally, we will work a bunch of examples where we will actually see directly how we are using this step-by-step strategy,0035
so we can see how the method gets applied on real word problems.0041
All right, let's go: first, what is a word problem?0044
The term word problem gets used a lot; it is pretty much just a catch-all.0048
It is for any problem that is primarily being described with words, as opposed to math symbols.0052
That doesn't mean very much; you can use words to describe a lot of things.0058
So, it means that there is no one way to approach word problems, because you could be talking about so many different kinds of problems.0061
Since there are so many different possible problems, we have to be ready to adapt when we are working on a word problem.0067
Trying to answer the question, trying to figure out the one way to solve word problems, would be like saying, "How do you play sports?"0073
There are a lot of different sports; there are a lot of different ways to play within each of those sports.0079
It depends on the specific situation; that is going to tell us what we have to do.0083
The best method depends on our specific situation, so there is no one way to do all word problems.0087
But we do know this one fact: pretty much all word problems are going to require us to think.0093
We have to pay attention and be creative, because we are being asked to do more than just follow a formula.0098
As opposed to a lot of problems that we will get in math, where it is just "here is the method that we do it;0103
let's apply that method one hundred times in a row."0108
It is going to be "understand this idea, and then do some critical thinking about it."0111
So, we have to actually be thinking and paying attention when we are working on word problems.0115
We should always be thinking, and we should always be paying attention, when we are doing everything in life.0118
But for word problems, it is going to be especially important that we are really thinking about what we are doing, because we are trying to understand it.0123
Otherwise, it is just not going to make sense.0128
Why are they so hard--why are people constantly thinking that word problems are the hardest kind out there?0132
Well, first, there is no simple formula for them: there is no one way that you do it--just plug things in it-- because each word problem can be different.0137
So, when we are working non-word problems, we normally have examples that we can follow step-by-step.0145
There is some formulaic method that we can just rely on.0149
But with a word problem, there is no such formulaic method.0152
It is up to us to be paying attention, to be ready to be creative and thoughtful.0155
The fact that it actually requires thought--it requires some creativity--that is one of the reasons why word problems are generally harder than non-word problems.0159
Second, it is a lot harder to teach word problems.0168
It is easy to teach simple, repeatable instructions--perhaps not easy all the time, but it is easier to teach simple, repeatable instructions,0171
things like formulas or step-by-step guides--anything like that where the idea is "do what I did; do what I did; do what I did."0179
That sort of thing is pretty easy, because you can just follow the "monkey see, monkey do" saying--you do what they led, and it works out.0186
But with word problems, you have to teach creativity; you have to teach an ability to understand what is going on.0194
You have to really teach analyzing a whole bunch of things at once, and understanding things.0198
As opposed to just teaching to follow a method, you have to teach how to understand and how to think.0202
That is a much bigger task than just teaching a few quick instructions.0206
So often, sadly, they are overlooked, because it is easier to teach that, and a lot of education is based0211
on taking standardized tests, so we end up seeing us teaching to standardized tests, as opposed to teaching0217
to a larger scale of thinking and understanding, which is the sort of thing0223
that is absolutely necessary if we are going to do well on word problems.0226
Don't let this make you be worried: don't despair--you can learn how to do them.0231
Just because they are a little bit harder, and you don't get much learning about them (usually),0237
doesn't mean you can't get great at them--you can get great at doing word problems.0241
It is just going to take some though, some imagination and patience.0246
Like everything else, it is going to be a skill that you can practice--you just need to practice the skill.0248
In this case, if, for example, your teacher doesn't assign you many word problems,0255
it might be a good idea to work an extra word problem out of your book.0259
Make sure it is one of the problems...most math textbooks have answers to the odd exercises, or sometimes the even exercises, in the back.0262
So, choose the one that you have the answer to, but that wasn't assigned to you,0272
so that you get the chance to have some extra experience with word problems.0275
So, if you know that you have difficulty with word problems, you have to focus on it; you have to work on it a little bit extra.0278
I know, I am asking for you to do more; but there is pretty much no other way to get better at something.0282
You get better at things because you practiced, not because you just want to be better at them.0286
So, word problems are a skill you can practice, and a skill you can definitely improve.0291
But you do have to practice it, if you want to improve.0295
But if you practice, you will definitely get better; so consider that.0297
If you have a lot of difficulty with word problems, just start tackling some easier word problems.0299
Work your way up to medium ones; work your way up to hard ones.0303
Just sort of do that in the background, as we are working on precalculus.0305
It will be a really useful skill that will really help you in a lot of things later--not just math, even--all sorts of things.0308
Why should we care about word problems?0315
While word problems may be difficult, they are also incredibly important.0317
In some way, they are the point of math: word problems turn math into something more than a meaningless series of exercises.0321
As opposed to just solving one equation after another, where it is just meaningless symbols, word problems "ground" what we are doing.0327
They give it context; they make the math mean something, which can make it, sometimes, beautiful, and usually important to what we are doing.0334
Well, when we use math in science, we can better understand and appreciate our universe.0342
It is through math that we can have physics be able to turn into equations that we can describe our world with.0350
It is through math that we can figure out specific things in chemistry and talk about quantities.0356
It is through math that we can do statistics in biology and sociology.0360
It is through math that allows us to understand the world around us.0364
When we use it in engineering, that allows us to build amazing feats of human ingenuity.0368
It allows us to build huge bridges; it allows us to build giant dams, skyscrapers...0373
It lets us build devices like phones that fit in the size in your hand...well, yes..."computers" is actually what I meant to say,0379
because most smartphones these days are becoming even more powerful and more powerful.0386
The incredible miniaturization of technology--what we have right now--the fact that I can be teaching you0389
when I am in a totally different place than you, and you are watching it right now...0394
Engineering: math has been put in all the engineering, and the computer things when we are building things--we need math0398
for us to be able to make all these things: math runs all of the technology that we have.0407
At heart, technology is based on what we have been able to figure out in math.0411
In many ways, it is applied math through other things.0415
And if we use math just on its own, we can find deep truths about the nature of logic and knowledge.0418
