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Posts tagged ‘Reducing fractions’

“Hack” for Simplifying Fractions


So c’mon … everything that can be said about simplifying fractions has been said … right?

Not quite! Here’s something that might just be original … a hack to smack those fractions down to size.

Suppose you’re staring at an annoying-looking fraction:  96/104, and it’s annoying the heck out of you, particularly because it’s smirking at you!

But it won’t smirk for long. For you open up your bag of hacks (obtained @ mathchat.me) and …

1st)  Subtract to get the difference between numerator and denominator. I also like to call this the gap between the numbers. Difference (aka, gap) = 104 – 96 = 8.

NOTE: Turns out that this gap, 8, is the upper limit for any numbers that can possibly go into BOTH 96 and 104. No number larger than 8 can go into both. And this is a … HACK FACT:  The gap represents the largest number that could possibly go into BOTH numerator and denominator. In other words, the gap is the largest possible greatest common factor (GCF).

2nd)  Try 8. Does 8 go into both 96 and 104? Turns out it does, so smack the numerator and denominator down to size:  96 ÷ 8 = 12, and 104 ÷ 8 = 13.

3rd)  State the answer:  96/104 = 12/13.

Is it still smirking? I think … NOT!

Try another. Say you’re now puzzling over:  74/80.

1st)  Subtract to get the gap. 80 – 74 = 6. So 6 is the largest number that can possibly go into BOTH 74 and 80.

2nd)  So try 6. Does it go into both 74 and 80? No, in fact it goes into neither number.

NOTE:  Turns out that even though 6 does NOT go into 74 OR 80, the fact that the gap is 6 still says something. It tells us that the only numbers that can possibly go into both 74 and 80 are the factors of 6:  6, 3 and 2. This, it turns out, is another … HACK FACT:  Once you know the gap, the only numbers that can possibly go into the two numbers that make the gap are either the factors of the gap, or the gap number itself.

3rd)  So now, try the next largest factor of 6, which just happens to be 3. Does 3 go into both 74 and 80? No. Like 6, 3 goes into neither 74 nor 80. But that’s actually a good thing because now there’s only one last factor to test, 2. Does 2 go into both 74 and 80? Yes! At last you’ve found a number that goes into both numerator and denominator.

4th)  Hack the numbers down to size:  74 ÷ 2 = 37, and 80 ÷ 2 = 40.

5th)  State the answer. 74/80 gets hacked down to 37/40, and that fraction, my dear friends, is the answer. 37/40 the final, simplified form of 74/80. 

O.K., are you ready to smack some of those fractions down to size? I believe you are. So here are some problems that will let you test out your new hack.

As you slash these numbers down, remember this rule. In some of these problems the gap number itself is the number that divides into numerator and denominator. But in other problems, it’s not the gap number itself, but rather a factor of the gap number that slashes both numbers down to size. So if the gap number itself doesn’t work, don’t forget to check out its factors.

Ready then? Here you go … For each problem, state the gap and find the largest number that goes into both numerator and denominator. Then write the simplified version of the fraction.

a)   46/54
b)   42/51
c)   48/60
d)   45/51
e)   63/77

Answers:

a)   46/54:  gap = 8. Largest common factor (GCF) = 2. Simplified form = 23/27
b)   42/51:  gap = 9. Largest common factor (GCF) = 3. Simplified form = 14/17
c)   48/60:  gap = 12. Largest common factor (GCF) = 12. Simplified form = 4/5
d)   45/51:  gap = 6. Largest common factor (GCF) = 3. Simplified form = 15/17
e)   63/77:  gap = 14. Largest common factor (GCF) = 7. Simplified form = 9/11

Josh Rappaport is the author of five math books, including the wildly popular Algebra Survival Guide and its trusty sidekick, the Algebra Survival Workbook. Josh has been tutoring math for more years than he can count — even though he’s pretty good at counting after all that tutoring — and he now tutors students in math, nationwide, by Skype. Josh and his remarkably helpful wife, Kathy, use Skype to tutor students in the U.S. and Canada, preparing them for the “semi-evil” ACT and SAT college entrance tests. If you’d be interested in seeing your ACT or SAT scores rise dramatically, shoot an email to Josh, addressing it to: josh@SingingTurtle.com  We’ll keep an eye out for your email, and our tutoring light will always be ON.

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(Divisibility) Practice Makes Perfect


As the saying goes, practice makes perfect.

And boy is that true in math! Of the standard school subjects, math requires the most practice, if you want to excel at it.

That being the case, this strikes me as a great time to practice the divisibility tricks we’ve just learned.

There are many skill areas where divisibility tricks are useful — solving proportions, factoring polynomials, multiplying fractions — but one of the most obvious is the critical skill of reducing fractions.

So now I’m offering you a chance to practice your divisibility skills for 2, 3, 4, 5 and 6. We will save the trick for 7 till we have a few more tricks “up our sleeves.”

For the following problems, answer these four questions:

1)  Which of these numbers — 2, 3, 4, 5 or 6 — divides evenly into the numerator (NM)?

2)  Do the same for the denominator (DNM).

