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8/7/2019 Chapter 3 Decimals
http://slidepdf.com/reader/full/chapter-3-decimals 1/45
Using Decimals
Review of comparing, rounding,
adding & subtracting, multiplying& dividing decimals
created by Alane Tentoni (copyright 2007)tentoni.weebly.com
8/7/2019 Chapter 3 Decimals
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What is a decimal?
A decimal is a dot t hat
goes after t he onescolumn.
It separates t he wholenumbers from t hepartial numbers.
8/7/2019 Chapter 3 Decimals
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About Decimals
Decimals as
we knowt hem werefirst used by
John Napier int he late 1500sin Scotland.
8/7/2019 Chapter 3 Decimals
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About DecimalsIn order to use decimals, you have to
understand place value.
1 2 3 4 . 5 6 7 8
To t he left of t he decimal, all t he numbers arewhole numbers. Each column is wort h ten times
t he column to its right.
8/7/2019 Chapter 3 Decimals
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About Decimals To t he right of t he decimal, all t he
numbers are like fractions. Each column is still wort h 10 of t he column tot he right.
1 2 3 4 . 5 6 7 8
8/7/2019 Chapter 3 Decimals
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Reading Decimals Zeroes t hat come at t he end of a decimal
dont add or take away any value.
.4 = .40 = .400 This is like saying fourtent hs = four tent hs and no hundredt hs
= four tent hs and no hundredt hs and not housandt hs.
8/7/2019 Chapter 3 Decimals
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Reading Decimals HOWEVER Zeroes t hat come between
t he decimal and t he ot her numbers are
VERY important!
.4 is four tent hs but .04 is four
hundredt hs. Would you rat her have fourdimes or four cents?
8/7/2019 Chapter 3 Decimals
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Comparing Decimals To tell if one decimal is
bigger t han anot her, you
have to compare t he samecolumn in bot h numbers.
The lengt h of t henumber does NOT
matter at all!!!!
8/7/2019 Chapter 3 Decimals
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Comparing DecimalsCompare t hese two numbers:
Which is larger?
.6 or .599823
All you need to do is look at t he tent hscolumn. 6 is more t han 5, so .6 is moret han .599823, even t hough .599823 has
more digits!
8/7/2019 Chapter 3 Decimals
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Comparing DecimalsAnot her comparison
Which is larger? .457 or .49?
The tent h columns are t he same (bot h 4),
but t he hundredt hs columns aredifferent. 9 is more t han 5, so .49 ismore t han .457.
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Rounding Decimals Rounding means cutting off
unnecessary digits.
Why would you use fewer digits t han
you know? Sometimes it is moreconvenient to give an approximateanswer.
8/7/2019 Chapter 3 Decimals
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Rounding DecimalsFirst, decide how many decimal places you
want in your answer.
Just t hrow away everyt hing behind t hat place. . .
Except! You willhave to decide whet her toincrease t he last digit or leave it alone.
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Rounding DecimalsLets round .576 to t he nearest hundredt h.
.576 is somewhere between .57 and .58. Which one is it closer to?
To decide, simply look at t he digit after t hehundredt hs place. Is it 5 or more? If so,round up. If not, leave it t he same.
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Rounding Decimals In our case, 6 is more t han 5, so .576
should be rounded up to .58.
What happens if you have a numberlike .398 to round to t he nearest hundredt h? (answer: .398 ~.40)
8/7/2019 Chapter 3 Decimals
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Rounding Decimals Be Careful!! Dont just
replace t he chopped
off numbers wit h zeroes! When youround, you are really
reducing t he number of digits behind t he
decimal!
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Rounding Decimals
Here are some numbers to round to t he
nearest hundredt
h.
1.3247 1.32
0.987 0.994.89721 4.90
Because we are rounding to
the nearest hundredth, each of
the numbers ends up with two
digits behind the decimal.
What if we had been roundingto the nearest tenth?
(answer: Rounding to t he nearest tent h leaves one decimal place. In t he
example: 1.3, 1.0, 4.9)
8/7/2019 Chapter 3 Decimals
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Adding & Subtracting
Decimals When you add decimals, line t he decimals
up one on top of t he ot her.
You have to add t he tent hs to t he tent hs,t he hundredt hs to t he hundredt hs, and so
on just as when you add wholenumbers, you add ones to ones and tensto tens.
8/7/2019 Chapter 3 Decimals
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Subtracting Decimals When you subtract, you may have to
annex zeroes to t he larger number soyou can borrow.
Example: 35.7 20.94= ?
35.70- 20.94
14.76
Annex a zero here soyou can borrow.
8/7/2019 Chapter 3 Decimals
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Multiplying Decimals
When you multiply
decimals, you should set t he problem up just as if
you were multiplyingwhole numbers
longest number on top,shortest on bottom.
8/7/2019 Chapter 3 Decimals
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Multiplying Decimals After you multiply t he numbers, you are
ready to put your decimal in place.
Count t he number of digits behind t hedecimal in bot h of t he multipliednumbers.
Put t hat many total digits behind t hedecimal in your answer.
8/7/2019 Chapter 3 Decimals
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Multiplying Decimals Heres an example:
1.2 one digit here
x 3.9 one digit here
108
_36_
4.68 two digits here
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Multiplying Decimals
Anot her example same numbers but
wit h
t he decimals in different places.
