Math Matters: Why Do I Need To Know This?
Bruce Kessler, Department of Mathematics
Western Kentucky University
Episode Twelve
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More on credit-card debt – The actual price of financed items
Objective: To illustrate the relevance of symbolic logic by showing some common
forms of “bad” logic in their symbolic representations.
Hello and welcome to “Math Matters: Why Do I Need To Know This?” It’s a show where
we take the entry-level mathematics that we teach in our general ed. courses and our teacher
ed. courses, and try to show you how it’s applicable to every day life. I’ve got a few neat
examples for you today.
I’m going to start out by kind of harping on an old topic and that is credit card debt.
I know I’m kind of on my soap box about this, but I can’t tell you how important it is for
young people to spend their money wisely, and not run up this huge credit card debt. And
in the effort of convincing you of this, I’m going to show you an example today. I want to
talk about what something actually costs. You may buy something on sale, but you put it
on a credit card where you’re carrying a balance, and what ends up happening there is that
you end up paying a lot more for the item than you thought you were paying. I just want
to take you through the mathematics of that, and show you how much your actually paying
on something. The bottom line is, a good deal is not going to be a good deal if you end up
paying more for it because of that credit card balance.
We’ve explained the credit card idea before, but you know I’ve been saying before lets go
through the ground rules of using credit cards. As long as you pay off your balance at the
end of the pay period, you’re not going to pay any interest; it’s almost like having an interest
free loan for that period of time. However, there’s a large segment of our population who
does carry a balance on credit cards, and when that happens, you will have to pay a finance
charge. That is, you take the monthly interest rate, that’s the annual divided by 12, and
you multiply that by the average daily balance. And what they do is they take the balance
at the end of each day in the pay period, average them up, and that is equivalent to paying
interest per day on what you owe. At the end of that pay period, you will have what’s called
a minimum payment. Now typically, that’s 4% of the balance or ten dollars, which ever is
bigger. There is another option: if you owe less then ten dollars, you’ll just pay off the
balance. You won’t have to pay ten dollars over what you owe, but you will have to make this
minimum payment, and that’s more then the finance charge. That’s important, too, because
otherwise, you’d never pay this card off. (Figure 1)
Now, I’m going to make some little assumptions here for what I’m going to do with you.
I’m going to assume our purchase was part of a larger balance, and the reason I’m doing
that is so I can use the big balance kind of rules, and it will take me longer to get to the
point where I’m just paying like ten dollars each month. I’m not going to get into this daily
average balance thing on this example, I’m just going to simply assume that we make the
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Figure 1, Segment 1
purchase early in the pay period and so we have to pay a full months interest on it and that
simplifies things greatly. And I’m also going to assume that I’m not adding to the balance,
that I’ve stopped using the card. If I make additional purchases, that lengthens the amount
of time it takes me to pay it off, and therefore the amount my item actually cost would be
greater then what I’m calculating. So I’m giving you the best case scenario here. When you
see the answer you say that’s the best that I could have done. (Figure 2)
Alright, here’s the example. Let’s say that you pay $100 for a pair of boots – it’s been a
while since I bought a pair of boots – I assume that’s a reasonable price for a decent pair of
boots. You got them 25% off and they were just darling or they just rock dude, depending
on your perspective there. Let’s say you’ve got a credit card that you pay for the boots with,
you pay $100 and your credit card has a 15% APR, that’s the annual percentage rate, and
after that purchase you decide to make the minimum payment each month. Well, a few
of the things we’ve got to get out of the way. We’ve got to calculate the monthly interest
rate, take the annual interest rate divide by twelve, and get the finance charge in the nth
month. Now, what I’m going to do instead of doing a continuous model, I’m just going to
say, “Okay, take the previous month’s balance, that’s what I mean by “Bn ,“ the balance of
the previous month, and then multiply by this monthly rate. The product of that will be the
finance charge. So, I’ll have to add that in at the end of each month. And then I have to
calculate the minimum payment. Now it’s typically going to be this, it’s going to be whatever
I get, here’s the previous balance I add the finance charge to it and then I pay 4% of that.