Geometry: when you were studying geometry, it was all just mathematical ideas.0423
They have applicability in the real world, but they are also just kind of beautiful, interesting ideas.0427
I have studied a lot of math, and I think there are some really, really cool things that you can see in math, that are just purely ideas in math.0431
But to describe them, we need words; just a bunch of symbols isn't going to do it.0438
And without language to connect math to these really deep, interesting ideas, we just have symbols.0442
So, we need language to be able to give us a deeper context, to ground what we are doing with our math.0448
It is through word problems that we find value in math.0455
Word problems are our connection between wanting to do things and learn things, and this interesting symbolic language that lets us solve things.0458
Word problems are the point of connection between wanting to know things and being able to solve things.0466
They are really, really important for that reason; in many ways, word problems are what gives us knowledge.0471
Now, not everyone is going to be convinced by my impassioned appeal to the inherent value of beauty and knowledge.0478
I think that wanting to learn, and the fact that learning is an amazing, really cool thing--that is a great reason to learn, in and of itself.0484
But if you would like a more material reason--grades.0492
You can get an A, and you can possibly even get a B, but you are never going to be able to achieve the highest marks0496
in a math class if you don't understand word problems.0500
To do your best in class, you need to be able to solve word problems.0504
Every math is going to include, or at least really should include, some word problems.0507
So, it is important that you know how to approach them.0512
With time and practice, you can understand them better, and you can improve your grades.0515
Remember what I was saying before: if you practice this, you will be able to improve at it.0518
And it is not just math class: any standardized test (like if you are going to take the SAT,0522
because you are interested in applying to college; or if you are currently in college,0526
and one day you want to take the GRE so that you can apply to graduate school)--0529
they are going to use word problems in there, as well.0533
Pretty much any test that you will take anywhere will have some word problem ideas going in it.0535
So, you want the ability to solve problems like that--being able to solve word problems is crucial in a lot of situations.0539
Also, if you want to do anything like engineer and build things, or do science,0547
or do anything that has really hard scientific connections, or is really analyzing0551
and measuring the world around us, you are going to need to understand math.0558
So, there are lots of really good reasons to understand math, in just a "material-benefit-to-us" way.0560
Finally, there is always that gem of an excuse, "I am never going to use this in real life!"0567
"I know that you might say it is useful in science, but when am I ever going to use partial fraction decomposition0572
(which you will learn about later)--I am never going to use this in real life, later on, so why should I learn it now?0577
Is that true--"I am never going to use this in real life"?--yes, honestly, you are probably right.0585
There is a very good chance that the things you will learn in this course, or any math course,0592
will never be the sort of thing that you get paid for doing later in life.0597
You will not be having to analyze functions, and that is going to be what gets you your paycheck every day.0601
That is not the point: that is not the point of learning these things.0607
You are not in school to learn the things that you are going to use later in life.0611
You are not in school just to learn those things that are going to get you a paycheck later in life.0615
You are there in school to learn how to learn.0619
Think about this: musicians do not play scales at concerts; boxers to fight punching bags in the ring; and surgeons do not operate on cadavers.0623
But the practice that they get by doing each of those things is absolutely necessary for them being able to perform well later in life.0632
So, you might not end up using this immediately in real life, or you might never use it in "real life" (whatever that means).0640
But what you are getting now is practice: whatever you do later in life, you are going to need to learn new things over and over.0648
So much of any interesting job, so much of whatever you do later in life, is going to be learning new things and getting good at those new skills.0657
School is your chance to practice this process, this skill of being able to learn something, get good at it, and use it.0665
This skill is what you are learning right now, and it is absolutely necessary if you want a job that pays well, is interesting, or is enjoyable.0673
And if you want one that is well-paying, interesting, and enjoyable, you are definitely going to have to be able0681
to learn lots of things quickly, in your situation, and do whatever you need to.0687
You need the ability to adapt, the ability to learn many things and operate in many conditions.0690
The way that you do that is: you practice it now.0695
You do not get better by just wishing you were going to be better; you get better by practicing it.0697
School, learning, is your chance to practice the learning that you will need for the rest of your life.0703
If you do not practice it now, because you are not going to use those things, yes, you are technically right.0709
But that is like saying, "I am not going to drive this car later in life; I am going to drive some other car; so I won't practice driving this first car."0713
To be able to drive that second car, you are going to have to have learned how to drive a car somewhere.0722
So, you can't skip learning how to drive the first car.0727
Even though there are going to be some differences between the two cars (they might be completely different cars),0729
there are a lot of parallels here--it is going to be very similar driving one car or the other.0734
It may be hard to see right now, but please trust me on this.0739
I am being as truthful as...this is just from the bottom of my heart: put effort into learning now.0742
The process of learning is going to give you skills that you will use for the rest of your life.0750
It can be difficult to see how valuable those skills are to you now.0755
But trust me, in 5 or 10 years, when you look back later, you are going to be so thankful0758
that you took the time and effort to really understand what you were doing--to go through0764
all of that practice of learning--because those skills of being able to learn quickly0767
and do well and understand things--they are going to make the rest of your life so much better and so much easier; it is really important.0771
All right, moving on to actually solving word problems: they are so important, so we really want to be able to solve them.0779
Now, as we discussed, there is this problem that we can't just solve all word problems with one method.0789
But luckily, there are some general guidelines that will help us work on them.0795
We are now going to see a four-step process for approaching word problems.0799
If you follow this method, you will have clear, concrete tasks that you can accomplish at every step.0802
Now, creativity and thought are still going to be absolutely necessary.0807
But these guidelines can be used on virtually any problem that you encounter, so it is a really useful thing here.0811
Let's look at it: the very first step is that you need to understand the problem--begin by figuring out what the problem is asking about.0816
You don't have to solve anything right now; you don't even have to figure out exactly what you are looking for.0823
But you need to figure out what the problem is asking about.0827
You just want to have some idea of what is going on--what is the general thing that is happening here?0834
Many word problems are going to unfold like a story, in some way; so you just want to understand--"What is this story telling me?"0839
