## Kiss those Math Headaches GOODBYE!

### How to Divide ANY Number by a Radical — Fast!!! (Math “hack” w/ full explanation)

Here’s a super-quick shortcut for  DIVIDING ANY NUMBER by a RADICAL.

Note: I’m using this symbol () to mean square root.
So √5 means the square root of 5;  √b means the square root of b, etc.
And … if you want to learn why this “hack” works, see my explanation at the end of the blog.

This “hack” lets you mentally do problems like the following three. That means you can do these problems in your head rather than on paper.

a)  12 / √3

b)  10 / √2

c)  22 / √5

Here are three terms I’ll use in explaining this “hack.”

In a problem like 12 divided by √3, which I write as:  12 / √3,

12  is  the dividend,

3  is  the number under the radical,

## The “Hack,” Used for  12 / √3:

1.  Divide the dividend by the number under the radical.
In this case, 12 / 3  =  4.
4 x √3  =  4√3

## Another Example:  10 / √2

1.  Divide the dividend by the number under the radical.
In this case:   10 / 2  =  5
2. Take the answer you get, 5, and multiply it by the radical.
5 x √2  =  5√2.  (Don’t forget to shake head in amazement!)

## Third Example:  22 / √5

1.  Divide dividend by number under the radical.
In this case,  22 divided by 5 = 22/5  (Yep, sometimes you wind up with a fraction or a decimal; that’s why I’m giving an example like this.)
2. Take the answer you get, 22/5, and multiply it by the radical.
22/5 x √5 =  22/5 √5.  [Note: the √5 is in the numerator, not
in the denominator. To make the location of this √5 clear, it’s best
to write the answer:  2√5 / 5].

## NOW TRY YOUR HAND by doing these PRACTICE PROBLEMS:

a)   18 / √3

b)   16 / √2

c)   30 / √5

d)   10 / √3

e)   12 / √5

– – – – – – – – – – – – – – – – – –

a)   18 / √3  = 6√3

b)   16 / √2  = 8√2

c)   30 / √5  = 6√5

d)   10 / √3  = 10√3/3

e)   12 / √5  = 12√5/5

– – – – – – – – – – – – – – – – – –

## WHY THE “HACK” WORKS:

It works because we rationalize the denominator of a fraction whenever the denominator contains a radical. Here’s the “hack” in general terms, with:

a  =  the dividend,

b  =  the number under the radical,

a / √b

=   a
√b

=   a     √b    =   a √b
√b   √b            b

Notice: we started with:  a / √b.

And keeping things equal, we ended up with  a √b / b.

This shows that the “hack” works in general. So it works in all specific cases as well!

– – – – – – – – – – – – – – – – – –

Final note: the number under the radical is called the radicand. But that term is so close to the term radical that I thought it would be less confusing if I just called this the number under the radical. I hope you are not offended.

### Find the LCM — FAST!

Here’s a video that goes with a blog entry that many people have found helpful: Find the LCM — FAST! This trick can be a real time-saver, so feel free to pass this around.

Josh Rappaport is the author of the Algebra Survival Guide and Workbook, which comprise an award-winning program that makes algebra do-able! The books break algebraic concepts down into manageable chunks and provide instruction through a captivating Q&A format. Josh also is the author of PreAlgebra Blastoff!, which presents an engaging, hands-on approach (plus 16-page color comic book) for learning the rules of integers. Josh’s line of unique, student-centered math-help books is published by Singing Turtle Press and can be found on Amazon.com

### How to Find the LCM for Three Numbers

Several readers have said they like my trick for finding the LCM described in the post “How to Find the LCM — FAST!” but wonder how to use the trick for finding the LCM for THREE numbers. Here is how you do that.

Essentially it involves using the same LCM trick three separate times. Here’s how it’s done.

Suppose the numbers for which you need to find the LCM are 6, 8, and 14.

Step 1)  Find the LCM for the any two of those. Using 6 and 8, we find that their LCM = 24.