I’ll say $0.50 is bigger then this. Now the reason I’m saying $0.50 instead of $10 is that I’m
assuming this is part of a larger balance maybe its like five percent of your balance. So the
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Figure 2, Segment 1
part of that $10 payment you make, you know, I’m assuming this is like 5% of that $10, and
that’s just an assumption. And if the balance is less than $0.50, we’ll just pay it off. That’s
the last option. (Figure 3)
Okay, so here’s the balance at the end of the nth month. You take the previous one, add
in the finance charge, and subtract the minimum payment. And I’ve plotted for you here the
balances over a period of time. These are months down here. It actually take 99 months
to pay off that pair of boots, that $100 pair of boots, and little quick calculation here, divide
by 12, will tell you that that’s 8 14 years. That’s a long time to be paying on boots, you’ve
probably worn them out by then. And what I’ve done here, I’ve actually gone and calculated,
I’ve gone and added all these finance charges up, and you end up paying$41.13 in finance
charges on that $100 pair of boots. So you end up paying a $141.13 for them. What was the
original price? Well, I got them 25% off, so 75% of the full amount was $100. Divide by
0.75, they started out at $133.33, so I’ve actually ended up paying more for the boots then I
would have paid at full price if I had waited till they went off sale. (Figure 4)
Now I do want to close with one comment about this. This business of paying four
percent of your balance as the minimum payment is new. The federal government, realizing
that many people were in a lot of debt, said that with the minimum payment being 2%, which
it used to be, it’s taking people forever to pay off their cards, so they’ve been prodding the
credit-card companies to come through to raise their minimum balances up to four percent.
If this had happened back when it was two percent, I just want to show you the math really
quick, I won’t do the formula with you again but I will show you the table which is generated,
okay, the plot of the balances. You see it lasted way longer then 99 months. Its out here
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Figure 3, Segment 1
Figure 4, Segment 1
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at about 230 months. That is getting close to 20 years. Twenty years to pay on a pair of
boots is ridiculous. Those boots are out of style badly – they may be back in style by then.
And if you add up the finance charges you’ve paid $129.28 in finance charges. That means
you’ve paid $229.28 for that pair of boots. Unbelievable. So be very careful with those kind
of purchases. (Figure 5)
Figure 5, Segment 1
I’ve got some summary pages here to recap what I’ve just talked about, and I’ll get set up
for the next segment.
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Summary page 1, Segment 1
Summary page 2, Segment 1
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Matrix multiplication – Resource management
Objective: To illustrate the relevance of matrices and matrix operations in realworld applications. We introduce the concept of a matrix and explain the operations of addition, subtraction, scalar multiplication, and multiplication.
The next thing I’d like to talk to you about is how matrices, how a matrix is used in our
everyday life, and how we could use that mathematics do some nice kind of everyday things.
This is a topic that occurs at the end of our college algebra course and we always catch some
of the “Oh, I’m never going to use this.” Well, I’m going to show you a nice example today
of where you could use it, you may not, but you could use it if you needed to.
I’ve got to do some background on this, “What is a matrix?” It sounds like a cheesy
science fiction movie “What is the matrix?” but the math stuff ’s not near as exciting as that
movie. I’m just talking about a rectangular set of values, that’s it. So anything that appears
like that, you know computer scientist call it an array, but if we can take a set of values and
put them into a rectangle, I call that a matrix. Now I’ll give you an example of one, kind of
how they look. Let’s flash this up. This is what I would consider a matrix. This particular
matrix has 3 rows, it has 4 columns. So we call this, we always do rows first, a 3 by 4 matrix.
That gives me reference to the number of rows and the number of columns and that mix of
numbers will become important as we start to manipulate things. (Figure 1)
Figure 1, Segment 2
We can add and subtract matrices in a very obvious way. The obvious way is we just take
things in the same location and add them up. So the 0 + 5, the 1 + 0, and so on. In this case
we started with a 2 by 3 matrix, we have to add a 2 by 3 matrix, or else it’s not defined, and
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we end up with a 2 by 3 matrix. Subtraction is exactly the same. If I subtract, we subtract
things in the like positions. So 0 − 5, 1 − 0, in this position 2 − 2, and the just carry out that
arithmetic as well. So adding and subtracting are not going to change the dimensions of the
matrix. I started with 2 by 3, I end with 2 by 3, okay? (Figure 2)
Figure 2, Segment 2
We can multiply a real number, or what some call scalar, times a matrix in a real obvious
way. If I say something like 3 times this matrix, well just go through each and every term
and multiply by 3. I mean, that’s what I’d do if nobody told me what to do, I’d say, well, this
makes sense. And just do the arithmetic and you end up with the same dimension matrix
when you’re done. (Figure 3)
Now the real operation I want to get to though is multiplying a matrix with another matrix.
