# Ratios

### Q:  The ratio of blue cars to non-blue cars in the parking lot is 2:7.  If there are 1260 cars in the parking lot, how many are blue?

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Solution by hand:

Number of blue cars = 2x

Number of non-blue cars = 7x

2x + 7x = 1260

9x=1260

x = 140

There are 2x blue cars, so there are 280 blue cars.

Solution using my TI-84 Plus Program RATIO:

Run RATIOHow?

Total number of items = 1260

Number of categories of items = 2  (blue and non-blue)

Ratio part for category 1 (blue) = 2

Ratio part for category 2 (non-blue) = 7

Distribution is: blue = 280 and non-blue = 980.

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# TI-84 Plus Programs

My TI-84 Plus programs are now available here.  Instructions and examples for each of the 40 programs are there so you can learn how and when to use them.  I’m in the process of cross-indexing them with the problems posted here.  Check back often and ask me questions!

You know that SAT book everyone has?  The blue one:  This one, or this one.  I’m about to start going through it page by page, pointing out the problems that can be solved using my programs.  Stay tuned!

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# Get a TI-84 Plus Calculator!!

## The BEST thing you can do to improve your SAT or ACT Math score is to get the TI-84 Plus calculator… and load it up with my programs.

It’s only \$108 today on Amazon, with free shipping! This is the best deal I’ve seen. By comparison, Walmart has it for \$119.27. Get it now: http://amzn.to/1vUA1IP

The TI-84 Plus is a great help to anyone taking the SAT or ACT. I have dozens of programs that will solve many typical problems in mere seconds. I can transfer these programs to your TI-84 Plus if you come to my office. BUT very soon they will be available for download on my website, with instructions and examples, so stay tuned!

By the way, there is no need to get any of the other fancier versions of this calculator. They won’t help you any more than this one does. But if you do want a certain color, well here are the other options:  http://amzn.to/1yi4bKd

(Yes, the TI-83 Plus is compatible with the programs, too, so that’s another option as listed in the above link, but the USB cable must be purchased separately.)

If all you need is the USB cable, here it is:

USB cable for  TI-84 Plus: http://amzn.to/1xAchur

USB cable for TI-83 Plus: http://amzn.to/1qAmCRx (Don’t buy the one above if you have the 83!!)

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# SAT – Probability

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### A: 1/10,000

Explanation:  Each digit is chosen from the integers 0 through 9, which is ten possible choices for each digit.  There are four digits to choose independently, so there are:

10 x 10 x 10 x 10 = 10,000 ways to choose the four digits.

Then, the one correct sequence of 4 digits, is one out of 10,000.  The probability is 1/10,000.

Another way to arrive at the number 10,000 is as follows:

The 4-digit sequence may be anything from 0000 to 9999.  So, how many is that?  9999 – 0000 +1 = 10,000.  Why the ‘plus one’ on the end?  If you just subtract the numbers, it tells you how many spaces there are between the numbers.  Add one to include either the first number or the last number, whichever one has not been already counted.

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# SAT Question Of The Day – estimating for hard geometry

### You’d be surprised how often you can really do this and get the right answer.  Follow me, and you’ll see lots of these!

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# Precalculus – Domain of a function

Q: What is the domain of this function?

‘Like’ if you got it!

Explanation:

The domain of a function is a description of the x-values that may be plugged into the function. For a square root function, the restriction is that the number under the radical must not be negative. Therefore,

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# So, how much change should I get?

I went to Costco today.  I almost never get out of Costco for less than \$100, but I wasn’t keeping a running total in my head.  It’s been one of those days, the kind where you just don’t even look at the prices (gasp, did I just admit that?!).   The cashier said the total was \$118.87, and that was fine.  I handed her \$120.00 and got to work packing my items in the cart.  The cashier was soon in total dismay when she discovered she had punched in ‘200.00’ instead of ‘120.00’.  Now, she’d have to get the manager!  The change amount was wrong and there was no telling how much it really should be!  And the line was so long already!  This was going to really slow her progress…  I piped up, ‘Just give me one dollar and 13 cents.’  She didn’t know whether to believe me.  She stared at the register display, which said ‘\$81.13’, and just couldn’t figure out what to do.  I continued, ‘You put in 80 dollars too much, so take 80 dollars back from the change amount.’  Her face said she believed I was telling the truth, but she was still puzzled.  I got the feeling she didn’t know how to take \$80 away from \$81.13, at least not on the spot, in her head.  She asked, ‘So how much is it?’  I repeated, ‘one dollar and 13 cents.’  She must have decided I was being honest, since I was asking for such a small amount.  She thanked me, presumably for saving her the call to the manager, and I was on my way with the correct change.