Now, this step may seem obvious, like everybody is going to think, "Well, of course I have to understand the problem!"0845
But so many people gloss over it, and they just try to hop right into solving the problem.0850
But you need to understand it first; how can you solve something if you don't even know what is going on?0855
You have to know what is happening in the problem, what the problem is about, what the ideas going on are,0861
before you are going to be able to have any chance of actually solving it.0867
So first, just get a sense of what is going on.0869
Once you have that, second, what are you looking for?0873
Set up your variables: what are you looking for?0876
Once you know what is happening in the problem, you want to figure out what you are trying to find.0878
What are the ideas that are central here, that you need to be able to work out this problem?0884
Now, almost always, especially for the next few years, before you get into really advanced math classes,0889
if you continue to study math in college--for precalculus, calculus, and even the next couple of years--0895
this will take the form of setting up variables: you will set up variables and define any variable that the problem asks for.0899
You might also need to define other variables, such as values that are talked about, but never explicitly given.0906
Now, I want to make absolutely sure that you write down what each variable means.0911
I really want to make sure that you do this; I am not kidding.0916
Until you are extraordinarily comfortable with word problems, you should be literally writing out a little reminder of what each variable means.0920
For example, if you had a problem where you had to talk about some number of tables, then you might say, before you started, "t = # of tables."0927
And then, you also might have to talk about chairs in the problem, so "c = # of chairs."0939
And that is enough right there; but very often, students will be working on it, and they will forget what this letter meant or what was really there.0945
This is a way of anchoring your work, so you know what you are searching for, what these ideas are about.0952
And by writing it down, it will make it that much easier to work on the rest of the problem.0957
Really, really, really: write down what the variable means--it should be completely obvious to you,0960
either because you will have a picture (which we will talk about later), or because you have literally written out what the variable means.0964
That way, you can't forget it while you are in the middle of working on the problem, and get confused.0970
So, you set up what you are looking for at this point: what you are looking for, what is unknown, what is going to help you get to figuring out the answer.0973
Third, you set up relationships: you want to figure out what the problem is telling you about what you are looking for.0982
What is the problem telling you about what you are looking for?0988
You are trying to solve something at this point, which you have figured out from #1.0993
And you know the sort of ideas that you are looking for; that was #2.0996
So now, you need to figure out what the problem is telling you about those things that you are looking for (those variables, if we are doing a math problem).1000
Use what you know; use what the problem gave you to set relationships between your unknowns and whatever information the problem gives you.1007
Now, in math class, especially for the next few years (once again), this will usually take the form of making equations.1015
This is what it is normally going to come down to: you will be making equations.1022
You will set up equations involving your unknown variable or variables, and whatever else you know from how the problem is set up.1025
The problem will tell you some information, and you will use that information to build equations.1031
And if it is something that is not specifically a math problem, or a more general form of math problem,1035
you might be just using it to set up things of what you know and what you can rely on as you start to work through it.1040
Sometimes, when you are working on these equations, you will realize that you have more unknowns than you originally thought.1045
That is perfectly fine; just make up new variables at this point.1049
You realize, "Oh, to use this idea, I need to also have this new variable introduced"--introduce a new variable!1052
Go back to step #2; make a new variable; write down what it means; and then plug it into the equation where it is going to go.1058
Figure out the equations that will involve that new variable that you realize you need to use.1063
Finally, once you understand what the problem is about, you know what you are looking for,1068
and you have all of your relationships (equations) set up, you are ready to solve it.1072
Normally, in many ways, this is going to actually be the easiest part, because it is just like doing any other exercise now.1076
You have equations to solve: you are looking to solve for something, and you have equations.1082
It is just going to be a matter of doing the math that you have been doing in every other section,1086
in every other part of the section that you are working on.1089
It will normally have many exercises, and it will only finally culminate in some word problems.1092
So, it is just like doing all of the exercises you have been doing before;1096
it is just that it had this framing of how we get to those equations--how we get to doing the exercise.1098
So, you have some equations to solve or something like that; roll up your sleeves and work on it like a normal non-word problem.1104
And as a general rule, to solve a problem, you need as many relationships as you have unknowns.1109
For example, if you need to find out three variables, then you will have to have three equations (or more) that relate the three variables together.1116
So, you will need the same number of relationships, the same number of equations,1125
as the pieces of information you are trying to find out.1129
So, it is like each equation is a key that you can fit into an unknown lock.1131
Each equation is a piece of knowledge that you can use to shine the light on some unknown thing.1138
In summary, complex problems can be approached by first understanding what you are working with;1145
then figuring out what you are looking to find (figure out what you want or need to know);1150
third, you work out relationships--how do the things that you know already from the problem,1157
and the things that you are looking to find--how are they connected to each other?1163
And then finally, you put it all together, and you get the answer.1167
So, that is our basic summary of method.1171
Now, it may seem surprising, but this method can actually be applied to pretty much any problem--any issue you have, not just math.1174
This is such a general tool for approaching complex ideas that you can use it1181
for pretty much anything where you are trying to figure out a way to solve an issue.1185
You are probably used to doing this in all sorts of situations already; it is just a good way to approach problem-solving in general.1189
So, let's actually see two examples that are not connected to math, just so we can get an idea of how we solve problems on a general scale.1194
So, consider these two situations: in the first situation, your car is making weird noises when you drive.1201
And in the second situation, your friend suddenly stops talking to you one day.1207
They are very different things, but both issues that you would want to deal with.1211
So, to begin with, our first thing is that we understand what is going on.1214
Your car is making weird noises when you drive: well, if my car is making weird noises, there is probably some sort of issue going on.1217
And so, now you also have...in a word problem in math, we are going to have a more specific being-told-what-to-do.1222
In real life, our issues are normally more amorphous, more uncertain in what we need to do to get to the right thing.1228
