Suppose you’re at the grocery store doing some shopping.
You’re hoping to pick up 300 pineapples with your friend Kathryn
who’s looking for 200 mangos.
Luckily, your other friend Melissa already has 75 kiwis at home.
But you can’t buy anything until you’ve figured out how much 103 pineapples,
56 and three-sixteenths mangos, and 27 kiwis cost.
Well this is how I always shop.
Today we’re going to talk about word problems,
math problems described by a short story or scenario
you have to translate back into math in order to solve.
Not everyone finds them terrible, but if you’ve had trouble with them in the past, you may
find that you look at one and just shut down.
So let’s talk about why that is and how to get past it.
G’Day!
I’m James Tanton, and this is Study Hall Algebra
presented by Arizona State University and Crash Course.
First, you might be confused because many word problems you find in textbooks sound
ridiculously artificial.
Why do we need 300 pineapples?
And why are we wondering how much they cost?
We’re at a store, where things have price tags!
None of this would ever happen in real life.
So they can feel hokey – or worse, like they were made up just to trick us.
And I don’t know about you, but if I feel like I’m being set up, I start feeling resentful
and suspicious,
and that can make it hard to think clearly.
But life does have problems.
And the more ambitious the problem, like in engineering or medicine or history,
the more likely it will contain numbers or patterns that relate to each other in complex ways.
We do use words to make sense of these problems and, ultimately, to solve them.
Now when solving problems, in algebra or in the rest of life,
there are two key steps that no one seems to talk about.
But that ends now.
I just need an example.
If I quickly consult my small collection of word problems...
“One gallon of 50% salt solution is mixed with three gallons of 10% salt solution.
What is the concentration of salt in the resulting four gallons?”
Hmm.
As you might be able to see, I’m a little flustered.
I’m going to acknowledge my emotional reaction, because how I’m feeling influences my problem-solving.
And that’s the first step: be human.
I’m feeling a little alarmed and unprepared.
I was talking quite confidently about word problems in general,
but now that I have a specific example, I’m a bundle of nerves.
And that’s ok.
Mathematics is a human experience, not just computations.
If you’re nervous about a problem, you might skip over important information when you read it.
If you’re excited or worried about something in another part of your life, you might feel
like algebra is totally irrelevant and find it hard to focus.
And If you’re feeling like you finally get the type of problem you’re working on,
you might feel confident enough to read ahead to the next topic so that you can keep that
good feeling going.
You don’t have to bring artificial enthusiasm – you just have to bring yourself,
however you’re feeling.
So acknowledge your reaction to the problem.
Take a deep breath.
And then step two is: do something!
Anything!
Read the question again.
Read the question backwards.
Maybe write down all the numbers you see in it.
Just do something, whether it’s related to the problem or not.
It sounds like we’re procrastinating, but we would waste a lot more time by just sitting
there doing nothing,
staring hopelessly at the problem.
Doing something helps overcome the emotional hurdle of getting started, hopefully leading
to some progress.
So I’ve acknowledged my reaction here: nervousness.
Time to do something.
I’m going to try drawing a picture to see if that can help me set up the problem.
Let’s go piece by piece and see what the problem gives us.
“One gallon of 50% salt solution.”
That sounds draw-able.
It doesn’t have to be fancy.
In fact, the simpler the better.
So this gallon is just a block of liquid, and I know that half, 50% of it, is salt.
That helps me see how I could start writing a math expression.
We need to think about the amount of salt.
Call it S. So far, S = 1*(0.5).
Half of that one gallon is salt.
Our first gallon is mixed with three more gallons of 10% salt.
So that’s three more blocks, and a tenth of each is salt.
Since we know we’re combining our liquids,
I can go ahead and divide the first gallon block into tenths to match the other gallon
blocks.
So that’s gonna add more salt to S, giving me the equation S = 1*(0.5) + 3*(0.1).
If we work that out, S = 0.8.
Huh.
Is that the answer?
0.8 seems like a lot of salt per gallon.
Almost too much.
Let’s reread the question and focus on what kind of answer we need to give.
“One gallon of 50% salt solution is mixed with three gallons of 10% salt solution.
What is the concentration of salt in the resulting four gallons?”
I already have four total gallons in this drawing, so I actually don’t need to draw
anything else.
I just need to figure out how much of the four gallons is salt.
So 0.8 gallons is the amount of the 4 gallons that’s salt.
Hah!
Now that seems more reasonable.
To match how the problem was written, we could translate to a percentage.
0.8/4 = .2.
20% salt seems like a logical amount for a 50% solution diluted with a lot of 10% solution.
And we can check our work with our picture.
I have five blocks of salt in the first gallon, one in the second, one in the third,
and one in the fourth.