And this is very important, the dimensions have to be correct. In this case I’ve started with a
3 by 2 matrix (k by m matrix,) I’m going to multiply this by a 2 by 3 matrix (m by n matrix).
Now what has to happen is this number (2, or m) and this number (2, or m) have to match
up, they have to be the same number or else you cannot do matrix multiplication with these
two matrices. What you will end up with is this (3, or k) by this (3, or n), it will be in this
case 3 by 3. So here’s the way you do it, I’m going to flash it up here, don’t lose your mind
cause this looks messy, but I’m going to walk you through it, okay? This is the product of
the two matrices, and here’s how I get the entries. To get this first entry, I go up to the first
row in the first matrix and the first column in the second matrix and I say “Okay, 3 times
1 plus 1 times 2,” and that gives me that first entry. (Figure 4) To get the second entry in
the first row second column, I get the first row in the first matrix and the second column in
the second matrix, 3 times 0 plus 1 times 1, and that gives you that entry. (Figure 5) If I
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Figure 3, Segment 2
go to the second row first column in the new matrix, use the second row of the first matrix
and first column of the second matrix, and I just pull that little stunt, − 2 times 1 plus − 2
times 2, that gives me that entry. (Figure 6) And you just do all that, do the arithmetic and
it gives you that answer. Now it’s not a hard trick, it could be a messy trick, but it’s not a
hard trick. (Figure 7)
Now the question is “What is that telling me?” I mean, dandy, I can do it, I can learn
to do it, but how is it useful to me? Well, let me give you an example. Let’s say that you’re
a restaurant manager, you’re a chef, you’re something like this and you mass produce food.
So you’ve got, you actually learned exactly how much of these resources you need for several
of the items on your menu: for white cake you need 2 thirds of a stick of butter, 4 cups flour,
4 eggs and 2 cups milk. And the eggs you probably don’t use the yolks cause its white cake
– see I know cooking. Coffee cake you have this mix of things, sugar cookies you have this
mix of things, bread you have this mix of things. Now what I can do is, this is a matrix in
a way, it’s just stated in table form. What I can do is peel all this other stuff off, and just
keep the numbers in that rectangular form, which is what I’ve done. I can go back and show
you, but those numbers match the ones I had earlier. (Figure 8 and 9)
Alright, now, let’s say that you go through and you know how much each of these items
cost. For example, butter is $0.50 a stick, flour is $0.30 a cup, and so on. Well, you can
put this information into a 4 by 1 matrix. Now that would have 4 rows and just 1 column.
So here’s the $0.50, here’s the $0.30 for the flour, you have to be consistent, you have to do
things in the same order and you have to use the same units of measure, which I am doing.
Now, what do I get if I look at A (the first matrix) times C (the second matrix)? (Figure 10)
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Figure 4, Segment 2
Figure 5, Segment 2
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Figure 6, Segment 2
Figure 7, Segment 2
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Figure 8, Segment 2
Figure 9, Segment 2
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Figure 10, Segment 2
Okay, and this is going to be matrix multiplication. This matrix – it’s just a one column
matrix – well, if I multiply these things, let me write it out, I would say 32 time this 0.50,
4 times this 0.30, 4 times that 0.12, and 2 times that 0.15. Add all that up to get the first
entry. Now what this is telling me is the cost of the items for that first thing, what was
it, white cake. This is how much the produce will cost me for a white cake. And then I go
through and do this again for the coffee cake, and so on. And I’ll spare you the gory details,
you just get these prices then. That’s the amount of goods that go into making these different
items, and you’ve got that instantly. And the cool thing is that, . . . that’s white cake, coffee
cake, sugar cookies and bread, cause I forgot what order they came in. (Figure 11)
Now the cool thing about this is, if there’s a change in price, okay, this is where it’s really
useful, all we have to do is change that C matrix. So, if the, what was that, butter? If that
goes up, you just change that one matrix and just multiply them again and all of these totals
change instantly. So this gives us a very slick way to recalculate the costs of things without
having to go through and do a bunch of tedious calculations. And maybe that seemed tedious
to you, but what I would advocate doing is, look these graphing calculators that we’re all
using can do matrix multiplication, plug them in there, and let it pop out the answer for you.
