Basics of Maths for all school students

8/18/2019 Basics of Maths for all school students

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http://www.ianswer4u.com/2011/05/integers-and-their-properties.html



What are Whole Numbers ?Whole numbers are the set of positive integers.They do not have any decimal or fractional part. Natural numbers along

with zero(0) are whole numbers.

What is Zero ?

Zero is the only whole number which is not a natural number.

NATURAL NUMBER

A natural number is a number that occurs commonly andobviously in nature. As such, it is a whole, non-neative number.

The set o! natural numbers, denoted N, can be de"ned in either o!

two ways#

N $ %&, ', (, ), ...*

N $ +', (, ), , ...*

Algebraic properties satisfied by the natural numbers

The addition () and multiplication (!) operations on natural numbers as defined above

have several algebraic properties"

• #losure under addition and multiplication" for all natural numbers a and b$

both a + b and a × b are natural numbers.

• %ssociativity" for all natural numbers a$ b$ and c$ a + (b + c) = (a + b)

+ c and a × (b × c) = (a × b) × c.

• #ommutativity" for all natural

numbers a and b$ a + b = b + a and a × b = b × a.

• &'istence of identity elements" for every natural number a$ a + 0 = a and a × 1

= a.

• istributivity of multiplication over addition for all natural numbers a$ b$ and c$ a ×

(b + c) = (a × b) + (a × c).

• No nonzero zero divisors" if a and b are natural numbers such that a × b = 0$then a = 0 or b = 0.

https://en.wikipedia.org/wiki/Closure_(mathematics)

https://en.wikipedia.org/wiki/Commutativity

https://en.wikipedia.org/wiki/Zero_divisor

https://en.wikipedia.org/wiki/Closure_(mathematics)

https://en.wikipedia.org/wiki/Associativity

https://en.wikipedia.org/wiki/Commutativity

https://en.wikipedia.org/wiki/Identity_element

https://en.wikipedia.org/wiki/Distributivity

https://en.wikipedia.org/wiki/Zero_divisor

http://searchsecurity.techtarget.com/definition/set



Natural Numbers are $*$+$,$-$... .../ and Whole numbers are 0$$*$+$... %ccording to

Wiipedia" 1n mathematics$ a natural number is either a positive integer ($ *$ +$ ,$ ...) or a

non2negative integer (0$ $ *$ +$ ,$ ...).

An integer +ronounced N-tuh-/er0 is a whole number +not a!ractional number0 that can be ositive, neative, or 1ero.

E2amles o! inteers are# -3, ', 3, 4, 56, and ),&).

E2amles o! numbers that are not inteers are# -'.), ' )7,).', .&5, and 3,8).'.

The set o! inteers, denoted Z , is !ormally de"ned as !ollows#

Z $ %..., -), -(, -', &, ', (, ), ...*

Rational (Fraction) Numbers

Rational Numbers" %ny number that can be written in fraction form is

a rational number . This includes integers$ terminating decimals$ andrepeating decimals as well as fractions. %n integer can be written asa fraction simply by giving it a denominator of one$ so any integer isa rational number .

%dditionedit/

Two fractions are added as follows"

3ubtractionedit/

4ultiplicationedit/

https://en.wikipedia.org/w/index.php?title=Rational_number&action=edit&section=8







https://en.wikipedia.org/wiki/Additive_inverse

https://en.wikipedia.org/wiki/Multiplicative_inverse



The rule for multiplication is"

ivisionedit/

Where c 5 0"

1nverseedit/

%dditive and multiplicative inverses e'ist in the rational numbers"

&'ponentiation to integer power edit/

1f n is a non2negative integer$ then

and (if a 5 0)"

The three types of fractions are :

Proper fraction

Improper fraction

Mixed fraction

Proper fraction:




Fractions whose numerators are less than the denominators are called proper

fractions.(Numerator < denominator)

For examples:

2323, 3434, 454, 56!, 67!", 292# 58$, 252, etc are proper fractions%

Improper fraction:

Fractions with the numerator either equal to or greater than the denominator are

called improper fraction. (Numerator & denominator or, Numerator ' denominator)

ractions i*e 544, 175+", 522 etc% are not proper fractions% These are improper

fractions% The fraction 77"" is an improper fraction%

Mixed fraction:

A combination of a proper fraction and a whole number is called a mixed fraction.

