Mathematics Decimal Notes

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    MATHEMATICS - FORM 1

    DECIMALS

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    DECIMAL AND FRACTION

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    1. A decimal is a fraction in !ic! t!e denominator 

      is 1" or a #oer of 1".

      For e$am#le %-

     &. T!e decimal #oint ' t!e dot ( se#arates

    t!e !ole n)m*er from its fractional #art.

      A n)m*er ritten it! a decimal #oint is

    +non as a decimal.

    For e$am#le%-

     ,. T!e n)m*er of diits after t!e decimal #oint

    is t!e same as t!e n)m*er of eros in t!e

    denominator.

      For e$am#le%-

     

    A( Re#resentin Decimals it! Diarams

    Decimals can *e re#resented it! diarams.

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    For e$am#le %-

      1 , /  1" 1"" 1 """

    '1 out of 10 ( ' 3 out of 100 ( ' 7 out of 1 000 (

    0( ritin Decimals *ased on i2en Diarams

    or+ed E$am#le

     

    Write a decimal to represent the shaded

    area in the diagram above.

    Solution

    The shaded area is 3 .

      10

    Therefore, the decimal is 0. 3.

    C) Conversion of Fractions into ecimals

     and vice versa

    1. ! fraction in "hich the denominator is

    10 or a po"er if 10 can be converted to a

      decimal mentall#. 

    Wor$ed %&ample 

    %&press the follo"ing as decimals.

    'a) ( 'c) 13

      10 1 000

    'b) *+

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      100

    Solution

    Wor$ed %&ample

    Change the follo"ing into fractions "ith

    denominators of 10 or po"ers of 10.

    'a) 0. 'b) 0.** 'c) 0.3(-

    Solution

    +. ! fraction can also be converted to a

    decimal b# dividing the numerator b#

    the denominator.

    3. i&ed numbers can also be e&pressed

      as decimals.

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    Wor$ed %&ample

    %&press the follo"ing fractions as decimals.

    'a) 3 'b) ( +  - *

    Solution

    or+ed E$am#le

    %&press the follo"ing as decimals.

    'a) 1 1- 'b) - *  100 +0

    Solution

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    or+ed E$am#le

    %&press the follo"ing decimals as mi&ed

    numbers.

    'a) 3. 'b) 11. 0+*

    Solution

    'a) 3. / 3 0.

      / 3

      10

      / 3   10

    3LACE 4AL5E AND DI6IT 4AL5E

     IN DECIMALS

    A( 3lace 4al)e and Diit 4al)e

    1. %ach digit in a decimal has its o"n

      place value and digit value.

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      oo$ at the decimal .+3* belo".

     Wor$ed %&ample

    State the place value and digit value

    of the underlined digit in each of the

    follo" decimals.

    'a) 13.- 'b) (.02*

    Solution

    'a) Tenths 0.-

    'b) Thousandths 0.00*

    +. ecimal places are the number of

    places occupied b# the digits after

    the decimal point.

      For e&ample 45

      'a) 0.3

      1 decimal place

      'b) 0.0

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      2 decimal places

      'c) 3.1*+

      3 decimal places

      'd) 2.(-+0

      4 decimal places

    6) 7epresenting ecimals on 8umber ines

      ! decimal can be represented on a number line.

      For e&ample 45

     

    Wor$ed %&ample

    State the decimal represented b# each

    of the letters on the number line above.

    Solution

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    9ne part / -.2* 5 -.0

      / 0.1*

    F  / -.2* 0.1* 0.1*

      / *.+*

    G  / *.+* 0.1*

      / *.-0

    Therefore, F / *.+* and G  / *.-0.

    C) Comparing T"o ecimals 

    When comparing t"o decimals, arrange the

    digits of the decimals according to their place

    values and then compare their digit values

    from left to right.

    Wor$ed %&ample

    Which is greater, +-.*3 or +-.3+

    Solution

    !rrange the decimals as sho"n belo".

