Kiss those Math Headaches GOODBYE!

Archive for February, 2010

How to Multiply by 25 in Your Head

This is a simple trick that anyone can easily learn. It is just a trick for
multiplying a number by 25.

If someone asked you what 25 times 36 equals, you’d probably be tempted
to reach for a calculator and start punching buttons. But remarkably, you’d
probably be able to work it out even faster in your head.

Since 25 is one-fourth of 100, multiplying by 25 is the same thing as
multiplying by 100 and dividing by 4. Or, even more simply:
first divide by 4, then add two zeros.

Here’s the example:

Problem: 36 x 25
First divide 36 by 4 to get 9.
Then add two zeros to get: 900.
That, amazingly enough, is the answer.

Another example: 88 x 25
First divide 88 by 4 to get 22.
Then add two zeros to get: 2,200.

Now try these problems in your head:

a) 25 x 12
b) 25 x 28
c) 25 x 48
d) 25 x 60
e) 25 x 84
f) 25 x 96

Here are the answers:
a) 300
b) 700
c) 1,200
d) 1,500
e) 2,100
f) 2,400

But, you say, what if the number you start with is not divisible by 4.
No problem. Just use this fact:
if the remainder is 1, that is the same as 1/4 or .25
if the remainder is 2, that is the same as 2/4 or .50
if the remainder is 3, that is the same as 3/4 or .75

So take a problem like this: 25 x 17
dividing 17 by 4, you get 4 remainder 1.
But that is the same as 4.25
Now just move the decimal right two places (same as multiplying by 100)
Answer is: 425

Another example: 25 x 18
dividing 18 by 4, you get 4 remainder 2.
But that is the same as 4.50
Now move the decimal right two places.
Answer: 450

Another example: 25 x 19
dividing 19 by 4, you get 4 remainder 3.
But that is the same as 4.75
Now move the decimal two places to the right.
Answer is: 475

Now try these in your head:
A) 25 x 21
B) 25 x 26
C) 25 x 35
D) 25 x 42
E) 25 x 63
F) 25 x 81

And here are the answers:

A) 525
B) 650
C) 875
D) 1,050
E) 1,575
F) 2,025

Debunking a Popular Website … w/ Algebra

A friend of mine sent me to a website that purports to be able to “read your mind” by guessing a product that you have in mind. It turns out that you can use algebra to debunk the premise of the site by finding out how the trick works.

First, check out the site here.

Now that you see how it works, here’s the algebra behind this “mind-reading” site.

Take any two-digit number.
Call the tens digit x; call the ones digit y.
Then the value of the two-digit number is 10x + y

As an example, take the number 73.
This is 10 x 7 + 3 = 70 + 3 = 73
See what I mean?

Then, if you subtract the individual digits from the
two digit number, that would be the same as writing down:

– x – y.

Putting the two expressions together, you get:

(10x + y) – x – y

Simplify that, as follows:

(10x + y) – x – y
= 10x + y – x – y
= 10x – x + y – y
= 9x

Now since x was a whole number to start with, the value of 9x must be a multiple of 9, such as 18, 27, 36, etc.

Now check it out. Do this process with a variety of two-digit #s
and you’ll see that the answer is always a multiple
of 9.

Now here’s the “kicker”: Look at the grid, and you’ll see that every time you see the grid, all products labeled with a multiple of 9 display the same product.

So that’s how this program “knows” your product.
They know it’s a multiple of 9, and they set up all
of those to have the same product.

Sneaky, and it works to trick lots of people. But not you, any more, since you have the power of algebra on your side.

How to Survive College Algebra

Why is algebra hard for college students? Let me count the reasons …

First, the college students who struggle with with algebra are usually the same folks who struggled with algebra in high school, only older now. They hated it then, and they dread it now. It didn’t make sense then, and it still doesn’t. So they are already predisposed to struggle with this class from painful past experiences.

Another problem stems from the vocabulary of algebra. The words that are used to describe algebra are — let’s face it — intimidating! Words like: polynomial, quadratic, radicals. This is a specialized language, written in annoyingly polysyllabic Latin. And when you start to dislike a subject it is natural that you start to dislike the vocabulary of that subject. And the vocabulary of algebra is somewhat remote and cold, easy to dislike.

Another problem is the tone of the textbooks that teach algebra. I mean, if you want to make a fire, you’d probably do well to burn an algebra book, for the simple reason that the text on the pages is so DRY. I mean, take a sentence from a typical algebra book, and it will sound like this (actual quote):
“The difference between two integers is defined as the absolute value of of the difference of the absolute value of the integers.” I mean, all you’d have to do is pull out a match. It will virtually light itself when placed next to this kind of prose.

The final reason has to do with the teachers who taught algebra back in high school. Now I am not picking on all teachers, but the ones I hear about over and over in my tutoring are those that droned on and on, “Then you subtract 17 from both sides, and finally you divide both sides by 3 …” just like the textbooks, never trying to make the ideas come to life. If you have a great algebra teacher, that can make the textbook bearable. But if the teacher is as dry as the textbook, it can be impossible for some people to make the critical connections.

So what is the solution? Good teaching in high school, and great teaching in college can make a huge difference.

For students who don’t have access to great teaching, however, there is my book, the Algebra Survival Guide. As a tutor who has worked with hundreds of high schoolers and scores of college students, I know how hard these students try, and how little progress they sometimes make. So I wrote a book in plain English, a book everyone can read as easily as you’d read a good novel. My goal was to take the edge off of the intimidating quality of algebra, and from the emailed responses I’ve received, it has worked.

I also threw some humor into the book. O.K., I’m not Jack Benny, but I do make a few jokes, here and there. I mean, we all need to have a little bit of fun, even doing math.

And I worked hard to connect the ideas of algebra to real life, to make them make sense.

For example, take the problem of – 8 + 3. I liken this situation to a tug of war. There are two teams in the tug of war, a Positive Team and a Negative Team. The – 8 means that the Negative Team has 8 people pulling. The + 3 means the Positive Team has 3 people pulling. All you have to do to solve the problem is answer two questions: first, which team will win, if everyone is equally strong? The Negatives, since they have more people pulling. And then, by how many does the larger team outnumber the other smaller team? By 5, since 8 is 5 more than 3. Put your answers together, and you’ll see that the Negatives win by 5, so the answer is – 5. See how easy it can be when you relate the concepts to real life?

The Algebra Survival Guide explains many ideas this way, and it gives algebra a friendly human face.

Here’s one example of a quote from a college student who found the book helpful:

I’m a returning college adult now in the 4th week of my College Algebra course. Your book has FINALLY filled in the gaps in my earlier education … thank you (to the third power)! Thanks for the great book … 30 years of math phobia gone in 3 hours of reading … really, thank you very much!
College Student, Mary Ellen Kirian, Lake Oswego, OR

If you’d like to check out this book, just go to this Amazon page,where you can read lots of reviews.

While you’re @ Amazon, don’t forget to check out the companion Algebra Survival Workbook, with thousands of additional practice problems, which take you from understanding to mastery. You’ll find that page here.

As they used to say on TV, “Try it, you’ll like it!”

— Josh