Profit = Revenue - Cost (the American way and Chinese now too)

So we left off with the slope = $3.50 (which most math prep math students would have gotten by just reading the original problem, but this is a struggling student and we needed to review the slope formula anyways).

Now we have to find the "b" in the y = mx + b formula.

So we plug in the slope and we get y = 3.50x + b.

and now we plug in one of our points (0,90) or (1,93.50) to find b.

90 = 3.50(0) + b or 93.50 = 3.50(1) + b

either one should give you b = 90 !! (or what the book calls "fixed cost")

so finally we have y = 3.50x + 90 as our "Cost linear function" or C(x).

Now we need to find the Revenue function and we see that Rev = $9x.

y = 9x because we are selling the shirts at $9 each. Or if you must find the 2 points (1,9) and (2,18)...if you sell 2 shirts you collect $18 !!

slope = m = (18-9)/(2-1) = 9 and "b" of the y = mx + b is 0 because if you sell 0 shirts you collect 0 money. Or just plug in one of our points.

My student plugs in one of the points and get b = 0. Excellent !!

Now the book asks us to plot the two graphs on the same set of axes.

Are you ready to play "battleship" ? Many parents ask me how can their kids learn math. I always suggest Monopoly (adding the dice numbers real fast, counting money and learning a little bit about the real world...especially when trading properties).
Battleship is another excellent game. It teaches about the xy plane and about strategy and good old trial and error. Any card game is good too. I am sure this all translate into the computer world. Although the saddest day of my life for me was when they computerized the score keeping at the bowling alley. What are they going to do next take George Washington off the one dollar bill ? Or Lincoln off the penny ?

So basically we plot (0,90) and (1,93.50) for Cost on the paper and connect the dots.

We plot (1,9) and (2,18) for Revenue on the paper and connect those dots.

The break-even point is where these two graphs cross. Where Cost = Revenue !!

I think we have reach the climax of our tutoring session when the student stops and does not go any further. Oh my gush i think the brain is full. No, it is his cell phone vibrating and it is his mother calling for him. The hour is up and she is waiting for him in the parking lot. (we have spent an hour on one problem)

He excuses himself and runs out of the room.

He returns and says his mother is giving him one half hour more. He is excited and wants to find the Break-Even point and wants to know if Juan is going to be successful selling t-shirts because he has a friend who sells t-shirts with sayings on them. (My Honors Calculus class just had t-shirt made up that said "Nuke it" on it. Referring to the use of the graphing calculator for the midterm exam.)

So now he wants to know how to finish the problem.

So i say well we have 2 equations and 2 unknowns (we used to call these variables but that is too confusing for students, so they changed it to unknowns)

My student happily says he is going to use ADDITION to solve this problem. I suggest substitution. Oh yea, how do you do that ? As he takes the last gulp of his Pepsi and finishes his candy bar (the breakfast of champions for 90% of college students).

So i have to lead him through the process of Cost=Revenue...so 3.50x + 90 = 9x

I explain how it is like substitution in a ball game. The number 3 batter goes in for the number 3 starter. Or the point guard goes in for the starting point guard.

I am pleasantly surprise when the student does his arithmetic (with a calculator) and gets x = 16.3636363636.

Then i am totally blown away when he says Juan has to sell 17 shirt to break-even.

Mission accomplished !! and he is a good businessman. My students in high school would say...."Juan has to sell 16.363636363636 t-shirts to break even !!"

The life of the math teacher is never easy....LOL

But the problem in the book is not finish. Part "f" says we now have to find the Profit linear function.

And again my college student surprises me when he pronounces that Profit = Revenue - Cost !!
(I chime in the "American way"...and he adds "and the Chinese way too". I get a tear in my eye. This boy or man in his 20's knows so much, but so little math).

So i ask him how are we going to find this equation of Profit. And he draws a blank.

We are now back in Math class and no where to go.

I ask again what do we do with 2 equations and 2 unknowns and he says ADDITION !!
I suggest substitution.

So now we get P = 9x - 3.50x + 90

the classic biggest mistake in algebra...the negative sign

we do not have time to play around and see some wrong answers, so i just suggest that he use parenthesis in his work.

P = 9x - (3.50x + 90)

The book now asks for how many t-shirts must Juan sell to make a $500 profit.

again a complete shut down by my struggling math student.

I suggest we get a couple of points...and he figures out that selling 0 shirts will give us a profit of -90. and he proclaims, hey we lost 90 dollars. (I cherish moments like that...of course a high school student would say, you can not have negative money...i did something wrong).

His second point he says if i sell 1 shirt, we still lose money. (notice the use of the word "we", he has taken ownership of this problem of Juan's).

So now he proclaims...if we sell 17 shirts we make $500. Taking his best guess as the extra 1/2 hour is running out.

Remember 75% of the population can NOT do this problem. And if we were watching Jay Leno "street walk" or "street talk" or "street smarts"...you would say 99% of the population can NOT solve this problem including many CEO's of companies.

With time running out, i suggest putting the $500 in for the Profit.

and my student successfully finishes the problem

500 = 9x - (3.50x + 90)

x = 107.2727

and he proclaims as he gets all his stuff together and throws away his pepsi can and candy wrappers (I told you he was a good boy), I better have 108 friends if i want to make $500 selling t-shirts.

mission accomplished.

But the book now wants him to graph the profit function P(x). Oh well maybe next week.