     

    !t the place value of hundredths, * is larger than 3.

    Therefore, +-.*3 is greater than +-.3+.

    Wor$ed %&ample

    Fill in each bo& "ith : ;: or :

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    Solution

    ) 9rder of ecimals

    We can arrange decimals in increasing or

    decreasing order according to their values.

    Wor$ed %&ample

    !rrange (.1, .-, .3, .1 in decreasing order.

    Solution

    Wor$ed %&ample

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    !rrange 11 , +.(3 and +.(3+ in increasing order.

      -

    Solution

    %) 7ounding 9ff ecimals. 

    ! ecimal can be rounded off to a "hole number

    or certain decimal places. The rules for rounding

    off decimals are as follo"s.

    'a) oo$ at the digit to the right of the digit that

    is to be rounded off.

    'b) =f the digit is more than or e>ual to *, add 1

      to the digit that is to be rounded off and ignore

      the rest of the digits after it.

    'c) =f the digit is less than *, maintain the digit

      that is to be rounded off and ignore the rest of 

      the digits after it.

    Wor$ed %&ample

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    7ound off .3(*+ correct to

    'a) 1 decimal place,

    'b) + decimal places,

    'c) 3 decimal places.

    Solution

    Wor$ed %&ample

    7ound off the follo"ing decimals to the

    nearest "hole numbers.

    'a) .1 'c) 3*+.-

    'b) 30.3+-

    Solution

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    ADDITION AND S50TRACTIONOF DECIMALS

    1. ecimals are added and subtracted in

      the same "a# as "hole numbers.

    +. When t"o decimals are added together

    or subtracted from each other, the deci5

      mal points must be placed directl# one

    belo" the other and the digits "ritten in

      the correct place value columns.

    !) !ddition of ecimals

    Wor$ed %&ample

    Calculate

    'a) 0.3+ -.+- 'b) * .

    Solution

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    Wor$ed %&ample

    Calculate

    'a) 3-.+ +.(3 2.2*

    'b) 2.+* *- 1(.31

    Solution

    6) ?roblem Solving involving

    !ddition off ecimal

    Wor$ed %&ample

    Salman is 1.(* m tall and !r@un is 0.12 m

    taller than Salman. Ao" tall is !r@un.

    Solution

    1. Bnderstand the problem

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      iven information 4

      Salman:s height / 1.(* m

      !r@un is taller than Salman b# 0.12 m

      Find 4 Aeight of !r@un

    +. evise a plan

      Bse addition.

    3. Carr# out the plan

      !r@un:s height

    / Salman:s height 0.12 m

      1.(* m 0.12 m / 1.2- m

      1 . ( *

    0 . 1 2

    1 . 2 -

    Therefore, !r@un is 1.2- m tall.

    -. Chec$ 

      1 . 2 -

    5 1 . ( *

      0 . 1 2 ' The difference )

    Wor$ed %&ample

    e"i collected +1.10 litres of late& on

    onda#, + litres on Tuesda# and

    +3.20 litres on Wednesda#. Find the

    total amount of late& collected.

    Solution

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    The total amount of late& collected "as

    2.20 litres.

    C) Subtraction of ecimal

    Wor$ed %&ample

    Calculate

    'a) 0.* 5 0.3*

    'b) 3 5 +.-+1

    Solution

    Wor$ed %&ample

    Calculate

    'a) *(.3 5 3*.+ 5 +.3-

    'b) 1 5 +-.20( 5 (.*

    Solution

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    ) ?roblem Solving involving

    Subtraction of ecimals

    Wor$ed %&ample

    SuDiela "eighs *+.03 $g and !Dman "eighs 0.-

    $g. Find the difference in their bod# mass.

    Solution

    1. Bnderstand the problem

      iven information 4

      SuDiela:s mass / *+.03 $g

      !Dman:s mass / 0.- $g

      Find 4 ifference in their bod# mass

    +. evise a plan

      Bse subtraction.