True story.  I wish it weren’t true.  I don’t blame her, though.  She’s a product of a broken system.  I wish our country put a much, much greater value on how to do basic math.  You don’t have to be one of the ‘smart ones’ to know how to do basic math.  Most everyone can master it, if it is made important.  My grandmother did computation by hand flawlessly, even at age 90.  Even though she once asked me where Mars was, and upon being told it was a planet out in space, asked whether we could drive there.  Growing up, my grandmother never had the educational opportunities children in our country have today.  She went to work after 7th grade.  She knew how to read, write, and do her arithmetic.  She did it well, and was proud of it.  Because it was made important.  It’s not that hard.

Well, it shouldn’t be that hard.  Actually, in today’s society it is hard.  It’s hard because we (well, not me, and hopefully not you either, but too many people) tell our little girls they are bad at math.  WHAT??  We (same ‘we’ as before) watch talk shows where our idolized celebrities tell us it’s cool to be ‘clueless when it comes to numbers’.  We hear other celebrities admit with embarrassment that they are ‘math nerds’.  We give tons and tons of money to sports ‘heroes’ while our teachers and researchers can barely scrape together enough food to eat.

And so, one day at a time, one student at a time, I try to change this collective mentality.  I tell my students and their parents that I’m in the business of putting myself out of business.  Please, go ahead and get your math sorted out perfectly.  Understand it so well that you won’t need me anymore.  I’ll be glad to get a different job!

I’m only one person, though I’m thankful to know so very many others who change the world daily, much more than I.  So, how about you hop on this train, too?  You don’t have to be a teacher or an accountant or a scientist to be on board.

Here’s something easy you can do in a few minutes, that just might make a difference.  The next time you see your 9-year-old niece, ask her to count by threes for a while.  Then write down the numbers in a list.  For the two digit numbers, ask her to add the digits together, and add the digits of the resulting total.  Get her to notice that the digits always eventually add up to 3, 6, or 9!  Try it with a three digit number, too:

12:   1 + 2 = 3

15:   1 + 5 = 6

18:   1 + 8 = 9

66:   6 + 6 = 12;   1 + 2 =3

819:   8 + 1 + 9 = 18;   1+8 = 9

Then point out that when the digits of a number add up to a number that is divisible by 3, then the original number is also divisible by 3.  (Use the word ‘multiple’ instead if it works better.)  Wait for it… ‘COOL!’  Now, ask her if the number 56,418 is divisible by 3.  She’ll be so proud to know the answer.  That’s how it works.  And welcome aboard!

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# Fun stuff!

Take a look at this set of beautiful animations that illustrate / explain common mathematical concepts much better than words ever could:

http://www.iflscience.com/brain/math-gifs-will-help-you-understand-these-concepts-better-your-teacher-ever-did

I especially love the water-pouring one with the squares on the sides of the right triangle.  This illustrates the Pythagorean Theorem (a² + b² = c²) in such a beautiful way.  You may have seen this diagram before – the three sides of a right triangle drawn into squares.  BUT the animation of pouring water from the two smaller squares into the larger square, filling it perfectly, is among the purest illustrations that math is truly an art.

What else?  The one about the exterior angles of a convex polygon always adding up to 360° is so simple and so perfect.  The algebraic proof of this is not too hard to understand, but it’s a lot of writing.  In a few seconds, you can see a great illustration of what happens when the polygon’s sides shrink down to nothing.  What is left is a bunch of angles that clearly total 360°.

I’ve chosen the two above examples to talk about mainly because they would be of the most illustrative use to my own students.  Still, I love each and every one of the lovely little animations there!

Let me know what you think!  Did something suddenly make sense to you?

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# ACT, Geometry – diagonal of a rectangular prism

Q: What is the (straight-line) distance between points A and B on the figure below?

Explanation:  We are finding the length of the green segment below:

This requires the three-dimensional version of the distance formula:

There is another way to do this, using a progression of two right triangles, with the green segment being the hypotenuse of the second triangle, but I find the above formula much simpler.  It is just an extension of the ‘usual’ (2-dimensional) distance formula we are so familiar with from Algebra and Geometry.

BUT if you want an even easier way, and I know you do, use my TI-84 Plus program ‘DIST3D’.

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# ACT , Algebra 1 – system of equations

MathPro Q&A Forum (ACT)

Q: For what value of A will the following system of equations have infinitely many solutions?

3x – 2y = 4

Ax + 4y = -8