But with this case, let's say we have figured out that our car has a problem.1234
We need to figure out what that problem is, and then we need to fix it.1238
Now, let's say, in this world, we don't want to spend the money on taking it to a mechanic.1241
So, that means we have to be the one to figure it out, and we have to be the one to fix the problem.1245
So, your car is making weird noises when you drive; that means that what we need to do here--1250
our understanding of it is that we need to figure out what the problem is, and then we need to fix it.1253
That is what our objective is: "Fix the problem."1258
In the version where we are talking about our friend who stops talking to us, what we want to do now is:1260
we ask ourselves, "Well, do I care about this friend enough to want them to still be my friend?"1265
We are going to assume that, yes, they are actually your friend; you want them to be friendly again.1269
But this is one of our things where we are trying to understand the problem.1273
We actually have to understand and think about what it is--how does it connect to the rest of what is going on?1275
All right, if your friend suddenly stops talking to you one day, but you actually just realized that you don't actually like them at all,1280
then they are no longer your friend; just make them no longer your friend.1284
But in this case, we are saying that, yes, our objective is to get our friend to be friendly again.1287
We want them to actually be our friend, not just be this person who won't talk to us.1292
All right, now we have our objectives in mind; now we can figure out what I need to know.1296
What are the things that I don't know that will help me get to that objective?1300
The second step: in the car example again, if our car is making weird noises, and it has some sort of problem (is our assumption),1305
we want to figure out what is making that noise; we also want to figure out1311
if the thing that is causing the noise is the result of some other issue.1317
And if there is some other root issue, has it damaged anything in the car?1320
And then finally, how can we fix any and all damage?1325
Now really, this is the only thing that matters: we don't really care...in truth, if some genie appeared out of nowhere,1327
and said, "I will fix your car, just because I am a nice guy," we would say, "Yes, genie, fix that car--yay!"1335
And they would fix our car; so really, if we could somehow get a genie to show up here at the end,1342
we wouldn't need to know the answer to these first three things.1346
But it is real life; it is likely going to be the case that these first three issues, these first three unknowns, would really help us to fix the damage.1348
If we can figure out what is causing the noise, if we can figure out if there is an issue that is making that noise,1357
and if we can figure out if that issue has damaged anything, those are all things that are going to help us1362
to fix the car and make it stop making that noise.1366
So, while our final unknown is the "how do we make it better?" we really want to know other things on the way,1369
if we are ever going to be able to figure out how to make it better.1375
In the version where we are looking at our friend, we might want to know why our friend is not talking to us, right?1378
And then, what has happened recently would probably be a good way to get a sense of why they are not talking to you.1386
And finally, what is the best method to make them friendly?1390
Once again, we might actually not care about all of the other things--although, with the first two things,1392
we would probably be curious about why your friend isn't talking to you.1399
But once again, if that genie appeared and said, "I will make your friend be your friend again--no problem!1401
I am a nice guy!" you would say, "Thanks, genie--you are the best!"1405
This genie would be pretty awesome, and they would make our friendship better.1409
But in real life, we have to deal with our problems; we have to make things better.1412
We have to figure out how to solve the problem; we don't have some genie who is just going to do it for us.1416
So, we have to figure out how to solve it.1421
So, while the only thing we really want to know is this final idea, to be able to get to the point1422
where we can figure out that final idea, it would probably be very useful to know1427
why our friend isn't talking to us, and what has happened recently that frames what is going on right now.1431
So, this is an idea of how we set up what are the unknowns that would help me answer that original question.1436
For car troubles, if we want to answer what those unknowns are--get a sense of those things--there are a bunch of things we could do.1442
We could look under the hood; we could get a service manual for the car.1448
A service manual tells us how a car is put together--how that car is put together.1451
We would talk to friends and family who know cars well; we might research the problem online,1455
or anything else we can think of that would tell us more about what is going on with the car.1459
These would help us answer the unknown questions of what is wrong.1462
What is the issue? Has there been any damage? And how do we fix whatever damage it is, and fix the underlying issue?1465
With the friend, we want to think about what happened to your friend recently.1471
We want to think about your recent behavior to your friend.1475
We want to ask our other friends if they know what is wrong--perhaps they have some information that we are unaware of right now.1477
We want to pay attention to how our friend is acting when we are not around, or when we are not interacting with them directly.1482
That will give us some sense of what is going on, and be able to help us understand the situation.1488
And we can use all of those pieces of information to help us figure out how to make them friendly again, and make things better.1492
With all of this information, we now just figure out what is wrong with the car and what we need to do to fix the issue; and then, we do it.1498
In the friend version, we use all of this information to figure out1504
what would put our friend in a good mood to be friendly towards us again; and we do it.1507
We use the information to give us many things that we can now to reach our objective.1511
That is what is going on in complex problem-solving, just generally all the time.1515
We figure out what the problem is; we figure out what we need to know to deal with that problem.1519
We figure out how we can answer those things that we need to know.1523
And then, we use all of that information to get what we want.1526
Math-specific method...that is really great for any complex problem, and it can be used on anything at all that you need to solve.1530
But what about what we are going to see exactly in math?1537
This general method is great, but let's see what we are going to use in the next couple of years.1540
Let's look at the specific approach that we will be using.1545
Once again, we understand what the problem is about: we want to just get a sense of what is going on.1547
Then, we set up and name variables: variables represent our unknowns, the things that we are going to need1552
and have to have a handle on to be able to answer the problem.1558
Then, we set up equations; we use the information that is given to us in the problem statement1562
to get us the equations that we need to be able to answer the question.1570
And then finally, we have equations; we solve for that answer.1574
We have equations; we know what we are looking for because we understood the problem in the beginning.1578
We just work it all out, and we get an answer.1582
There are going to be a few word problems, probably around 5 to 10 percent (it depends).1585
But these are going to be mostly concept questions and proofs.1590
So, there are a few word problems that won't use this exact formula of "Understand, Variables, Equations, Solve."1592
All right, that is our formula in general when we are approaching word problems.1599