+13+3, 213+3, 3252, 4252, ++110++, #1315+3+ and +2353 are exampes of mixed

fraction%

T-o 12+2, ma*e a -hoe%

1212 1212

1212 + 1212 = 2222 = 1

.hat -i you /et if you add one more 12+2 to a -hoe0

1212 + 1212 +1212 = 1 + 2222 =

11212

No-, you ha1e three haf or you can say that you ha1e a -hoe and a haf or 12+2%

Numer such as +12+2 is a mixed numer%

1n mathematics$ an irrational number is a realnumber that cannot bee'pressed as a ratio of integers$ i.e. as a fraction. Therefore$ irrational

numbers$ when written as decimal numbers$ do not terminate$ nor do theyrepeat.

xampes:



% real number is any element of the set 6$ which is the union of the set ofrationalnumbers and the set of irrational numbers. 1n mathematical

e'pressions$ unnown or unspecified real numbers are usually

represented by lowercase italic letters u through z.

LCM and HCF

9rime Number # 9rime number is a number which has no !actors e2cetitsel! and Unity.

• Ex :2, 3, 5, 7, 11, 13, 17 etc are prime numbers

• Common Multiple : A Common Multiple of two or more numbers is a number

which is exactly divisible by each of them.

o Ex : 12 is a common multiple of 2, 3, 4 and 6

• Least Common Multiple (LCM) : The LCM of two or more given numbers is

the Least Number which is exactly divisible by each of them.

• Highest Common Factor (HCF) : The HCF of two or more numbers is the

Greatest Number which divides each of them Exactly.

o It is also Called Greatest Common Divisor (GCD)

:;< E2amle

;onsider the numbers '( and '3# The !actors o! '( are # ', (, ), , 8, '(.



The !actors o! '3 are # ', ), 3, '3.

' and ) are the only common !actors +numbers which are !actors

o! both '( and '30.

There!ore, the hihest common !actor o! '( and '3 is ).

Example Questions

Work out the answers to these questions and click the uttons marked to see

whether !ou are correct"

(a) #ind the hi$hest common %actor o% 20 and &0 ! %ollowin$ the steps elow'

=hat are the !actors o! (&>

=hat are the !actors o! )&>

=hat is the hihest common !actor o! (& and )&>

() #ind the hi$hest common %actor o% 1 and 12 ! %ollowin$ the steps elow'

=hat are the !actors o! '>

=hat are the !actors o! '(>

=hat is the hihest common !actor o! ' and '(>

The lowest common multile +L;M0 o! two whole numbers

is the smallest whole number which is a multile o! both.

L;M E2amle

;onsider the numbers '( and '3 aain#

The multiles o! '( are # '(, (, )8, 4, 8&, 6(, 4, .... The multiles o! '3 are # '3, )&, 3, 8&, 63, 5&, ....

8& is a common multile +a multile o! both '( and '30, and there are no

lower common multiles.

There!ore, the lowest common multile o! '( and '3 is 8&.

Example Questions

Work out the answers to these questions and click the uttons marked to see

whether !ou are correct"

(a) #ind the lowest common multiple o% and * ! %ollowin$ the steps elow'



=hat are the "rst ten multiles o! 3>


=hat is the lowest common multile o! 3 and 6>

() #ind the lowest common multiple o% and 10 ! %ollowin$ the steps elow'


=hat are the "rst ten multiles o! '&>

=hat is the lowest common multile o! 8 and '&>

We want to %ind the ,-# and .-/ o% the numers 0 and *2"

tart ! writin$ each numer as a product o% its prime %actors"

0 = 2 2 &

*2 = 2 2 2 & &

o make the next sta$e easier3 we need to write these so that each new prime %actor e$ins in

the same place'

0 = 2 2 &

*2 = 2 2 2 & &

4ll the 525s are now ao6e each other3 as are the 5&5s etc" his allows us to match up the

prime %actors"

he hi$hest common %actor is %ound ! multipl!in$ all the %actors which appear in oth lists'

o the ,-# o% 0 and *2 is 2 × 2 × & which is 12"

he lowest common multiple is %ound ! multipl!in$ all the %actors which appear

in either list'

o the .-/ o% 0 and *2 is 2 × 2 × 2 × & × & × which is &0"



irst -e find the east common mutipe (%5%M%) of 2$, 3! and 4%

Therefore, east common mutipe (%5%M%) of 2$, 3! and 4 & 2 6 2 6 3

6 3 6 6 " & +2!