That’s eight blocks of salt out of 40 total blocks.
And 8/40 is equal to 2/10, or 1/5, or 20%.
My equations are just confirming what my drawing is already telling me.
Let’s try another one.
The only way to get better at word problems is to practice.
“A farmer has 200 meters of fencing and needs to build a pen for their goat,
which keeps wandering off and eating their neighbor’s laundry.
One side of the rectangular pen will consist of the wall of their orchard,
which is 80 meters long.
The goat is 1 meter long from nose to tail.
What is the area of the finished pen?”
Is that a small goat?
How much laundry could it really eat?
I wish I had a pet goat.
What’s my emotional reaction?
Daydreaming.
My mind is wandering.
If I acknowledge that, I can redirect and focus.
I’m going to do something different than last time and start by making a list of the
numbers we have and the numbers we want.
The numbers we have are: 200 meters of fencing, 80 meters of wall,
and 1 meter of goat.
Do we need that last one?
Nope.
No one’s ever told me to find the length of a goat
before calculating the area of a rectangle.
Sometimes word problems come with extra information that won’t help us solve the problem.
And the numbers we want are: just one number, the area of the pen.
It’s a rectangular pen, so we just need length and width.
Our first list shows we can call the length 80 meters.
Now we need a width.
How do we find the width?
Well, we started with 200 meters of fencing, and we’ll use 80
for the side opposite the orchard wall.
So we have 200 minus 80, which leaves 120 for the other two sides.
120 divided by 2 is 60.
Let’s check that works.
60 + 60 + 80 = 200 meters of fencing.
Yes!
I feel triumphant.
But wait.
We want the area of the finished pen, not just the side lengths.
Sometimes solving a tricky bit in the problem can feel like we’ve solved the whole thing,
but we always have to check what the question was actually asking for.
80 meters times 60 meters is 480 square meters.
I solved this problem by gathering my information and asking myself questions
about what information I still needed.
Now everyone claims algebra shows up in real life, but sometimes it can be
quite hard to notice.
We can put on our Math Goggles and use our X, Y, and Z-ray vision to see better.
So you’re probably not ever going to buy 300 pineapples, but you might go to the movies
with some friends…
At the theater, two of your friends picked up sodas and popcorn while the rest of you
nabbed some good seats.
One friend got three popcorns and two sodas and paid $23.75.
Your other friend got two popcorns and one soda for $14.75.
On the way home, you’re trying to figure out how much you owe them for your soda.
But the movie theater didn’t itemize the receipts.
You know you all had the same sizes of sodas and popcorn, so it should be possible
to figure out what each item costs.
Equations with xs and ys seems like overkill, but maybe you can just visualize the situation.
Good thing you have your Math Goggles on.
So you start drawing on the back of the receipt...
Three popcorns and two sodas equals 23.75, while two popcorns and one soda equals 14.75.
If you had 5000 sodas and 3000 popcorns, you wouldn’t be able to draw a picture of each
one.
But because you have such small quantities, a quick sketch can help you notice patterns.
And you do!
In both equations you have one soda and two popcorns.
You know that’s equal to 14.75.
Subtract the first equation from the second, and you get a new equation.
One popcorn and one soda are equal to 9.
Adding that to your list of original equations helps you notice something else.
One soda and one popcorn show up in the second equation, so you can subtract your new equation
from that one and see one popcorn equals 5.75.
So close!
We could plug that into any of our equations, but the 3rd equation seems easiest.
Aha!
One soda equals $3.25.
You’ve just solved this algebra problem without any xs or ys at all.
Solving problems visually might seem like a weird thing to do in an algebra class.
But in fact, noticing patterns and experimenting with them is the essence of mathematics.
So when faced with a word problem -- whether in a textbook or on the street
-- what can you do?
Draw a picture to help you set up your equation.
Sometimes drawing a picture will solve your equation.
And if you’re working on a problem where it initially seems like you need to assign
lots and lots of variables,
see if you can describe everything based on just one variable.
For example, let’s say you have a problem involving Alice, Bennett, and Carl’s ages.
Instead of assigning a, b, and c, see if you can describe them in terms of a, a + 2, and
2a minus 7.
Then you just have one unknown to solve for instead of 3.
Next time we’ll look at other confusing areas of math people rarely explain,
including negative numbers and some ideas about why they behave the way they do.
Until then...
Cheers!
Thanks for watching Study Hall: Algebra,
which is produced by Arizona State University and the Crash Course team at Complexly.
If you liked this video and want to keep learning with us here in Study Hall, be sure to subscribe.
You can learn more about ASU and the videos produced by Crash Course
in the links in the description.
See you next time!