(Figure 12)
Again, I’m going to summarize for you what I just talked about in a few slides and I’ll
take a small break and get set up for the next segment.
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Figure 11, Segment 2
Figure 12, Segment 2
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Summary page 1, Segment 2
Summary page 2, Segment 2
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Summary page 3, Segment 2
Summary page 4, Segment 2
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Systems of three linear equations – Three equations,
three unknowns
Objective: To illustrate the importance of being able to solve linear systems
of equations by showing its application to a real-world situation. We will discuss different methods for solving systems of three linear equations with three
unknowns that generalize to larger linear systems.
The last example goes back to something we talked about last week. Last week, I looked at
linear equations. Those are where you have an x and a y, whatever, and they’re not raised
to any powers, they’re just to the first power. Those are called linear equations, and I looked
at situations where you had two linear equations and two unknowns and talked about how to
solve that system of equations to find an x and a y that satisfied both of those equations at
the same time.
Now the reality of life is, if you’re dealing with things like that, then you’re not limited to
two variables and you have to know how to handle more general situations. So I’m going to
take you through an example of that today I’m just working with three by three, but everything
I’m talking about can be generalized to 4 by 4 systems, 5 by 5 systems and so forth. But
before I’m done I’m actually going to whip back in the matrix idea to show you that that’s a
way to solve these systems.
Let me give you an example and I’m stretching here to give you examples that don’t, you
don’t have to be a high-powered college grad to use. Let’s say you inherited some money. And
you thought it was wise, it was a pretty big lump sum, so you thought it was wise to have
someone kind of manage that for you. And I hope you never find yourself in this position,
but your financial guy has left town with the records and you’re now wondering where your
money is. You do know this, your’re trying to go back and find out what happened. So you
do know, you have limited information, you do know that you started with $30,000. You
know that it went into three accounts, a 3% and 4% account, two accounts there, and the
amount that went into those was proportional to the rate, so the 4 percent had a little bit
more than the 3 percent. And then the rest of it went into a high risk account, so you’re
getting a higher interest rate but there’s a chance you might lose it. And that’s the one that
wasn’t at a bank so you’re kind of worried about that one.
So you can recoup from your tax forms that you earned in that first year, that you earned
$2,656 in interest. So the question is, how much was invested with each rate, how much can
I track to a bank, and how much is kind of floating around out there somewhere, probably a
down-payment on a condo for my financial advisor.
So again I’ve got three amounts, I’ll call them x, y, and z. I’ll do them in increasing
order so we can keep them straight. And I actually have three equations. I know that the
total of these three amounts has to be 30,000, okay, that was the total inheritance. And then
the proportion business, the proportion is just an equality of ratios. The amount in the first
account to the amount in the second account is 3 to 4; it’s proportional to the rates. so you
do the cross multiply trick: 4x then equals 3y and I’ll move everything to one side because
I want to have this x, y, z kind of structure to my system just for my own convenience. So
there’s my second equation based on that proportion. And then I have some information
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about the interest: 3% of the first amount plus 4% of the second amount, plus 12% of the
third amount I know that that adds up to 2,656. So there is my last equation and I just went
through and multiplied everything by a hundred to clear it of those decimals, clear it of those
fractions. So I do end up with some kind of large numbers here, but they are workable, we’re
okay. (Figure 1)
Figure 1, Segment 3
Let me tell you what’s going on with this, give you a little background. A linear equation
in three variables represents a plane in 3-space, and we live in 3-space, so a plane is just
a flat surface. So in order to have a solution to these three equations I have to have three
planes that all intersect at one point. And that happens here, I’m going to illustrate it for
you.
The first plane, if I graph it looks like this, and it actually kind of leaves my viewing
rectangle here at the bottom. (Figure 2) And the second equation, the plane for that, I’m just
graphing these things, actually graphing these with a piece of software called Mathematica.