But sometimes we are going to have to prove things; or sometimes it is asking a general concept question1602
of "Is this true? Is this false? Why does this happen?" and in those situations, we won't be able to use this exact method.1606
But we can fall back on that more general method, where we just think in large-scale, complex terms.1611
Tips: let's start talking about tips that help us improve the efficacy of this method.1618
We understand how the method works now; it is "Go through and understand;1623
set up the things that you are looking for (your variables); set up what you know (your equations);1626
and then finally put it all together and solve it."1630
Here are some tips that will help us to use this method.1633
First, draw pictures: this is probably the single most useful tip in all of this.1636
Draw pictures: it may not be possible for every problem--some problems are going to be completely abstract, and you won't be able to do this.1642
But when you can, it is going to help massively.1647
Basically, if a problem is asking about geometry, shapes, or something that is physically happening--1649
if something is a real-world thing that we can imagine--it is really useful to draw a picture.1655
If the problem doesn't give you a picture right away, draw it yourself--just draw a picture.1660
And don't worry about making the picture look nice--you just want something that you can sketch in a few seconds, and that makes sense to you.1664
Cars can become boxes; people can become little stick figures; you can change things to dots and just straight lines.1670
You don't have to draw an entire road, if you are talking about a road; it is enough to just have a street line.1675
You just want something that you can use as a reference point, so that you can see it in your mind's eye.1680
It is not absolutely necessary to do this; but a visual representation of a problem can help so much.1684
When you are not sure how something works or what to do, sometimes a quick sketch1689
(so we can see what is going on) can help clear things up so much--it can be really, really useful.1693
Drawing pictures is a great way to approach word problems.1697
It really helps us understand it, helps us figure out what we are looking for, helps us set up our equations, and it can help us solve it.1700
It is really, really useful stuff.1706
The next tip is breaking things into pieces: Imagine you have a plate in your hands.1709
You drop it, and it breaks into many pieces.1714
If you collect all of the pieces and you put them back together, you will have a plate again, right?1716
And if you pull apart the rebuilt plate, you will have all of the pieces again.1720
You can take something into the pieces that make it up, and you can put it back together to make the whole.1723
The whole is made out of its parts, and the parts make up the whole.1727
The sum of the parts equals the whole; the whole equals the sum of the parts.1730
That is what this idea is expressed by; it is a simple idea, but it is important, and it comes up in math a lot.1734
Consider this problem right here: if we want to figure out what the area of the shaded area is--1740
we want to figure out how much shaded area there is--by breaking this figure into its pieces, we can figure out how to do this.1745
We see that that shaded part and that semicircle--they come together to form a square.1752
Well, I know what the square is; I can figure out what the semicircle is.1757
And so, I can use that information to figure out the shaded area.1760
You can break things into their pieces; and this will normally be applied to geometrical things, where we are talking about shapes and objects.1763
But once in a while, it will actually be applied to mathematical things, where it is just in terms of letters and things.1769
And we will actually break things into their pieces then, and use each of the pieces on its own.1773
We will see those things later on, potentially; and we will talk about it then, if it happens.1778
But normally, you are going to end up seeing it in pictures, where we are dealing with shapes, geometry, and things like that.1782
But it is a really useful idea and comes up in a lot of different places.1789
Next, try out hypothetical numbers: sometimes, it can be hard to find out what is going on in a problem,1792
because we are not working with numbers--we are really good at working with numbers,1797
because we have been doing it for a long time at this point.1800
So, you can try plugging in hypothetical numbers to help you understand what is going on.1803
Now, if you try a hypothetical number, make sure you try out numbers that follow the rules set in your problem.1808
If your problem says that the number is even, you want to make sure whatever hypothetical number you use1813
is something like 2, 4, 6, 8, so that you are following the rules that they tell you, so that you can go along with that path.1818
This is a great way to test the equations you set up.1825
We can set up an equation and say, "Well, this should tell me how much money the company made."1827
And what we do is plug in some numbers and say, "Well, what if they made 10 widgets?"1832
Well, we could probably figure out what it should be for 10 widgets, and so we could make sure that our equation1838
is giving out the same thing that we know 10 widgets should make them, in terms of money.1842
And so, we can make sure that, yes, our equation seems to be checking out.1846
It is working when I use it on simple numbers, that I can easily understand.1849
So, now it is up and running, and we can use it on variables and trust it.1852
So, if you have difficulty setting up equations, try plugging in hypothetical numbers to understand what is going on in there.1857
It is easy to make mistakes when you are setting up equations, so check them afterwards by plugging in hypothetical numbers, if you are not sure.1862
You can plug in a number where you understand what should happen there.1868
And then you will see that that makes sense; and we will see some examples when we work on the examples part of this lesson.1871
If things don't work out with your hypothetical number, you know you made an error in your equation, or possibly an error in your hypothetical number.1877
And there is something you need to go back and work on; make sure you fix it before you move on and try to solve it.1882
Student logic: this is an interesting idea.1888
The hardest part often can be figuring out what relationships are going on.1891
But you have a secret weapon--you are a student.1896
What I mean by this is that you can use student logic: you can be pretty much certain that whatever your problem is going to be about,1899
it is going to use what you are currently learning.1906
If you are a student, they are not going to have you do something that you haven't learned.1909
And they are probably mainly going to be focusing on the things that you have been working on.1914
So, whatever you are currently learning is exactly what you will probably use to set up the relationships.1917
For example, if you are currently studying parabolas, you know that the problem1922
can almost certainly be solved by using something that you learned about parabolas.1925
So, you will set up your word problem, and then you will say, "Oh, right, what do I know about parabolas?"1929
"Oh, the top of a parabola happens at this equation," and so you will see if that would be useful here; and it probably would be.1932
Try to figure out how the problem is related to what you have recently been studying.1938
How is this connected to what you have been working on in this section, in this lesson, in this chapter?1941
And then, use those ideas to help you set up the equation.1946
You don't have to normally worry about things that are completely unrelated to what you have just been learning,1949
because your teacher is normally trying to teach you the things that you have been working on.1953
So, they are going to be using those ideas in the word problems, as well.1957
Jump in: this is a good suggestion--you don't want to just get paralyzed.1961