Methods of finding HCF :

• HCF by factorization :

1.Express each of the given number as the product of Prime Factors

2.Choose common factors

3.Find the Product of Lowest Power of these Factors.

• This Product is the required HCF of the given Numbers

Ex : Find the HCF of 84, 540

If you find this method till confusing, don't worry. There is another method to find

HCF.



• HCF by Method of Division :

o Consider two different numbers.

o Divide the longer number by the smaller one.

o Now divide the divisor by the reminder.

o Repeat this process of dividing the preceding divisor by the last

reminder obtained, till you get the reminder "0"

o The LAST DIVISOR is the HCF of the given TWO numbers

Thats it. Now the answer is 14 :)

#7N8&63T17N3

+ hour & ! minutes

+ minute & ! seconds

+hour & ! minutes & 3! seconds (! 6 !)

+ day & 24 hours

+% 7o- many minutes are there in a year0

8oution:

+ year & 3! days%

+ day & 24 hours%



+ hour & ! minutes%

8o one year & (3! 6 24 6 !) minutes%

& ($"! 6 !) minutes%

& 2! minutes%

2% 7o- many hours are there in a year0

8oution:

.e *no-,

+ year & 3! days%

+ day & 24 hours%

8o in one year & (3! 6 24)

& "$! hours%

3% 7o- many minutes in ! hours0

8oution:

.e *no-,

+ hour & ! minutes%

8o ! hours & (! 6 !) minutes%

& 3! minutes%

4% 5on1ert 22 minutes to hours and minutes%

8oution:

.e *no- that ! minutes & + hour

22 minutes & (229!) hours

& 3 hours 4 minutes%



.rite the decima as a mixed numer -ith + as the denominator%

"0& = &

100



.rite the decima as a fraction -ith + as the denominator% educe thefraction to simpest form%

0"

=

10 = 7 2

10 7 2

= 2

"0 & = &

100 is -ritten in simpest form%.rite the decima as a fraction -ith + as the denominator% educethe fraction to simpest form%



TRIANGLE













Alebra - Basic ?e"nitions

=hat is an E@uation

An e@uation says that two thins are e@ual. t will have an e@uals sin $ lie

this#

x + 2 = 6

That e@uation says# what is on the left (x + 2) is eual to what is on the

right (!)

Co an e@uation is lie a statement this e@uals that

"arts of an #uation

Co eole can tal about e@uations, there are names !or diDerent arts +better

than sayin that thiny there0

:ere we have an e@uation that says 4x ; " e@uals , and all its arts#



A $ariable is a symbol !or a number we donFt now yet. t is usually a letter lie

2 or y.

A number on its own is called a Constant.

A Coe%cient is a number used to multily a variable +&x means & times x,

so & is a coeGcient0

Cometimes a letter stands in !or the number#

#xam'le ax2 + bx + c

x is a variable

a and b are coeGcients

c is a constant

An 'erator is a symbol +such as H, I, etc0 that shows an oeration +ie wewant to do somethin with the values0.

A *erm is either a sinle number or a variable, or numbers and variablesmultilied toether.

An #x'ression is a rou o! terms +the terms are searated by H or J sins0

Co, now we can say thins lie that e2ression has only two terms, or the

second term is a constant, or even are you sure the coeGcient is really >

#x'onents

http://www.mathsisfun.com/algebra/polynomials.html




http://www.mathsisfun.com/algebra/like-terms.html

http://www.mathsisfun.com/exponent.html



http://www.mathsisfun.com/algebra/like-terms.html




The exponent +such as the ( in 2(0 says how man

times to use the value in a multilication.

E2amles#

,2 - , . , - !&

/ - . .

20 - . . 0

E2onents mae it easier to write and use many multilications

E2amle# &02 is easier than . . . . 0 . 0, or even 00

"olnomial

E2amle o! a 9olynomial# /x2 + x 1 2

A poynomia can have constants, ariables and the ex'onents

345424/4666

But it never has division by a variable.

Monomial4 7inomial4 *rinomial

There are secial names !or olynomials with ', ( or ) terms#

Li8e *erms

i*e Terms are terms whose variables +and their exponents such as the

( in 2(0 are the same.

n other words, terms that are lie each other. +Note# the coe%cients can be

diDerent0

#xam'le



+'7)02y( -(2y( 82y(

Are all li8e terms because the variables are all x2

5) 3x−4(x+4)=2

Aly the distributive roerty.