Step 2)  Find the LCM for another pair from the three numbers. Using 8 and 14, we find that their LCM = 56.

Step 3)  Find the LCM of the two LCMs, meaning that we find the LCM for 24 and 56. The LCM for those two numbers = 168.

And that, my good friends, is the LCM for the three original numbers.

So, to summarize. Find the LCM for two different pairs. Then find the LCM of the two LCMs. The answer you get is the LCM for the three numbers.

Here are a few problems that give you a chance to practice this technique.

Find the LCM for each trio of numbers.

a)  10, 25, 30

b)  16, 28, 40

c)  14, 32, 40

The LCMs for each trio are:

a)  150

b)  560

c)  1,120

Josh Rappaport is the author of the Algebra Survival Guide and Workbook, which comprise an award-winning program that makes algebra do-able! The books break algebraic concepts down into manageable chunks and provide instruction through a captivating Q&A format. Josh also is the author of PreAlgebra Blastoff!, which presents an engaging, hands-on approach (plus 16-page color comic book) for learning the rules of integers. Josh’s line of unique, student-centered math-help books is published by Singing Turtle Press and can be found on Amazon.com

### “Hacks” for Slaying Proportions, Part 1: the Amazing Horizontal Canceling Trick

Proportions can seem intimidating, but they’re actually one of the easiest types of word problems to master. In this series I’ll offer a number of tips that help you conquer algebraic proportion problems.

But first, a cool shortcut you can use whenever you’re facing down an algebraic proportion …

In working with proportions, I’m amazed that few students know how a canceling process that would help them find the solution more quickly and efficiently.

So I want to share the trick, for all who’ve never seen it.

Of course, given a problem like:  6/x  =  24/32,

most of us know that we can cancel vertically with the two numbers in the fraction on the right, to get:

6/x  =  3/4

Then we just cross-multiply to get:

3x  =  24, and see that x  =  8.

In other words, we know we can cancel vertically given a proportion, just as we can cancel vertically with any fraction.

What many people don’t know though, is that there’s another way we can cancel when solving proportions — horizontally!

— What? you say.

Horizontally, I say. And no, I’m not joshing.

For example,  given the proportion:  7/4  =  21/x

you can cancel horizontally with the two numbers in the numerator:  the 7 and the 21. These reduce to 1 and 3.

The proportion then becomes:

1/4  =  3/x  [I’m really not kidding.]

Cross-multiplying, you get the answer in one quick step:   x = 12.

What’s really convenient is that you can also cancel both vertically and horizontally in the same problem. For example, in

6/x  =  42/28,

you could first cancel horizontally, to get:

1/x  = 7/28

Then you can cancel vertically, to get:

1/x  =  1/4

Cross-multiplying, you get the answer in just a step:  x = 4

I find that when students cancel before cross-multiplying, they’re more apt to get the right answer, and to get less frustrated, for the numbers they deal with remain small.

For example, in the last problem, if the student had not canceled at all, he would have a cross-multiplication mess of:

6 x 28 = 42x

That sort of problem just opens up the door to arithmetic mistakes. But canceling before cross-multiplying shuts that door since it makes the numbers smaller and easier to manage.

So now you get a chance to practice horizontal cancelling!

First use horizontal cancelling to get the answer to these
proportions. Those who’d like an added challenge might like to try them in their head:

a)   x/12  =  3/4

b)  3/7  =  x/35

c)   z/48  =  7/12

d)  y/56  =  7/8

Now go really wild! Use both horizontal and vertical canceling to make quick work of these proportions:

e)  x/9  =  16/36

f)   x/22  =  30/66

g)  32/56  =  y/14

h)  13/q  =  65/35

And here are the answers to all of these problems:

a)  x  =  9

b)  x  =  15

c)   z  =  28

d)   y  =  49

e)   x  =  4

f)   x  =  10

g)   y  =  8

h)  q  =  7

### Multiplication Trick #5 — How to Multiply Two-Digit Numbers by 11

This is the fifth in my series on multiplication tricks. I suggest that you make mental math “tricks” a steady part of your math instruction. Benefits students will reap include:

—  delight with the tricks themselves

—  enhanced confidence in working with numbers

—  students who otherwise don’t like math — or don’t like it much — often find the tricks irresistibly fun and interesting

TRICK #5:

WHAT THE TRICK LETS YOU DO: Multiply two-digit numbers by 11.