    3. Carr# out the plan

    0.- $g 5 *+.03 $g

     

    Therefore, the difference in their bod#

      mass is .3( $g.

    -. Chec$ 

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      * + . 0 3

      . 3 (

      0 . - 0

    Wor$ed %&ample

    Eaiton bought .3 m of cloth. She used +.1* m

    to ma$e a shirt and 3.+ m to ma$e a pair of

    pants. Ao" much of the cloth remained

    Solution

    .3 m 5 +.1* m 5 3.+ m / 0.( m

     

    Therefore, 0.( m of the cloth remained.

    M5LTI3LICATION AND DI4ISION OF DECIMALS

    A( M)lti#lication of Decimals

    i - 6enerel m)lti#lication of decimals

    1. To find t!e #rod)ct of decimals7 m)lti#l8 t!e

      n)m*ers in t!e same a8 as for !ole n)m-

      *ers first. T!en7 #)t in t!e decimal #oint.

    &.T!e n)m*er of decimal #laces in t!e anser 

      m)st corres#ond to t!e total n)m*er of decimal

      #laces in t!e decimals *ein m)lti#lied.

    or+ed E$am#le

    Calc)late

    'a( ".9, $ ".9

    Sol)tion

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    ii( M)lti#l8in a decimal *8 a #oer of 1"

    !en m)lti#l8in a decimal *8 1"7 1""7

    1 """7 ect.7 e mo2e t!e decimal #oint 17

    &7 ,7 ect. #laces res#ecti2el8 to t!e ri!t.

    or+ed E$am#le

    Calc)late

    'a( 1.:& 'c( ".;

    '*(

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      ".17 "."17 ".""17 ect.7 e mo2e t!e decimal

    #oint 17 &7 ,7 ect. 3lace res#ecti2el8 to t!e

    left.

    or+ed E$am#le

    Calc)late

    'a( &:.; $ ".17

    '*( =,.

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    0( 3ro*lem Sol2in in2ol2in M)lti#lication

    of Decimals 

    or+ed E$am#le

    Dalina needs "./= + of flo)r to ma+e a ca+e.

    Ho m)c! flo)r is needed to ma+e ; ca+es >

    Sol)tion

    1. 5nderstand t!e #ro*lem

      6i2en information %

      Amo)nt of flo)r to ma+e 1 ca+e ? "./=

      Find % Amo)nt of flo)r to ma+e ; ca+es

    &. De2ise a #lan

      5se m)lti#lication.

    ,. Carr8 o)t t!e #lan

    ; $ "./= + ? 9.

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      M)lti#l8 aain to 2erif8 t!e anser.

    or+ed E$am#le

    A a##le costs RM".9". T!e #rice of a @ac+fr)it

    is :.; times t!e #rice of t!e a##le. !at is t!e

    cost of = @ac+fr)its >

    Sol)tion

    Cost of a a##le ? RM".9"

    Cost of a @ac+fr)it ? :.; $ RM".9"

    Cost of < @ac+fr)its ? = $ :.; $ ".9"

      ? RM1

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    or+ed E$am#le

    Calc)late

    'a( ".9 B 1"7 'c( ,;.:< B 1 """.

    '*( &,1.: B 1""7

    Sol)tion

    ii - Di2idin a !ole n)m*er *8 a !ole n)m*er 

    or+ed E$am#le

    Calc)late ,= B :

    Sol)tion

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    T!erefore7 ,= B : ? ;.9=

    or+ed E$am#le

    Calc)late ; B , and i2e 8o)r anser correct

    to : decimal #laces.

    Sol)tion

    T!erefore7 ; B , ? &.

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    '*( ".&& B &&

    Sol)tion

    &. !en di2idin a !ole n)m*er *8 a decimal7

      con2ert t!e di2isor to a !ole n)m*er first *8

      mo2in its decimal #oint to t!e ri!t.

    or+ed E$am#le

    Find t!e 2al)e of eac! of t!e folloin.

