chapter 3 decimals

8/7/2019 Chapter 3 Decimals

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Using Decimals

Review of comparing, rounding,

adding & subtracting, multiplying& dividing decimals

created by Alane Tentoni (copyright 2007)tentoni.weebly.com



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.



About Decimals

Decimals as

we knowt hem werefirst used by

John Napier int he late 1500sin Scotland.



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.



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



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.



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?



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!!!!



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!



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.



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.



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.



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.



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)



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!



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)



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.



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.



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.



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.



Multiplying Decimals Heres an example:

1.2 one digit here

x 3.9 one digit here

108

_36_

4.68 two digits here



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



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.



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



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.



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.



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.



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.



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



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



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?



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.



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



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!



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.



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.



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.



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.



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.



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



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.



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



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!

chapter 3 decimals

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