HOW YOU DO IT:  To multiply a two-digit number by 11, first realize that the answer will have three digits. The first (left-most) digit of the answer is the first digit of the number; the last (right-most) digit of the answer is the last digit of the number; and the middle digit is the sum of the first and last digits.

But those are just words … here’s a living, breathing example …

Example:  11 x 25

Look at 25. The first digit is 2; the last digit is 5.

First digit of answer is 2, so thus far we know the answer looks like:  2 _ _

Last digit of answer is 5, so now we know the answer looks like:  2 _ 5

Middle digit is 7, since 2 + 5 = 7.

The answer is the three-digit number:  2 7 5, more casually known as 275.

It’s that easy!

ANOTHER EXAMPLE:  11 x  63

First digit of answer is 6, so thus far we know the answer looks like:  6 _ _

Last digit of answer is 3, so now we know the answer looks like:  6 _ 3

Middle digit is 9, since 6 + 3 = 9.

The answer is the three-digit number: 6 9 3, or just 693.

Try these for practice:

11 x 24

11 x 31

11 x 52

11 x 27

11 x 34

11 x 26

11 x 62

11 x 24 = 264

11 x 31 = 341

11 x 52 = 572

11 x 27 = 297

11 x 34 = 374

11 x 26 = 286

11 x 62 = 682

NOTE:  If you’re clever (and we’re sure that you are), you have probably realized that this trick, as described, works only when the digits add up to 9 or less. So what do you do when the digits add up to 10 or more? Some of you may figure this out on your own. For those who need a little help, the answer to this will be included in an upcoming blog post.

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information!

### Multiplication Trick #3 — How to Multiply by 25 FAST!

Here’s the third in my series of multiplication tricks. The first was a trick for multiplying by 5. The second a trick for multiplying by 15, and now this one, a trick for multiplying by 25. Anyone see a pattern?

TRICK #3:

WHAT THE TRICK LETS YOU DO: Quickly multiply numbers by 25.

HOW YOU DO IT:  The key to multiplying by 25 is to think about quarters, as in “nickels, dimes, and quarters.”

Since four quarters make a dollar, and a dollar is worth 100 cents, the concept of quarters helps children see that 4 x 25 = 100.

Since four quarters make one dollar, children can see that twice that many quarters, 8, must make two dollars (200 cents). And from that fact children can see that 8 x 25 = 200.

Following this pattern, children can see that twelve quarters make three dollars (300 cents). So 12 x 25 = 300. And so on.

Fine. But how does all of this lead to a multiplication trick?

The trick is this. To multiply a number by 25, divide the number by 4 and then tack two 0s at the end, which is the same as multiplying by 100.

A few more examples:

16 x 25. Divide 16 by 4 to get 4, so the answer is 400. [In money terms, 16 quarters make \$4 = 400 cents.]

24 x 25. Divide 24 by 4 to get 6, so the answer is 600. [In money terms, 24 quarters make \$6 = 600 cents.]

48 x 25. Divide 48 by 4  to get 12, so the answer is 1200. [In money terms, 48 quarters make \$12 = 1200 cents.]

Try these for practice:

20 x 25

32 x 25

36 x 25

16 x 25

24 x 25

44 x 25

52 x 25

76 x 25

20 x 25  =  500

32 x 25  =  800

36 x 25  =  900

16 x 25  =  400

24 x 25  =  600

44 x 25  =  1100

52 x 25  =  1300

76 x 25  =  1900

But wait, you protest … what about all of the numbers that are not divisible by 4? Good question! But it turns out that there’s a workaround. You still divide by 4, but now you pay attention to the remainder.