I talk briefly with his Mom in the parking lot and she is so proud of her boy. She has to drive him to work now and he can not be late. And how do you think he will do in this math class at the college ?

I lie as usual. Some how the college and this student have gotten him this far. He will probably earn a C. He gets a bunch of points for at least trying to do his homework. They give him points for attendance and even one take home test that he should get 115 pts on. And if the students help the teacher with some surveys, they get 5 more points. A few teacher presents should seal the deal. And a lot of hard work staring at his homework and trying to take notes in class. Plus he says there is this nice Asian-Indian (thank God for the Asians) grad student who is helping him in the math lab every night. My tutoree is only taking one class per semester while he works during the day.

Extra Help

Today (Sat.) i was tutoring a student like i usually do on Sats for the past 33 years. It always amazes me what goes through these kids minds as i try to help them. This particular student is a college student. A very successful businessman and very much liked by everyone he meets. I would say he has a very HIGH social IQ. But

He just does not get math.

Today we were working on the equation of a line... y = mx + b.

If just seeing that gets your palms sweating, you are in good company.

Why does the education world think people need to know Algebra to succeed in this world. Percents, yes ! maybe some simple ideas of discounts, yes ! getting the correct change at the store, absolutely. But unless a student is going to college, why waste everyone's time and energy. It could be better spent on learning a trade or real life skills. Remember only 50% of the population TRIES to go to college. My generation it was 20% and my parents generation only 10%....

with 50% NOT making it through college...that leaves only 25% of the new 20 and 30 something population with a college degree. And many only learned how to drink and failed at that too...lol

So this college student in what they are calling Calculus but really is only Algebra 2 at best, is struggling with the words Cost, Revenue and Profit. They also throw in there marginal cost, fixed cost and breakeven point, and Supply and Demand to make the word problems sound impressive.

But like i tell all my students, no matter how many fancy words they are using, they are basically giving you 2 points !! and with those 2 points, you need to find the slope and where it crosses the y axis. Something most schools teach in 6 or 7th grade and reinforce all through algebra and geometry.

Today's problem was Juan Valdez (yes, most textbooks are published in California) is making t-shirts. It cost him $3.50 to buy the t-shirts and a fix amount of $90 to set up his shop. He sells the shirts for $9.00. What is his "break-even" point ?

Let me know when you get the answer. And yes, you all got A's and B's in algebra because they do not give out C's anymore.

(For a discussion on slope, check out my other blog on "Fun Math" and Thales.)

So the book ask first to find the Cost linear function. Right off the bat, my student does not know what "linear" means. The biggest problem with word problems in the public schools is that the students do not know how to read. Many in the 10th grade are at a 4th or 6th grade level reading wise.

Ok so we need to remember that Slope is Rise/Run (rise over run). How much did the line rise divided by how much the line ran. In this case with our boy Juan, the slope is $3.50. (of course the selling price of $9 a shirt gets in the way. We call this "white noise"...or too much information.)

Since the book asked for the cost equation, we do not need to worry about the selling price for a little while.

At this point when helping a student, they need a plan of attack that will work with ALL word problems of this type, not just cost and profit problems. (Sometimes this is a double edge sword. I have learned over the past 30 years of tutoring struggling students especially with limited tutoring time, that to "get through the test" sometimes you just have to memorize certain problems and hope the teacher just puts those certain problems on the test. In my classes I call them "classics" and yes, the t-shirt problem is a classic in Algebra 1 classes and sometimes in Prealgebra at the prep schools.)

So i teach my tutorees to put down 2 points. Always go back to the 2 points. And then memorize the slope formula (change in y over the change in x). Struggling students can memorize better than think. Of course the downside of this strategy is that they forget what they have learned when they memorize something else. But who needs algebra in the real world anyways.

Ok the first point is x=0 then y=90 (what we call the "fixed" cost....if you sell 0 t-shirts you will still have to shell out the $90 to set up the shop or operation.)

At this point my college student who has passed precalculus in college (probably more of review of algebra one and geometry) asks me:

y = C(x) = mx + b

"that extra C(x) gets in my way !!", "I am stuck, i do not know what to do ?"

So i have to simple say..."get rid of it"...do you know y = mx + b ?

Now we go onto our second point of the problem...what if you sell 1 shirt ?

The cost is $90 plus $3.50 for the shirt.

The student says, "so is the cost of the first shirt $93.50 ?" good question.

Well that would be one expensive shirt, but true...if you sell no more shirts, that first shirt would have cost Juan $93.50...the reason most small businesses (95% in fact) go out of business in the first year. I wish we could have a nice discussion about the real world, but our time together is limited. (Most businesses goal should be to breakeven by the third year...that is why you have to have DEEP pockets to make a small business work. Most people blow ALL their money on the first year and then lose it all.)

So again we need to get AWAY from the wording and back to the 2 points (0,90) and (1,93.50).

therefore the slope is (93.50-90)/(1-0) = 3.50. The book tries to call this marginal cost, but this only confuses struggling students. m = 3.50 they understand.

(My student says...."oh, M in y = mx + b stands for marginal cost"...and I have to think for a second...in 33 years no one has made that connection to me...i told you he was a successful businessman. (Ok, Dr. Mead is that correct ? Dr. Morris ? how about you Dr. Zerla my math history prof ? Or Mr. Blount my trusty sidekick who likes to look every thing up.)