    'a( < B ".&

    '*( ,,".11

    Sol)tion

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    i2 - Di2idin a decimal *8 a decimal

    !en di2idin a decimal *8 a decimal7 )se t!e

    idea of e)i2alent fractions to con2ert t!e di2isor 

    to a !ole n)m*er. S!ift t!e decimal #oints t!e

    same n)m*er of #laces in t!e di2idend and di2i-

    sor to ma+e t!e di2isor a !ole n)m*er.

    or+ed E$am#le

    Find t!e 2al)e of eac! of t!e folloin.

    'a( =.=& B ".<

    '*( "."9&  ".;

    Sol)tion

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    2 - Di2idin a decimal *8 a fraction and 2ice 2ersa

    !en di2idin a dec$imal *8 a fraction7

    c!ane t!e o#eration from di2ision to

    m)lti#lication and in2ert t!e fraction at

    t!e same time.

    or+ed E$am#le

    Find t!e 2al)e of eac! of t!e folloin.

    'a( ,.9& B ,  :

    Sol)tion

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    D( 3ro*lem Sol2in in2ol2inDi2ision of Decimals

    or+ed E$am#le

    A+mal *)8s < tic+ets for a circ)s s!o

    costin RM;=.;". Ho m)c! does eac!

    tic+et cost >

    Sol)tion

    1. 5nderstand t!e #ro*lem

    6i2en information %

      N)m*er of tic+ets *o)!t ? <

      Cost of < tic+ets ? RM ;=.;"

      Find % Cost of 1 tic+ets

    &. De2ise a #lan

      5se di2ision

    ,. Carr8 o)t t!e #lan

      RM;=.;" B < ? RM1:.,"

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    T!erefore7 t!e cost of 1 tic+et is RM1:.,".

    :. C!ec+

      1 : . , "  $ <  ; = . ; "

    COM0INED O3ERATIONS

    OF 7 -7 $7 B OF DECIMALSA( Com*ined O#erations of Additionand S)*traction

    or+ed E$am#le

    Find t!e 2al)e of eac! of t!e folloin.

    'a( 9.:1

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    0( 3ro*lem Sol2in in2ol2in Addition and S)*traction

    or+ed E$am#le

    A 2essel ei!in "./; + contains &=.;

    + of rice. If 1".= + of t!e rice is )sed7

    find t!e total mass of t!e 2essel it! t!e

    remainin rice.

    Sol)tion

    1. 5nderstand t!e #ro*lem

    6i2en information %

      Mass of 2essel ? "./; +

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      Mass of rice ? &=.; +

      Mass of rice )sed ? 1".= +

      Find % Mass of t!e 2essel and remainin rice

    &. De2ise a #lan

    3erform addition folloed *8 s)*traction.

    ,. Carr8 o)t t!e #lan

    "./; &=.; - 1".=

      ? &

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    Sol)tion

    'a( 9 $ " . / B :  =

      ?

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      * & -. + G 11  +

      / * & -. + G 3

      +

      / 12+. G 3

    +

      / 12+. & +

      3

      / 3*. + G 3

      / 1+. -

    -. Chec$ 

      3 & 1+. - / 12+.

      +

      12+. G 3

      / -. +

    Wor$ed %&ample

    !m# bought +0 eggs costing 73. 0.

    issa bought 3* of those eggs. Ao"

    much did issa pa#

    Solution

     

    Cost of +0 eggs / 73.0

    Cost of 3* eggs

    / 3.0 G +0 & 3*

    / 0.12 & 3*

    / .*

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    Therefore, issa paid 7.*.

    %) Combined 9perations of !ddition, 

    Subtraction, ultiplication and ivision

    Wor$ed %&ample

    Calculate each of the follo"ing.

    'a) 1+. * 5 0. + & +.

    'b) 3. + & - 5 10. - G -

    'c) *. 1+ 5 +. G 1

    -

    Solution