3x+(−4x−4K4)=2

3x+(−4x−16)=2

3x−4x−16=2

Cubtract4x!rom 3xto et −x.

−x−16=2

−x=16+2

Add 16and 2to et 18.

−x=18

x=−18

2) −3̂3

Raise 3to the ower o!3to et 27.

−1K27

/) (9+8)K7

Add 9and 8to et 17.

17K7

Multily 17by 7to et 119.

119

&) 9(7−11x)

https://www.mathsisfun.com/square-root.html








9(7)+9(−11x)

−99x+63

uadratic E@uations

An e2amle o! a uadratic E@uation#

The name uadratic comes !rom @uad meanin s@uare, because the variable

ets suared +lie 2(0.

https://www.mathsisfun.com/algebra/expanding.html



https://www.mathsisfun.com/algebra/brackets.html

https://www.mathsisfun.com/algebra/degree-expression.html

https://www.mathsisfun.com/algebra/definitions.html


https://www.mathsisfun.com/algebra/brackets.html



t is also called an E@uation o! =e/ree ( +because o! the ( on the 20

Ctandard <orm

The Ctandard <orm o! a uadratic E@uation loos lie this#

a, b and c are nown values. a canFt be &.

2 is the 1ariae or unnown +we donFt now it yet0.

:ere are some more e2amles#

(2( H 32 H ) $

& n this one a$(, b$3 and c$)

2( J )2 $ &

This one is a little more tricy#

=here is a> =ell a$', and we donFt usually write '2(

b $ -)

And where is c> =ell c$&, so is not shown.

32 J ) $ &

os This one is not a @uadratic e@uation# it is missin 2(

+in other words a$&, which means it canFt be @uadratic0

:idden uadratic E@uations

Co the Ctandard <orm o! a uadratic E@uation is

ax2 > x > c &

But sometimes a @uadratic e@uation doesnFt loo lie that <or e2amle#

n disuise n Ctandard

<orma, b and c

2( $ )2 J 'Move all terms to le!t hand

side2( J )2 H ' $ &

a$', b$J),

c$'

(+w( J (w0

$ 3

E2and +undo

the bracets0,

and move 3 to le!t

(w( J w J 3

$ &

a$(, b$J,

c$J3

https://www.mathsisfun.com/algebra/factoring-quadratics.html

https://www.mathsisfun.com/algebra/completing-square.html






1+1J'0 $ ) E2and, and move ) to le!t 1( J 1 J ) $ &a$', b$J',

c$J)

:ow To Colve t>

The solutions to the uadratic E@uation are where it is e@ual to 1ero.

There are usually ( solutions

They are also called roots, or sometimes 1eros

There are ) ways to "nd the solutions#

'. =e can actor the ?uadratic +"nd what to multily to mae the uadratic

E@uation0

(. =e can 5ompete the 8uare, or

). =e can use the secial uadratic <ormula#

Oust lu in the values o! a, b and c, and do the calculations.

About the uadratic <ormula

9lus7Minus

The @ means there are T= answers#

:ere is why we can et two answers#

But sometimes we donFt et two real answers, and the ?iscriminant shows why

...

?iscriminant

?o you see b( J ac in the !ormula above> t is called the ?iscriminant, because

it can discriminate between the ossible tyes o! answer#

https://www.mathsisfun.com/numbers/real-numbers.html

https://www.mathsisfun.com/numbers/complex-numbers.html



when b( - ac is ositive, we et two ea solutions

when it is 1ero we et /ust NE real solution +both answers are the same0

when it is neative we et two 5ompex solutions

Complex solutions? LetFs tal about them a!ter we see how to use the !ormula.

Usin the uadratic <ormula

Oust ut the values o! a, b and c into the uadratic <ormula, and do the

calculations.

E2amle# Colve 32P H 82 H ' $ &

;oeGcients are# a $ 3, b $ 8, c $ '

uadratic <ormula# 2 $ −b ± √(b2 − 4ac)(a

9ut in a, b and c# 2 $ −6 ± √(62 − 4×5×1)(I3

Colve# 2 $ −6 ± √(36 − 20)'&

2 $ −6 ± √(16)'&

2 $ −6 ± 4'&

2 $ J&.( or J'

Answer# 2 $ J&.( or 2 $ J'

And we see them on this rah.