If the remainder is 1, that’s like having 1 extra quarter, an additional 25 cents, so you add 25 to the answer.

Example:  17 x 25. Since 17 ÷ 4 = 4 remainder 1, the answer is 400 + 25 = 425.

If the remainder is 2, that’s like having 2 extra quarters, an additional 50 cents, so you add 50 to the answer.

Example: 26 x 25. Since 26 ÷ 4 = 6 remainder 2, the answer is 600 + 50 = 650.

If the remainder is 3, that’s like having 3 extra quarters, an additional 75 cents, so you add 75 to the answer.

Example:  51 x 25. Since 51 ÷ 4 = 12 remainder 3, the answer is 1200 + 75 = 1275.

Now try these for practice:

9 x 25

11 x 25

14 x 25

19 x 25

22 x 25

25 x 25

34 x 25

49 x 25

9 x 25  =  225

11 x 25  =  275

14 x 25  =  350

19 x 25  =  475

22 x 25  =  550

25 x 25  =  625

34 x 25  =  850

49 x 25  =  1225

Happy teaching!

—  Josh

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information!

### Fraction divided by a fraction

Hi,

This one is going out to all of you … on station MCHT.

Does anyone know any good tricks for the situation where a fraction divides another fraction. In other words, for a problem like [(2/3)÷(4/5}, does anyone know of a col way to make this much easer than the way most people learn this in a school?

If so, send me your thoughts In any case, after I get a bunch of your ideas, I’ll share mine. Then we can vote on which approach we like the most.

If you’ve got an idea, send it to:

into@SingingTurtle.com

Make the subject line: Dividing a Fraction by a Fraction.

Have fun!

— Josh

### Multiplication Trick: x 15 — How to Multiply by 15 FAST!

Here’s the second in my set of multiplication tricks. (The first was a trick or multiplying by 5.)

TRICK #2:

WHAT THE TRICK LETS YOU DO: Multiply numbers by 15 — FAST!

HOW YOU DO IT: When multiplying a number by 15, simply multiply the number by 10, then add half.

EXAMPLE:15 x 6

6 x 10 = 60

Half of 60 is 30.

60 + 30 = 90

That’s the answer:15 x 6 = 90.

ANOTHER EXAMPLE:15 x 24

24 x 10 = 240

Half of 240 is 120.

240 + 120 = 360

That’s the answer:15 x 24 = 360.

EXAMPLE WITH AN ODD NUMBER:15 x 9

9 x 10 = 90

Half of 90 is 45.

90 + 45 = 135

That’s the answer:15 x 9 = 135.

EXAMPLE WITH A LARGER ODD NUMBER:23 x 15

23 x 10 = 230

Half of 230 is 115.

230 + 115= 345

That’s the answer:15 x 23 = 345.

15 x 4

15 x 5

15 x 8

15 x 12

15 x 17

15 x 20

15 x 28

15 x 4=60

15 x 5=75

15 x 8=120

15 x 12=180

15 x 17=255

15 x 20=300

15 x 28=420

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information!

### How to Find the GCF — FAST!

Time-saving tips are great, right? So I’d like to share a time-saving tip for math.

This tip SIMPLIFIES the process of finding the greatest common factor (GCF) for two numbers, a good thing to know when simplifying fractions, reducing proportions, etc.

First, have you ever noticed that when students search for a GCF, they sometimes don’t know when to stop searching? This tip alleviates that problem, for it tell students exactly when they can stop testing numbers.

It turns out that students can stop testing when they reach the DIFFERENCE between the two numbers whose GCF they’re trying to find.

As an easy example, let’s say you need to find the GCF for 16 and 20.

All you do is subtract 16 from 20, to get the difference, 4, and this number — 4 — is the largest number that could POSSIBLY go into both 16 and 20 evenly.

Once you know that, just test 2, 3, and 4 to find the highest one that goes into 16 and 20. Of course that would be 4, so you got the GCF right off the bat, in this case.