3)  Then choose the largest number that divides into both NM and DNM. For these problems, this number will be the GCF.

4)  Finally, reduce the fraction by dividing both NM and DNM by this number.

Here’s an example that shows what you’d write:

ex)  24/42

1)  NM:  2, 3, 4, 6
2)  DNM:  2, 3, 6
3)  GCF = 6
4)  Answer:   4/7

NOW TRY THESE PROBLEMS:

a)  20/24
b)  25/40
c)   18/48
d)  26/60
e)  21/72
f)  30/85
g)  36/66
h)  56/92
i)  84/102
j)  99/141

ANSWERS:

a)  20/24
1)   NM:  2, 4, 5
2)  DNM:  2, 3, 4, 6
3)  GCF =  4
4)  Answer:   5/6

b)  25/40
1)   NM:  5
2)  DNM:  2, 4, 5
3)  GCF =   5
4)  Answer:  5/8

c)   18/48
1)   NM:  2, 3, 6
2)  DNM:  2, 3, 4, 6
3)  GCF =  6
4)  Answer:  3/8

d)  26/60
1)   NM:  2
2)  DNM:  2, 3, 4, 5, 6
3)  GCF =  2
4)  Answer:  13/30

e)  21/72
1)   NM:  3
2)  DNM:   2, 3, 4, 6
3)  GCF =   3
4)  Answer:   7/24

f)  30/85
1)   NM:  2, 3, 5, 6
2)  DNM:  5
3)  GCF =  5
4)  Answer:  6/17

g)  36/66
1)   NM:  2, 3, 4, 6
2)  DNM:  2, 3, 6
3)  GCF =  6
4)  Answer:  6/11

h)  56/92
1)   NM:  2, 4
2)  DNM:  2, 4
3)  GCF =  4
4)  Answer:  14/23

i)  84/102
1)   NM:  2, 3, 4, 6
2)  DNM:  2, 3, 6
3)  GCF =   6
4)  Answer:   14/17

j)  99/141
1)   NM:   3
2)  DNM:  3
3)  GCF =  3
4)  Answer:   33/47

How to Find out if 6 Divides in Evenly – Divisibility by 6


Award numeral 6

Image via Wikipedia

So far we’ve learned fun & easy divisibility tricks for the numbers 3 and by 4. Learning these tricks helps us reduce fractions with serious speed, and it helps us perform other math operations with a lot more ease. So let’s keep the learning
process going.

[Note:  If this is the first of these divisibility blogs that you have seen, search this blog for posts about divisibility by 3 and by 4; that way you’ll get caught up with the flow of these posts.]

The trick for 5 is incredibly simple:  5 goes into any number with a ones digit of 5 or 0. That is all you need to know. Not much else to say about 5.

And here is the trick for 6:  6 divides into any number that is divisible by BOTH 2 and 3. In other words, for the number in question, check to see if both 2 and 3 go in evenly. If they do, then 6 must also go in evenly. But if EITHER 2 or 3 does NOT go into the number, then 6 definitely will NOT go in. So you need divisibility by BOTH 2 AND 3 … in order for the trick to work.

Here’s an alternative way to say this trick, a way some kids find easier to grasp:  “6 goes into all even numbers that are divisible by 3.”

EXAMPLE 1:  74 — 2 goes in, but 3 does not, so 6 does NOT go in evenly.

EXAMPLE 2:  75 — 3 goes in, but 2 does not, so 6 does NOT go in evenly.

EXAMPLE 3:  78 — 2 and 3 BOTH go in evenly, so 6 DOES go in evenly.

Notice that since the tricks for 2 and 3 are quite simple, this trick for 6 is really quite simple too. It is NOT hard to use this trick even on numbers with a bunch of digits.

EXAMPLE 4:  783,612 — 2 goes in, and so does 3, so 6 DOES go in evenly. [checking for 3, note that you need to add only the digits 7 & 8. 7 + 8 = 15, a multiple of 3, so this large number IS divisible by 3.]

Now give this a try yourself with these numbers. For each number tell whether
or not 2, 3 and 6 will divide in evenly.

PROBLEMS:
a)  84
b)  112
c)  141
d)  266
e)  552
f)  714
g)  936
h)  994
i)  1,245
j)  54,936

ANSWERS:
a)  84:  2 yes; 3 yes; 6 yes
b)  112:  2 yes; 3 no; 6 no
c)  141:  2 no; 3 yes; 6 no
d)  266:  2 yes; 3 no; 6 no
e)  552:  2 yes; 3 yes; 6 yes
f)  714:  2 yes; 3 yes; 6 yes
g)  936: 2 yes; 3 yes; 6 yes
h)  994: 2 yes; 3 no; 6 no
i)  1,245:  2 no; 3 yes; 6 no
j)  54,936: 2 yes; 3 yes; 6 yes

 

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! 

Divisibility: Find out if 3 divides evenly into an integer


Quick:  What English word has 12 letters, almost half of which are are the letter “i” — well, 5 of the 12, to be exact?