;hec -&.(# 3I+J&.(0P H 8I+J&.(0 H '

$ 3I+&.&0 H 8I+J&.(0 H '

$ &.( J '.( H ' $ &

;hec -'# 3I+J'0P H 8I+J'0 H '

$ 3I+'0 H 8I+J'0 H '

$ 3 J 8 H ' $ &


https://www.mathsisfun.com/numbers/imaginary-numbers.html







2 $ −b ± √(b2 − 4ac2a

! ne"ati#e bo$ %as thin&in" $es o' no about "oin" to a pa't$

at the pa't$ he tal&e to a s*ua'e bo$ but not to the 4 a%esome "i'ls+t %as all o#e' at 2 am,

;omle2 Colutions>

=hen the ?iscriminant +the value b( J ac0 is neative we

et 5ompex solutions ... what does that mean>

t means our answer will include Ima/inary Numers . =ow

E2amle# Colve 32P H (2 H ' $ &

;oeGcients are# a $ 3, b $ (, c $ '

Note that the ?iscriminant is neative# b( J ac $ (( J I3I' $ -'8

Use the uadratic <ormula# 2 $ −2 ± √(−16)'&

The s@uare root o! -'8 is i

+i is Q-', read mainary Numbers to "nd out more0

Co# 2 $ −2 ± 4i'&

Answer# 2 $ J&.( &.i

The rah does not cross the 2-a2is.

That is why we ended u with comle2numbers.

n some ways it is easier# we donFt need more calculation, /ust leave it as ;%2 @

%4i.

Cummary

uadratic E@uation in Ctandard <orm# a2( H b2 H c $ &

uadratic E@uations can be factored



uadratic <ormula# x & −b ± √(b2 − 4ac2a

=hen the ?iscriminant +b(Jac0 is#

ositive, there are ( real solutions

1ero, there is one real solution

neative, there are ( comle2 solutions

Simultaneous Equations

5oncept

8imutaneous euations are a set of euations -hich ha1e more than one un*no-n 1aues%?uestions in1o1in/ simutaneous euations reuire students to find the un*no-ns% irst, -e ha1e

to represent the euations -ith different numers or etters for cear expanation% Then -e proceed

-ith the eo- steps%

There are /eneray t-o methods to so1in/ simutaneous euations%

A By sustitution

A By eimination

It may e etter to use one method o1er the other for certain type of simutaneous euations

uestion% Cny -ith practice, -i you e ae to deduce -hich method est suits that specific

uestion%

xampe

8o1e the simutaneous euations 2x > y & and x > 2y & "

ememer that -e first ha1e to represent the euations -ith proper symos%

(+) 2x > y &

(2) x > 2y& "

.e sha so1e it first -ith the sustitution method and sho- the eimination method at a ater

sta/e%

Method of 8ustitutionIn the method of sustitution, -e express x in terms of y in one

euation and sustitute it into the other%

rom (2) , x & " D 2y (3)

8ustitute (3) into (+)%

2("D2y) > y &



y & 3

8ustitute y & 3 into (+) or (2)%

x & +

Method of iminationet us earn ho- to so1e simutaneous euations usin/ eimination%

Ta*in/ the ao1e exampe, -e can choose to eiminate x%

2x > y & (+)

x > 2y & " (2)

To eiminate, say, x:

Mutipy to otain the same numer of xEs on oth euations and cance the xEs y sutraction%

This ea1es y%

2 x (2) : 2x > 4y & +4 (3)

(3) D (+) : 3y & #

y & 3

F/ain y sustitutin/ y & 3 into (+) or (2),

-e arri1e at the same concusion that x & +%

Both methods are euay suited to the soution of this proem%In any case, the aim is to reduce

the t-o un*no-ns to one throu/h manipuation of oth the euations%

Simultaneous Equations

3imultaneous e9uations and linear e9uations$ after studying this section$ you will be able to"

• solve simultaneous linear e9uations by substitution

• solve simultaneous linear e9uations by elimination

1f an e9uation has two unnowns$ such as *y ' : *0$ it cannot have uni9ue solutions. Two

unnowns re9uire two e9uations which are solved at the sametime (simultaneously) ; but even then

two e9uations involving two unnowns do not always give uni9ue solutions.