1.2 one digit here
x .39 two digits here
108_36_
.468 t hree digits here
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WHOA! Hang on! Did t hat last problem say 1.2 x .39
= .468?
Question: How can you multiply 1.2 bysomet hing and get an answer less t han 1.2?
Answer: Anytime you multiply by somet hingless t han 1, t he answer is smaller t han t henumber you started wit h.
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Multiplying Decimals If t he answer doesnt have enough digits,
you will have to put zeroes between t he
decimal and t he first number..12 two digits here
x .39 two digits here
108_36_
.0468 four digits here
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8/7/2019 Chapter 3 Decimals
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Dividing Decimals Dividing decimals is a lot like
dividing whole numbers, but we
need a way to get t he decimalsin t he right place in t he answer.
Before we start dividing decimals,
lets look at dividing some wholenumbers.
8/7/2019 Chapter 3 Decimals
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Dividing Decimals
42 ÷ 6 = 7 And 420 ÷ 60 = 7
In t he second equation, bot h 42 and 6 havebeen multiplied by ten. Because bot h numbers were multiplied by t he samet hing, t he quotient did not change.
8/7/2019 Chapter 3 Decimals
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Dividing Decimals We can use t hat trick to divide numbers
wit h decimals.
Because moving t he decimal to t he right isjust like multiplying by ten, if we move t hedecimal t he same number of places in
bot h
numbers, our quotient stays t hesame.
8/7/2019 Chapter 3 Decimals
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Dividing DecimalsHeres an example: .132 ÷ .12:
.12 .132
If t hese were whole numbers, you would say, How manytimes will 12 go into 13? But itsharder to t hink of .12
and .13.
If you could move t he decimal of t he divisor (.12) over 2places, you would have a whole number. You can dot hat as long as you move t he decimal of t he dividend
over 2 places as well.
8/7/2019 Chapter 3 Decimals
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Dividing DecimalsSo now our problem looks like t his:
NOTICE: The decimal movedstraight up from t he dividend tot he quotient.
Lining up t he number in t hequotient and t he dividend is VERY important because if t hey arewrong, your decimal will be in t he
wrong place.
12. 13.21.1
-12
1 2-1 2
0
8/7/2019 Chapter 3 Decimals
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ALWAYS Check! Now that we have an answer, we need to check our
work.
Multiply the quotient by the divisor. You should getthe dividend back.
1.1
x.1222
11.132
1 digit
2 digits
3 digits
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Hang on! How can we take two
small numbers like.12 and .132 anddivide t hem and get a bigger number?
Doesnt dividingalways mean you get a smaller number?
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Dividing Decimals
Anot her way to look at .132 ÷ .12 is tosay, How many groups of .12 does it take to make .132?
.12 + .012 = .132
It takes one and a little more, so ouranswer of 1.1 looks reasonable.
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Dividing Decimals Let¶s try another example:
1.25 ÷ .4 .4 1.25
First of all, let¶s estimate how many .4¶s it would
take to make 1.25
.4 + .4 + .4 = 1.2 so it will take 3 groups of .4 plus
a little more to make 1.25
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Dividing DecimalsFirst, move t he decimal in t hedivisor and t he dividend.
4. 12.5
3.1
-1205-4
1
In t his case, we have pulleddown all our numbers, but westill have a remainder.
DO NOT tack your remainderonto t he end of your answer!
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8/7/2019 Chapter 3 Decimals
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Annexing Zeroes-
4. 12.5000
3.125
-1205-4
10-820-20
0
When you get a
remainder of zero,you can stop pullingdown zeroes.
8/7/2019 Chapter 3 Decimals
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Check Your Work!The original problem was 1.25 ÷ .4.
The quotient was 3.125
Check: 3.125
x .4
1.2500
3 digits
1 digit
4 digits
Since 1.2500 = 1.25, our answer is correct.
8/7/2019 Chapter 3 Decimals
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Dividing Decimals Sometimes when we divide, the quotient of
the two numbers makes a pattern that
never stops!
This is called a ³repeating decimal.´
The kind that does stop is called a³terminating decimal.´ If you can workyour problem to a remainder of zero, youhave a terminating decimal.
8/7/2019 Chapter 3 Decimals
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Dividing DecimalsTip:
Divisors that have factors of alltwos or fives will definitely
terminate.
(like 2, 4, 5, 8, 10. . .)
Everything else can repeat ± it
depends on the dividend.
8/7/2019 Chapter 3 Decimals
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Dividing Decimals Here is a repeating decimal.
.3 5.56 First, move t he decimal.
3. 55.6
.
Put t he decimal on t hequotient line.
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Repeating Decimals
When youve pulleddown all yournumbers and you stillhave a remainder, you
need to annex zeroesand keep going.
3. 55.6
18.5
-325
-2416
-15
1
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Repeating Decimals
3. 55.6000
18.533
-325
-24
16-15
10-9
10
From here on, no
matter how manyzeroes we pull down,we will always get 10and t he next numberwill always be 3. The3 is repeating.
8/7/2019 Chapter 3 Decimals
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Repeating Decimals To show t hat a number repeats, place a
bar over all t he numbers t hat form t he
pattern.
In our example, only t he 3 wasrepeating:
18.53
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Get t he point? Decimals are a pretty convenient way to
represent fractional values.
Decimal rules are not difficult, but event hough you know t he rules, you must practice t hem until t hey are second nature!