Why it’s the word “D-I-V-I-S-I-B-I-L-I-T-Y” — a great thing to understand if you’re going to spend any amount of time doing math. And guess what:  virtually ALL students do a fair amount of math, so everyone would do well to master the tricks of divisibility.

With the tricks for divisibility in your command, you will have a much easier time:

—  reducing fractions
—  multiplying fractions
—  dividing fractions
—  adding and subtracting fractions
—  finding the GCF and LCM
—  simplifying ratios
—  solving proportions
—  factoring algebraic expressions
—  factoring quadratic trinomials
—  Need I say more?

I’m sure  you get the point — divisibility tricks are handy to know.

Since the tricks of divisibility are fun and interesting, too, I’ll share as many as I can think of. If, after I’m done, you know tricks I have not mentioned, feel free to share them as comments. Or, if you know any additional tricks for the numbers I’m covering, share those! It’s always fun to learn ways to get faster at math.

Today, I’ll share the trick that tells us whether or not a number is divisible by 3. Now many of you probably know  the basic trick. But even if you do, don’t skip this blogpost. For after I show how this trick is usually presented, I’ll share a few extra tricks that most people don’t know, tricks that make the basic trick even easier to use.

Here’s how the trick is usually presented.

Take any whole number and add up its digits. If the digits add up to a multiple of 3 (3, 6, 9, 12, etc.), then 3 divides into the original number. And if the digits add up to a number that is not a multiple of 3 (5, 7, 8, 10, 11, etc.), then 3 does not divide into the original number.

Example A:  Consider 311.

Add the digits:  3 + 1 + 1  = 5

Since 5 is NOT a multiple of 3, 3 does NOT divide into 311 evenly.

Example B:  Consider 411.

Add the digits:  4 + 1 + 1  =  6

Since 6 IS a multiple of 3, 3 DOES divide into 411 evenly.

Check for yourself:

311 ÷ 3 = 103.666 … So 3 does NOT divide in evenly.

But 411 ÷ 3  =  137 exactly. So 3 DOES divide in evenly.

Isn’t it great how reliable math rules are? I mean, they ALWAYS work, if the rule is correct. In what other field do we get that level of certainty?!

Corollary #1:

Now, to make the rule work even faster, consider this trick. If the number in question has any 0s, 3s, 6s, or 9s, you can disregard those digits. For example, let’s say you need to know if 6,203 is divisible by 3. When adding up the digits, you DON’T need to add the 6, 0 or 3. All you need to do is look at the 2. Since 2 is NOT  a multiple of 3, 3 does NOT go into 6,203.

So now try this … what digits do you need to add up in the following numbers? And, based on that, is the number divisible by 3, or not?

a)  5,391
b)  16,037
c)   972,132

Answers:

a)  5,391: Consider only the 5 & the 1. DIVISIBLE by 3.
b)  16,037: Consider only the 1 & 7. NOT divisible by 3.
c)   972,132: Consider only the 7, 2, 1 & 2. DIVISIBLE by 3.

Corollary #2:

Just as you can disregard any digits that are 0, 3, 6, and 9, we can also disregard pairs of numbers that add up to a sum that’s divisible by 3. For example, if a number has a 5 and a 4, we can disregard those two digits, since they add up to 9. And if a number has an 8 and a 4, we can disregard them, since they add up to 12, a multiple of 3.

Try this. See which digits you need to consider for these numbers. Then tell whether or not the number is divisible by 3.

a)  51,954
b)  62,497
c)  102,386

Answers:

a)  51,954: Disregard 5 & 1 (since they add up to 6); disregard the 9; disregard the 5 &4 (since they add up to 9). So number is DIVISIBLE by 3. [NOTE:  If you can disregard all digits, then the number IS divisible by 3.]
b)  62,497: Disregard 6; disregard 2 & 4 (Why?); disregard 9. Consider only the 7. Number is NOT divisible by 3.
c)  102,386: Disregard 0, 3, 6. Disregard 1 & 2 (Why?). Consider only the 8. Number is NOT divisible by 3.

See how you can save time using these corollaries?

Using the trick and the corollaries, determine which numbers you need to consider, then decide whether or not 3 divides into these numbers.

a)  47
b)  915
c)  4,316
d)  84,063
e)  25,172
f)  367,492
g)  5,648
h)  12,039
i)  79
j)  617
k)  924

ANSWERS:

a)  47:  Consider the 4 and 7. Number NOT divisible by 3.
b)  915:  Consider no digits. Number IS divisible by 3.
c)  4,316:  Consider the 4, 1. Number NOT divisible by 3.
d)  84,563:  Consider only the 5. Number NOT divisible by 3.
e)  71,031:  Consider the 7, 1, 1. Number IS divisible by 3.
f)  367,492:  Consider only the 7. Number NOT divisible by 3.
g)  5,648:  Consider only the 5. Number NOT divisible by 3.
h)  12,039:  Consider no digits. Number IS divisible by 3.
i)  79:  Consider only the 7. Number NOT divisible by 3.
j)  617:  Consider the 1, 7. Number NOT divisible by 3.
k)  927:  Consider no digits. Number IS divisible by 3.