3olve the two simultaneous e9uations"*y ' : < /

y : *' */



from */ y : *' 2 = subtract from each side3ubstituting this value for y into / gives" *(*' > ) ' : < ,' > * ' : < = e'pand the bracets

-' > * : < =tidy up-' : 0 =%dd * to each side' : * =?y dividing both sides by - the value of ' is found. 3ubstitute the value of ' into y : *' > givesy : , 2 : + 3o ' : * and y : +

N7T&"

• 1t is a good idea to label each e9uation. 1t helps you e'plain what you are doing ; and may

gain you method mars.

• This value of ' can be substituted into e9uation / or */$ or into the e'pression for y" y : *' ;

.

• #hoose the one that is easiest@

• %s a chec$ substitute the values bac into each of the two starting e9uations.

•

The second method is called solution by elimination!

N7T&"The method is not 9uite as hard as it first seems$ but it helps if you now why it wors.

1t wors because of two properties of e9uations"

• 4ultiplying (or dividing) the e'pression on each side by the same number does not alter the

e9uation.

• %dding two e9uations produces another valid e9uation"

e.g. *' : ' 0 (' : 0) and ' ; + : A (' also : 0).

%dding the e9uations gives *' ' ; + : ' 0 A (' also : 0).

The obBect is to manipulate the two e9uations so that$ when combined$ either the ' term or the y term

is eliminated (hence the name) ; the resulting e9uation with Bust one unnown can then be solved"

Cere we will manipulate one of the e9uations so that when it is combined with the other e9uationeither the ' or y terms will drop out. 1n this e'ample the ' term will drop out giving a solution for y. Thisis then substituted into one of the otiginal e9uations. Dabel your e9uations so you now which one your are woring with at each stage. &9uation / is *y ' : <

&9uation */ is y : *' 6earrange one e9uation so it is similar to the other.



*/ y > *' : 2 also * ' / gives ,y *' : E which we call +/ */ y > *' : 2

+/ ,y *' : E */ +/ gives -y : -so y : + substituting y : + into / gives (+) : *'so *' : ,$ giving ' : * and y : +

"ultiplying and #i$iding #ecimals

Introduction %s with whole numbers$ sometimes you run into situations where you need to multiply ordivide decimals. %nd Bust as there is a correct way to multiply and divide whole numbers$ so$too$ there is a correct way to multiply and divide decimals. 1magine that a couple eats dinner at a Fapanese steahouse. The bill for the meal is G-<.+*

Hwhich includes a ta' of G,.E,. To calculate the tip$ they can double the ta'. 3o if they nowhow to multiply G,.E, by *$ the couple can figure out how much they should leave for the tip.

http://void%280%29/

http://void%280%29/

http://void%280%29/



CereIs another problem. %ndy Bust sold his van that averaged *0 miles per gallon ofgasoline. Ce bought a new picup truc and too it on a trip of E,.*- miles. Ce used +.-gallons of gas to mae it that far. id %ndy get better gas mileage with the new trucJ ?oth of these problems can be solved by multiplying or dividing decimals. CereIs how to doit."ultiplying #ecimals

4ultiplying decimals is the same as multiplying whole numbers e'cept for the placement of

the decimal point in the answer. When you multiply decimals$ the decimal point is placed in

the product so that the number of decimal places in the product is the sum of the decimal

places in the factors. DetIs compare two multiplication problems that loo similar" *, • +E$

and *., • +.E.

('

2 )8'(48(&9493&

('.

2 ).8'(48(&9963&

Notice how the digits in the two solutions are e'actly the same > the multiplication does not

change at all. The difference lies in the placement of the decimal point in the final answers"

*, • +E : A$A0,$ and *.,• +.E : AA.0,.To find out where to put the decimal point in a

decimal multiplication problem$ count the total number of decimal places in each of the

factors.

*., the first factor has one decimal place

+.E the second factor has one decimal place

AA.0, the product will have : * decimal places

Note that the decimal points do not have to be aligned as for addition and subtraction.

E%ample

Kroblem &!'( )!* + ?

+.0,' E.

+0,<*,0<-,,

3et up the problem. 4ultiply. %dd.

#ount the total numberof decimal places in the

factors and insert thedecimal point in the



+.0,' E.

