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Posts tagged ‘How to reduce fractions’

How to Convert Percents to Decimals — Flawlessly


My last post showed how you convert decimals to percents.

Now I’ll dare to do the obvious with a post on how to convert percents to decimals.

Since converting percents to decimals is the opposite of converting decimals to percents, it makes sense — does it not? — that we’d use the opposite procedure. And that is the case.

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Not only that, but we do the opposite procedure in the opposite order, too. How’s that for going totally opposite?

Since the final step in converting decimals to percents is tacking on a percent symbol (%), the very first step in converting percents to decimals is taking off that symbol.

And since the first step in converting decimals to percents is nudging the decimal point two places to the right, the last step in converting percents to decimals is pushing the decimal point two places to the left.

Let’s take a look at the process with this example.

Problem:  convert 73.2% to a decimal.

Step 1:  Take off the percent symbol. 73.2% changes to just 73.2

Step 2:  Move the decimal point two places to the left. 73.2 changes to .732 That’s all there is to it. This tells us that 73.2% is the same as .732, or 7 hundred thirty-two thousandths.

If you don’t recall the steps that you’re turning around, here’s a quick way to remember the process of converting percents to decimals. As I said in my last post, we can make use of alphabetical order, setting up the words for decimal and percent, in order, like this:

D-Decimal                              P-Percent

Then we draw an arrow showing that we’re converting from percent form to decimal form. The arrow shows the direction of the conversion: percent to decimal.

D-Decimal    <————–    P-Percent

This arrow points to the left, and that tells us that we move the decimal point to the left when we convert a percent to a decimal.

Let’s look at the process again, this time focusing now on how we use the arrow’s direction to help us.

Problem:  Convert  4.782% to a decimal.

Step 1:  Rip off the percent symbol. 4.782% changes to 4.782

Step 2:  Give the decimal point two shoves in the arrow’s direction. Since a percent to decimal conversion makes the arrow point left, we shove the decimal point two spaces to the left.  4.782 changes to .04782

This tells us that .04782 is the same as 4.782%

Note: if there are no digits showing to the left, we’re free to add 0s on the left side of the leftmost digit to create a place where the decimal point lands, after being shoved to the left.

In the last example, we had to tack a 0 on the left of 4.782 — making it 04.782, to get a digit (0) to the left of which we placed the final decimal point. Be confident that you can write as many 0 digits as you need to the left of a number’s leftmost digit. For example, it is just fine (though admittedly strange) to write 4.3 as 0004.3. You’d do this weird maneuver if you need that many zeros to the left of the 4. This occurs in converting numbers to scientific notation, for example.

So, now that you know the process, try your hand at converting the following percents to decimals: (Answers at the bottom of this post.)

a)  38%                                                                                                                           b)  19.3%                                                                                                                         c)   4.2%                                                                                                                         d)  175%                                                                                                                         e)  398.6%
f)  2,400%

Answers to the problems in the last post, converting decimals to percents:

a)  8590%                                                                                                                     b)  416.2%                                                                                                                     c)  20873.5%
d)  4.7%
e)  207,465%
f)  28.3%
g)  .569%

Answers to problems in this post:

a)  38%        =  .38
b)  19.3%     =  .193
c)   4.2%      =  .042
d)  175%      =  1.75
e)  398.6%   =  3.986
f)  2,400%    =  24.0, or just 24

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


Award numeral 6

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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! 

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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.

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