Working on word problems can sometimes freeze you, and you are thinking,1967
"I don't know what to do! I don't know what to do!" so just calm down and try something.1969
You are not sure where to start; you don't know exactly what you are looking for; you don't know how to solve it, so you get scared.1974
But instead--that is OK; that happens to everyone; instead, throw in something; try to do something.1979
If you can't set up the equation right the first time, that is OK.1985
If you pick the wrong variable, that is OK; if you plug in the wrong hypothetical number, that is OK.1987
You are probably going to learn from your mistake.1992
Normally, by making a mistake, we think, "Oh, well, that didn't work..."1995
Oh, but because that didn't work, this other thing would work...and then you are right on the way to doing it.1999
As long as you pay attention to what you are doing, and you think about what makes sense and what doesn't make sense,2003
and you are double-checking your work and thinking "Is this sane? Is this a reasonable thing to be doing?"--2007
you are normally going to see where you went wrong.2012
And by seeing where you went wrong, you can realize what you need to do in the problem.2015
The fastest way you can learn is often by just making a mistake.2018
So, reach in; get your hands dirty; it is OK if things go wrong at first, because eventually they will work out.2022
If you just stand back and keep thinking, "I don't know how to do this! I don't know how to do this!"2027
yes, you are right--you are not going to know how to do it, because you are just saying one thing; you are pulled back.2030
You need to try something; trying something is almost always going to work better than doing nothing.2035
So, just jump in and get started, if you get stuck.2040
All right, let's look at some examples: a four-part method: first, understand; 2) Set up variables; 3) Set up equations; 4) Solve the thing.2044
So, first, we need to understand this; #1--we read through it.2053
"Sally has a job selling cars. Her monthly base pay is $2,000, along with a 1.7% commission on all the cars she sells."2057
"If she earned $5,315 in March, what was the total cost of the cars she sold in March?"2065
So, the first thing we have to do is understand what is happening here.2070
Sally has a job selling cars--that makes sense.2073
She gets paid some base pay; she gets paid some amount, and then..."along with"...she gets paid $2000, plus some other thing, this commission.2076
Now, we might say, "I don't know what a commission means."2085
So, what do we do? We look it up! You put the word "commission" into an Internet search, or you look up "commission" in a dictionary.2088
It would probably be a good first step.2095
If you don't know what the word "commission" is, you look it up.2097
We look up "commission," and we find out that it is normally a percentage fee that you get for selling something.2099
So, if somebody has a 10% commission, if they sell $100 of things, they personally get $10 back.2105
So, a commission is a way of making a profit off of what you sell to other people.2112
A salesman gets a commission on their sales, generally, to encourage them to sell more.2116
So, she gets a 1.7% commission: 1.7% of whatever she sells, she gets back; OK.2120
So, she gets $2000, plus 1.7% of the amount of the cars that she sold in the month.2128
If she earns $5,315 in March, what was the total cost of the cars she sold in March?2134
Oh, so we have a piece of information about March; we are looking to know about the cost of that.2137
So, we are looking for a connection between how we know her money breaks down and what the cost of those cars must have been.2142
All right, so number 1 is done: we understand what it is about.2147
#2: What are the things that we need to know?2151
Well, we want to be able to talk about her pay--how much does she get paid?2153
So, we want to have some way of being able to talk about how that varied based on her commission.2158
How about we make c equal, not commission, but the cost of cars sold.2163
This is the amount of money she makes off of the cars she sells.2170
So now, let's figure out a way of being able to make this $2000, along with 1.7% commission, into something.2174
She gets $2000; so #2 is done--we figured out the things that we need to talk about, her pay and the amount of the cars that are sold.2180
Those are the two variables that we are really looking at.2188
#3: We set up equations--we want to have some way of being able to talk about her pay.2190
$2000 + 1.7% of the cost of those cars that are sold...how does that work out?2195
You might say, "Well, I don't really remember how percent works," so let's test some things.2203
We know that 10% of 100 should end up being 10; we know that 5% of 100 should end up being 5.2207
We know that 1% of 100 should end up being 1; so, we might say, "Oh, right, you move the decimal 2 over, and then you multiply."2216
0.017; let's check and make sure that that makes sense.2224
If she sold $100 car (a cheap car, but...) if she were to sell $100, 1.7% of 100 would be one dollar and 70 cents.2228
So, she should make $1.70: .017...let's check that out: 0.017 times 100 would end up being...2240
we move the decimal 2 over...we get 1.7, or $1.70, which would be a dollar and 70 cents.2249
That makes sense, so .017 times the cost of the cars sold is equal to the amount that she gets paid in any given month.2258
Now, what month are we looking at specifically? March.2267
We know that p, in this case, is equal to $5,315.2269
So, we take this information; we plug it in here; we have 2000 + 0.017 times the cost of the cars = 5315.2275
From here, it is just step #4: we set up our equations; we know how the things interact.2286
We know all this; now we just have an equation to solve.2294
We want to know c; c is what we are looking for; it asks what was the total cost.2297
We are looking for c = ?, so we just solve for what c has to be.2302
Subtract 2000 from both sides; we get 0.017c = 3315.2308
And then, we divide both sides by .017; so we get c = 3315/0.017.2318
We plug that into a calculator to make it easy for us, and we get 195,000 dollars.2334
So, she makes $195,000 in terms of what she sells, which brings her a commission of $3315, so a total pay of $5315.2340
The total cost of all the cars she sold was $195,000; great.2350
Example 2: the first thing we want to do is understand what is going on.2355
The very first thing...we have a semicircle of radius 8 inscribed inside of a rectangle.2360
So, we might say, "What is a semicircle?" Well, we look at the picture--a semicircle is half of a circle.2364
Semi- means half; half of a circle makes sense.2370
So, it is radius 8; from center to edge is 8; great--it is inside of a rectangle, so it is inscribed.2373
We see exactly what it looks like; what is the area of the shaded portion?2380
We want to figure out how much is the shade in here.2384
We understand what we are looking for: we have half a circle held inside of a rectangle, so it barely touches the edges;2387
and we are looking to figure out what is the part that isn't the circle, but is still contained in the rectangle--what is that shaded portion.2393
So, #1 is done; #2--what would allow us to know this?2399
Well, if we knew what the area of the rectangle was (area of rectangle); if we know what the s for the shaded area;2403
and if we knew the circle's area; we would be in pretty good shape to be able to figure this out.2421
So, in #2, those are the three things we are looking for.2429
R equals the area of the rectangle; S equals the shaded area; C equals the circle's area.2432
#3: we start looking for some equations--can we figure out what the area of a rectangle is?2439
We say, "Oh, yes, it is length times width"; so R = length times width.2445
So now, let's go back, and let's look at our picture.2451
Well, if this is 8 from center to edge, look, over here is the edge of the circle; so this is 8;2453
over here is the edge of the circle, so this is 8; so the entire length is 16.2460