+0,

<*,0<.-,,

==

=

product. * decimal places. decimal place.

+ decimal places.

Answer +.0, • E. : <.-,,

3ometimes you may need to insert zeros in front of the product so that you have the rightnumber of decimal places. 3ee the final answer in the e'ample below"

E%ample

Kroblem '!'&, '!'- + ?

0.0+A' 0.0<*LE

3et up the problem. 4ultiply.

0.0+A' 0.0<

0.00*LE

==

=

#ount the total numberof decimal places in thefactors and insert thedecimal point in theproduct. + decimal places.* decimal places.

- decimal places. Answer 0.0+A • 0.0< : 0.00*LE Note that you needed to

add zeros before *LE toget the - decimalplaces.

1f one or more zeros occur on the right in the product$ they are not dropped until after thedecimal point is inserted.

E%ample

Kroblem .!'( *!/0 + ?

*.0,' .L-

0*0<+E0*0,00+LA<0

3et up the problem. 4ultiply.

%dd.

*.0,' .L-

0*0

==

* decimal places.* decimal places.



<+E0*0,00+.LA<0

=

, decimal places.

Answer *.0, • .L- : +.LA< %nswer can omit thefinal trailing 0.

"ultiplying #ecimals To multiply decimals"

• 3et up and multiply the numbers as you do with whole numbers.

• #ount the total number of decimal places in both of the factors.

• Klace the decimal point in the product so that the number of decimal places in theproduct is the sum of the decimal places in the factors.

• Meep all zeros in the product when you place the decimal point. ou can drop thezeros on the right once the decimal point has been placed in the product. 1f the

number of decimal places is greater than the number of digits in the product$ you caninsert zeros in front of the product.

"ultiplying by Tens

Tae a moment to multiply ,.,EL by 0. Now do ,.,EL • 00. Oinally$ do ,.,EL • $000.

Notice any patterns in your productsJ

.852 '&.85&

.852 '&&8.5&&

.852 '&&&85.&&&

Notice that the products eep getting greater by one place value as the multiplier (0$ 00$

and $000) increases. 1n fact$ the decimal point moves to the right by the same number of

zeros in the power of ten multiplier.

,.,EL • 0 : ,,!EL

P

,.,EL • 00 : ,,E!L

P

,.,EL • $000 : ,,EL!

P

ou can use this observation to help you 9uicly multiply any decimal by a power of ten (0$

00$ $000$ etc).

E%ample

Kroblem '!'& *'' + ?

0.0+ • '' : J 00 has two

http://void%280%29/



zeros.

0.0+ • 00 : + 4ove the

decimal point

two places to

the right to find

the product.

Answer 0.0+ • 00 : +

"ultiplying a #ecimal by a 1o2er of Ten

To multiply a decimal number by a power of ten (such as 0$ 00$ $000$ etc.)$ count the

number of zeros in the power of ten. Then move the decimal point that number of places

to the right.

Oor e'ample$ 0.0-, • 00 : -.,. The multiplier 00 has two zeros$ so you move the

decimal point in 0.0-, two places to the rightHfor a product of -.,.

#i$iding #ecimals

To divide decimals$ you will once again apply the methods you use for dividing whole

numbers. Doo at the two problems below. Cow are the methods similarJ

Notice that the division occurs in the same wayHthe only difference is the placement of the

decimal point in the 3uotient.

http://void%280%29/

http://void%280%29/



E%ample

Kroblem *-!&. 4 - + ?

SSSSSSSS 8 ) 1 8.3 2

Cet u theroblem.

(.( 5 8 ) 1 8.3 2

-' 8 2 3

-' 8 7 2

- 6 ( 0

?ivide.

(.( 5 8 ) 1 8.3 2

9lace decimaloint in the@uotient. tshould belaceddirectly abovethe decimaloint in thedividend.

Answer <.+* Q < : *.*L

1n cases lie this$ you can use powers of 0 to help create an easier problem to solve. 1n this

case$ you can multiply the di$isor $ 0.+$ by 0 to move the decimal point place to the right.

1f you multiply the divisor by 0$ then you also have to multiply the di$idend by 0 to eep

the 9uotient the same. The new problem$ with its solution$ is shown below.

E%ample

Kroblem .)'!* 4 '!& + ?

SSSSSSSS 0.3 ) 2 6 0.1

Cet u theroblem.