These two planes intersect and the intersection of these planes is a line where the two planes
meet, okay? (Figure 3) And now, for this to have a solution the other plane needs to come
through here and cut through that line, and I do believe that happens, here’s the last one, I
graphed it. Yep, it comes through, cuts through that line, I actually tried to draw a big black
dot at the position where they all crossed. (Figure 4)
So this will have a solution. The problem is that I can’t get it very well from the graph;
it’s hard to tell exactly what it is. It’s a 2-D representation of a 3-D graph, which is kind
of ridiculous. So what I really need here is some way to solve it exactly. I’m going to give
you two methods, one is fairly elaborate on paper, the other is one that we can do with our
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Figure 2, Segment 3
Figure 3, Segment 3
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Figure 4, Segment 3
graphing calculator and a little bit of knowledge of matrices.
Okay, let’s do the on paper one first. Its we call the substitution method; we did this with
the 2 by 2 systems last week, where you go through, you solve for one variable, you plug it
into the others. Then, and let me show you what it looks like, and this particular thing, let
me flash it up here. There’s my system, I’m going to solve for one variable in one equation,
and I’m going to start with this top equation and solve for x. That was just my choice, okay?
So I just move all this other stuff to the other side, and then I’ll take this and just substitute
it into the two other equations. Okay, which kind of looks like this, I’ve cleaned it up here,
and I’ve spared you some of the gory details. And then I plugged it into the other and I
really spared you the gory details here, because it’s a little messy, but you can clean that up,
it’s just a little bit of algebra. And now what I have is a system of two equations with two
unknowns. So I’ve reduced the size of the system by one. (Figure 5)
And then just do it again. I’m going to choose to solve for y in this equation. Actually,
I’m avoiding fractions is what I’m doing, I’m dodging places that have a coefficient of one.
So I solve for y here and then I’ll jam this in this equation here. (Figure 5) So that looks
like that, cleaned it up for you. And then you solve for the last equation, it’s now a 1 by 1
system, which I know how to solve. So z turns out to be 18,800, that’s the amount, if I’ve
done things correctly, that’s the amount that’s in that shady account out there somewhere.
To get the others just back substitute. What I mean by that is we had a nice formula for y
in terms of z, plug z into it and solve for y. So y, the 4% account had $6,400 dollars , and
then take both of those and plug it in for x, and that works out to be $4,800. So that tells
me the amounts invested in each one. And that’s the actual solution. (Figure 6)
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Figure 5, Segment 3
Figure 6, Segment 3
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Now very quickly, and I say quickly cause I don’t want to get into all the gory details,
but I do want to show you how to do this with matrices. Some square matrices will have an
inverse which you can calculate with your calculator, and I use a superscript negative 1, for
that acts like a 1. If you multiply these two together it gives you a matrix with all 1’s down
the middle and when you multiply by that its like a matrix 1. (Figure 7)
Figure 7, Segment 3
So what you do is you take the system, you transfer that into a matrix equation. Now
what I’ve done here I’ve just taken my coefficients and just moved them down and this times
that gives me the first equation and so on. and then you can calculate the inverse of this
matrix, you do it on the calculator, don’t do it by hand, we don’t know how to do that yet,
and then multiply the two, this is all calculator stuff, and it, too, will give you the answer.
(Figure 8)
Okay, that’s the quick way to do it, I’ll let you experiment with your calculator to get the
right answer, I’ve got some pages that kind of summarize what I just talked about, we’ll be
right back.
Closing
I hope you’ve enjoyed the examples I’ve shown you today. I really kind of searched for
some good kind of practical examples where you don’t have to be a specialist in some kind of
scientific area to use the math we’re talking about. If you have ideas, or you’ve seen some
math and you say, and you think to yourself “There’s no way I’m ever going to use that,”
please contact me. There’s an email address on the screen right now that you can email
me at. You can go to the webpage and contact me through the webpage and we’ll flash that
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Figure 8, Segment 3
Summary page 1, Segment 3
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Summary page 2, Segment 3
Summary page 3, Segment 3
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address up in just a few moments. I would love to hear from you, I need your good ideas and
I’m up to the challenge, I love it. Thanks for tuning in this week and with that I am done
and I’ll see you later.
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