Vertically, you have...we go from here, and we go directly up; then this is 8 here, so it must be 8 on this side, as well.2469
So, we have length times width; we know that it is 16 times 8, which comes out to be 128.2478
So now, we know what the area of the rectangle is; great.2487
What about the area of the shaded area--do we know what that is?2490
No, that is what we are looking for--it is what the problem asked, so that is our question mark--it is our unknown.2494
C: can we figure out what is the semicircle's (let me write this out...it is the semicircle, not the circle,2498
because it is half of a circle)...C equals the semicircle's area.2504
Now, we say, "Well, what is the area for a circle?"2508
If I knew the area for a circle or a semicircle, I would be good.2512
Well, we probably remember that the area for a circle is πr^{2}.2515
And even if we don't remember what the area for a circle is, we say, "I have learned this before; let's type it into an Internet search!"2518
You type in "area of circle"; you have it right there.2526
Or if you have a math book, you can possibly look in the cover, and it will already have that formula right there.2530
The area of a circle is πr^{2}; so the circle equals πr^{2}; the area for a circle is equal to πr^{2}.2535
Now, notice: this is r, not this capital R; the area of our rectangle is very different than r.2547
What is little r here? We look once again, and try to remember--what is it?2553
Oh, right, it is the radius of the circle, from the center of the circle out.2556
The radius of our circle is 8, from the center of the circle out.2560
Now, there is one difference between the area of a circle and the area of our semicircle.2565
What is a semicircle? It is half of a circle, so it is 1/2 times πr^{2}, so 1/2 times π times 8^{2}.2569
That equals 1/2 times π times 8^{2}...8^{2} is 64, so that gets us 32π.2582
Now, that is 2 pieces of information: we know that r equals 128, and we know c equals 32π.2589
But we are still looking for this other thing: we have three unknowns to start with, R, S, and C.2598
And we have two pieces of information; so we need some way to connect S to R and C.2602
Well, we look at this, and we think, "Oh, the area of the rectangle, the shaded area, and the semicircle--they are all connected!"2606
The area of the rectangle contains both of the other ones, put together, so it is S + C.2612
So, at this point, we are ready to go on to step 4: we have all of our equations set up.2619
Step 4: R = S + C--we have numbers, so let's get S on its own, because that is what we are looking to solve.2623
So, R - C = S; the area of the shaded area is equal to the rectangle's area, minus the area of the semicircle.2629
So, we plug in the area for the rectangle; that is 128, minus the area for the semicircle--that is 32π; and that is equal our shaded area.2639
And that is our answer, because there is no way to combine 32π and 128--they speak different languages.2649
If we want, we could plug it into a calculator and get 32 times 3.14, and then get something.2654
But 128 - 32π...that is a great answer; there we are.2659
The next example: Tobias has precisely 17 coins in his pocket; so the first thing we are doing is understanding what is going on.2662
#1: Tobias has some coins in his pocket--that makes sense; 17--cool.2669
Coins come in three types, so he has three different coin types: quarters, nickels, and pennies.2674
Now, if we didn't remember, we would say, "Oh, what is a quarter?" and we would look it up; a quarter is 25 cents.2678
What is a nickel? 5 cents. What is a penny? 1 cent.2683
He has a total of 2 dollars and 17 cents in coins; if he has 2 pennies, how many quarters does he have?2687
Let's see...everything here makes sense; the kid has some change in his pocket.2694
They come in three different types of coins that make up that change.2699
And he has a total amount of money, and now we want to find out what the specific number of quarters is.2703
So, what might be useful to know? First, we are certainly going to need to know what is the number of quarters.2708
That is our very first thing; so here in step 2, the first unknown that we definitely have to figure out is number of quarters.2714
Well, we will probably also want to know how many nickels and how many pennies he has,2720
because they are connected to the other pieces of information right here.2723
n = number of nickels; and finally, p = number of pennies.2726
Those little reminders serve us to understand what is going on as we work through.2737
Now, #3: How can we connect these ideas together?2740
Now, we notice...well, how can I talk about the number of coins in his pocket?2744
Well, I know what the number of quarters is; I know what the number of nickels is; I know what the number of pennies is;2749
do I literally know what they are?--no, but I have names for them.2752
So, we can talk about...well, how can we say how many coins he has, in terms of these variables?2756
Well, he is going to have his number of quarters, plus his number of nickels, plus his numbers of pennies.2760
That should describe the amount of coins--the number of coins in his pocket.2766
Let's check and make sure: let's do a really quick check.2770
What if he had 2 quarters, one nickel, and one penny?2772
Then that is 2 quarters for q, 1 nickel, and 1 penny (p); so 2 + 1 + 1 is 4 coins, and he would have 4 coins.2776
Great; that makes sense; in this case, we are told that he had 17 coins, so q + n + p = 17.2784
What else were we told? We were also told that he had $2.17.2792
Oh, so how can we figure out how much money he has?2794
Well, if he had two quarters, that would be 50 cents.2797
Let's make things easy, and let's, in fact, instead of talking in terms of dollars...we will talk about 217 cents.2800
We will think just in terms of cents: if he has just one quarter, how many cents does he have? 25.2808
If he has two quarters, he has 50 cents; it is 25 times the number of quarters, so 25 times q is how much money, how many cents, he has from quarters.2812
What about nickels? Well, 5 times the number of nickels--let's check and make sure.2822
3 nickels would be 15 cents; 5 times 3 would be 15; it works out.2826
So, 5 times nickels, plus...what if we had pennies? Well, what number would we multiply by pennies?2830
Oh, pennies would just be by 1; so it is just the number of pennies.2837
That is going to be the amount of money in his pocket; in this case, we know how many cents he has--he has 217 cents, $2.17.2841
2.17 dollars is the same thing as 217 cents; we are thinking about this in terms of cents.2850
That way, we just don't have decimal numbers.2855
Finally, do we have any other information?--because currently, we have three unknowns and three relationships.2857
Oh, yes, he has two pennies--so we know immediately that p = 2.2861
Three pieces of information; three unknowns; we are good to go.2868
So, #4: we start working things out--well, we are going to have to somehow use p = 2 in one of these two.2873
So, let's plug it into this one first: so q + n + 2 = 17; that means q + n = 17, or...2879
let's go with n; we will solve for n, because we want to work out q over here.2889
So, we will plug it in here, and we can keep working up through substitution.2892
n = 17 - q; great; we will take that information, and we plug it in over here.2896
So, 25q + 5...remember, we have to substitute with quantities...(17 - q) + p = 217.2903
Do we know what p is? Yes, we certainly know what p is; p is just 2.2917
So, 25q...we will distribute that 5...+ 5(17)...5 times 17 is...oops, I made a mistake here...2922
sorry about that; hopefully you caught that and were thinking, "What are you doing?"...2935
q + n...we subtracted 2 here; this is a good example of why we have to make sure we do the exact same thing on both sides.2940
15; 15; so this shouldn't be 17 - q over here; it should be 15 - q over here.2947
Great; so 25q + 5(15 - q)...we get 75 - 5q; and then, we will also subtract 2 right now, just to get rid of it.2954
So, we will get...equals 215; great, 25q + 75 - 5q.2964