Keep in mind that that greatest possible greatest common factor is not necessarily the true, greatest common factor. But it does set an upper limit for GCFs, and having that upper limit really reduces kids’ stress.

Another example:  find the GCF for 25 and 35.

35 – 25 = 10, so 10 is the greatest possible GCF. But of course 10 does not go into 25 and 35, so 10 is not the GCF. Check the numbers less than 10, and you’ll see that 5 is the GCF. But no more checking above 10, as kids are likely to do, unless you tell them when to stop.

I have dubbed this mathematical object the GPGCF, for Greatest Possible Greatest Common Factor, and I’ve found that students really appreciate learning it’s there — to alert them when it’s “quitting time.”

Try it out yourself, whenever it next flows with your lesson. Let me know what kind of reaction you get from the kids, and good luck.

By the way, if you’d like to explain to your students why this trick works, here’s a way to look at it. If you think about this situation via the number line, the GPGCF is simply the distance between the two numbers whose GCF you’re trying to find. Let’s go back to our first example: searching for the GCF for 16 and 20. The difference between 20 and 16, 4, is the distance between 16 and 20 on the number line. So if any number does go into both 16 and 20, it cannot be larger than 4, since that’s the space between the numbers.

To see this clearly, imagine that you wonder for a moment if 8 might be the GCF for 16 and 20. Well it is true that 8 does go into the first of these numbers, 16. But the next number that 8 goes into evenly must be 8 greater than 16, or 24. In other words, 8 is going to “leap past” 20, by hitting 24, when it goes into its next multiple. So the space between the numbers — 4 in this case — gives you the biggest number that could possibly fit into both numbers.

DIRECTIONSs:  Given each pair of numbers, first find the GPGCF. Then use the GPGCF to help you find the GCF.

a)  6, 10

b)  8, 12

c)  12, 15

d)  12, 20

e)  14, 28

f)  18, 26

g)  27, 36

h)  36, 48

i)  42, 60

j)  72, 80

a)  6, 10   GPGCF = 4   GCF =  2

b)  8, 12   GPGCF = 4   GCF =  4

c)  12, 15   GPGCF = 3   GCF =  3

d)  12, 20   GPGCF = 8   GCF =  4

e)  14, 28   GPGCF = 14   GCF =  14

f)  18, 26   GPGCF = 8   GCF =  2

g)  27, 36   GPGCF = 9   GCF =  9

h)  36, 48   GPGCF = 12   GCF =  12

i)  42, 60   GPGCF = 18   GCF =  6

j)  72, 80   GPGCF = 8   GCF =  8

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information!

### How to Multiply Two Teen Numbers — FAST

Ever wondered if there’s a quick way to multiply two numbers in the teens, like:
14 x 17, or 18 x 16?

Turns out, there is. Stick around a minute, and you’ll learn it. Let’s try it for:
18 x 16.

The trick involves doing two operations with the numbers in the ones place of the teen numbers. The numbers in the ones place are 8 (in the ones place of 18) and 6 (in the ones place of 16).

First, ADD the two digits in the ones place.
8 + 6 = 14.

Take that sum, 14, and add it to 10:   14 + 10 = 24.

Tack a zero on to the end. 24 becomes 240. (Keep that number in mind.)

Next MULTIPLY the two digits in the ones place:
8 x 6 = 48.

Now just add this product, 48, to the 240.
240 + 48  = 288.

That’s the answer. This may seem a little tricky and a little weird, at first, but it gets easy after a few times. Trust me …

O.K., fine, don’t trust me. But just try it one more time, with 14 x 17, and see for yourself.

4 + 7 = 11. 11 + 10 = 21.

21 becomes 210.

Then 4 x 7 = 28, and 210 + 28 = 238.

That’s all there is to it.

Now try these:

a) 13 x 16
b) 12 x 17
c) 14 x 19
d) 12 x 19
e) 13 x 14
f) 17 x 18
g) 19 x 17
h) 15 x 19
j) 16 x 17
k) 18 x 19