SSSSSSSSSS

3. ) 2 6 0 1.

Multily

divisor anddividend by



'& to create awhole numberdivisor.

4 8 63 ) 2 6 0 1

-( 2 0

-' 4 2 1

- ( ' 0

?ivide.

!ns%e' (8&.' &.) $ 486

7ften$ the dividend will still be a decimal after multiplying by a power of 0. 1n this case$ the

placement of the decimal point must align with the decimal point in the dividend.

E%ample

9roblem 5:629: ; /62: - <

SSSSSSSSSS 3.2 5 ) 1 5.2 7 5

3et up the problem.

SSSSSSSSSSS 3 2 5.) 1 5 2 7.5

4ultiply divisor and

dividend by 00 to

create a whole number

divisor.

.63 2 5.) 1 5 2 7.5 -' ) & & 2 2 7 5

-( ( 6 3 0

ivide. +*- goes into

-*A four times$ so the

number , is placed

above the digit A.

The decimal point in

the 9uotient is placed

directly above the

decimal point in the

dividend.

!ns%e' '3.(63 ).(3 $ .6



#i$iding #ecimals

#i$iding #ecimals by Whole Numbers

ivide as you would with whole numbers. Then place the decimal point in the 9uotientdirectly above the decimal point in the dividend.

#i$iding by #ecimals

To divide by a decimal$ multiply the divisor by a power of ten to mae the divisor a whole

number. Then multiply the dividend by the same power of ten. ou can thin of this as

moving the decimal point in the dividend the same number of places to the right as you

move the decimal point in the divisor.

Then place the decimal point in the 9uotient directly over the decimal point in the

dividend. Oinally$ divide as you would with whole numbers.

#i$iding by Tens

6ecall that when you multiply a decimal by a power of ten (0$ 00$ $000$ etc)$ the

placement of the decimal point in the product will move to the right according to the number

of zeros in the power of ten. Oor instance$ ,.* • 0 : ,.*.

4ultiplication and division are inverse operations$ so you can e'pect that if you divide a

decimal by a power of ten$ the decimal point in the 9uotient will also correspond to the

number of zeros in the power of ten. The difference is that the decimal point moves to theright when you multiplyR it moves to the left when you divide.



1n the e'amples above$ notice that each 9uotient still contains the digits ,,ELHbut as

another 0 is added to the end of each power of ten in the divisor$ the decimal point moves an

additional place to the left in the 9uotient.

#i$iding by 1o2ers of Ten

To divide a decimal by a power of ten (0$ 00$ $000$ etc.)$ count the number of zeros in

the divisor. Then move the decimal point in the dividend that number of decimal places

to the leftR this will be your 9uotient.

E%ample

Kroblem &*!'0 4 *' + ?

+.0- Q ' : J 0 has one

zero.

+.0- Q 0 : +.0- 4ove the

http://hotmath.com/hotmath_help/topics/rounding-decimals.html



decimal point

one place to the

left in the

dividendR this is

the 9uotient.

Answer +.0- Q 0 : +.0-

=iiding with =ecimals

8i6idin$ with decimals is a it more di%%icult" hese da!s3 most teachers don9t mind much i%

!ou use a calculator" :ut it9s $ood to know how to do it !oursel%3 too3 and !ou alwa!s need to

e $ood at estimatin$ the answer3 so !ou can make sure the calculator9s answer is reasonale"

;ecall that in the prolem x 7 y = z 3 also written

x is called the dividend3 y is the divisor3 and z is the quotient"

Step 1: Estimate the answer ! roundin$" <ou9ll use this estimate to check !our answer later"

Step 2: % the di6isor is not a whole numer3 then mo6e the decimal place n places to the ri$ht

to make it a whole numer" hen mo6e the decimal place in the di6idend the same numer o%

places to the ri$ht (addin$ some extra >eros i% necessar!")

Step 3: 8i6ide as usual" % the di6isor doesn9t $o in e6enl!3 add >eros to the ri$ht o% the

di6idend and keep di6idin$ until !ou $et a 0 remainder3 or until a repeatin$ pattern shows up"

Step 4: ?ut the decimal point in the quotient directl! ao6e where the decimal point now is in

the di6idend"

Step 5: -heck !our answer a$ainst !our estimate to see i% it9s reasonale"

Example:

8i id

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Basics of Maths for all school students

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