Let's consolidate that: we will get 20q, and then let's also move the 75 over to the other side; -75, -75...2971
So, we get 20q =...215 - 75 is 140, so q is equal to 7.2979
The number of quarters he has in his pocket must be 7 quarters, because we were able to work things out from those original equations.2987
And if we wanted to figure out what the number of nickels he had was, we could just plug it right here, and we would get,2994
"Oh, he must also have 8 nickels"; so if n = 8 and p = 2, q = 7, we could do a really easy check.2998
Is 7(25) + 5(8) + 2 equal to 217; yes, it ends up being that that does check out.3007
If we multiply the number that the quarters should be worth and that and 7 + 8 + 2,3014
that equals 17, so everything checks out; our answer makes sense; q = 7; great.3019
Final example: we have a tank for holding water.3024
The first step we need to do: we need to understand what is going on.3027
The tank for holding water is shaped like a circular cone with its point towards the ground.3030
What does that look like? Oh, yes, OK...you draw something circular, and then it comes to a point, like a cone.3035
That makes sense; the tank is 10 feet tall, and has a diameter of 6 feet at the top.3045
Let's put that into this: it is 10 feet tall, and has a diameter that is 6 feet across.3049
What is the volume when the tank is full?3057
And then, we have a second part, which is "What is the volume when the water is only 5 feet deep?"3060
So, let's just start by breaking it here, and we will answer, "What is the volume in the tank when it is full?"3063
Our first thing makes sense: we have this cone full of water; we fill it up to the top; how much water is going to be in it?3068
The second part, second idea: we want to get what we need to know here.3076
It seems like it would be useful to talk about the height; well, we actually know what the height is.3081
Height h will equal the height of the cone; and let's say d equals...we were told the diameter, so let's say the diameter,3085
even though these are really just going to be values; we can talk about them as if they are like that.3094
And that seems like everything we need to know right now, so let's go and let's see...3099
do we have a good way to relate these two things together?3103
How can we relate the height and diameter of a cone to its volume?3106
When I think, "Well, how do I get the volume of a cone? I have learned this before--they told me in geometry..."3110
Right, it is this: we remember the formula--or maybe we don't remember the formula, like,3115
"Well, I know I have been told it..." if you have been told it, it is out there on the Internet, right?3120
Or it is in a math book; either crack open a math book, or do a quick web search, and you will be able to find it really quickly.3125
And you find out that the volume of a cone is equal to 1/3 times what the volume of its cylinder would have been.3132
The volume of its cylinder would have been the area of the top, πr^{2}, the circle, times the height of the cylinder.3140
OK, now at this point, we think, "Wait a second; we are talking about volume; we want to know what volume is."3149
So, volume = ?; that is what we are really searching for.3155
But did we have r show up before? No, we didn't have r show up before, so let's make a new one: r = radius.3157
We were told some things here: we were told the diameter is 6 feet, so how does a radius connect to that?3165
Well, we say, "Oh, right, the diameter is just double the radius; the radius is half the diameter; so r equals 3."3171
Halfway would just be...if the whole thing is 6, then halfway is going to be 3; so r = 3.3179
Our h equals...our height was 10 feet tall; so h = 10; now we have only one unknown left in this equation here.3188
V = 1/3 times πr^{2} times height: we know what the radius is; we know what the height is;3198
π is just some number; 1/3 is just some number; the only thing we are looking for is the volume--really easy.3203
We just plug our numbers in at this point; so, the fourth step is just solving it.3207
Our volume is equal to...plugging in the values...1/3 times π times 3^{2} times 10.3212
That is going to end up being...the 1/3 here will cancel the squaring that we have here.3220
So, we have π times 3 times 10, or 30π; and we have to talk about it in terms of units.3226
So, if it is volume, and we have been doing feet before, it must be cubic feet.3232
And there is our answer; all right.3237
Now, what if we wanted to this other portion of it?3240
We will change over to a new version; now, it is nice, because it is going to follow a lot of parallel ideas.3244
So, we can just use what we have already figured out.3251
Instead of being 10 feet tall, it is only 5 feet tall; so, the water really only comes up to here.3254
Now, if that is the case, if the diameter is 6 up here, is the diameter going to be the same down here?3261
No, that makes no sense: the diameter can't be the same, because it is a cone.3269
It shrinks down, the farther down we get.3274
So, if we were all the way at the bottom, the diameter would be 0; if we go all the way up to the top, it would be 10.3277
It makes sense that the diameter is going to be half of what it would have been before, because we are now at half the height.3280
So, the diameter in the middle is 3; so that means our radius is equal to 1.5; our height is 5; and that same formula from before,3286
our volume formula for a cylindrical cone, still works; so volume equals 1/3 times πr^{2} times height.3296
We plug in 1/3 times π times (1.5)^{2} times 5.3306
We work that out: 1/3 times π times 11.25 simplifies to 3.75π; and it is in cubic feet.3316
And there is our answer: we are basically following the same outline we did before.3335
So, we don't have to worry about doing it step-by-step, because we can just work from our previous idea.3341
We figured out how it was done in the complex way; now it is just a matter of using new values3346
and making sure that the values we are using are right.3350
r changes, because we are at a different place; height changes because we are at a different place.3353
But all of the relationships are still the same, which makes sense--we want to be thinking about it,3357
but it makes sense that all of the relationships are the same, because we are still just looking3360
to figure out what is the volume inside of this cone.3364
It is just now a cone inside of a cone.3367
All right, I hope that made sense; I hope that gives you a slightly better understanding of how to approach word problems.3371
Don't be that scared by them; it is just a matter of breaking it down and understanding what is going on.3376
setting up what you are looking for, figuring out the relationships that connect3380
what you are looking for to what you know, and then finally just solving it like a normal math problem.3384
All right, we will see you at Educator.com later--goodbye!3389
1 answer
Last reply by: Professor Selhorst-Jones
Tue Nov 4, 2014 11:18 AM
Post by Magesh Prasanna on October 27, 2014
unable to watch full lecture.
1 answer
Last reply by: Professor Selhorst-Jones
Sat Jul 5, 2014 4:01 PM
Post by Thuy Nguyen on July 5, 2014
Hi Vincent, I think explaining similar triangles for the water tank problem would have been valuable in showing us how to figure out the radius in other situations when the height of the smaller cone isn't conveniently 1/2 of the bigger cone.
You are so right about the importance of learning word problems. In my engineering class we're not asked to find x, but we're asked to set up the solution to a word problem. I would recommend the book, "How to Solve Algebra Word Problems" by Mcgraw Hill, it changed my life by helping me see how to translate word problems into math equations, and from then on I was hooked on solving word problems.
2 answers
Last reply by: Taylor Wright
Sun Jul 21, 2013 10:43 PM
Post by Najmeh Haidari on April 29, 2013
Is it just my laptop are we actually not allowed to skip some of the video to the place we need.
Btw do you talk about Loci's? If so, what topic is it under?
thanks