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Mathematics Class V
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UNIT 6
FACTORS AND MULTIPLES
• DEFINITION OF FACTORS AND MULTIPLES. • TESTS FOR DIVISIBILITY. • PRIME AND COMPOSITE NUMBERS • PRIME FACTORISATION. • LCM AND HCF. • RELATION BETWEEN LCM AND HCF • WORD PROBLEM OF LCM AND HCF
DEFINITION OF FACTORS AND MULTIPLES
• FACTORS: A factor is a number that can divide another number without leaving a remainder. For e.g.. 24 is divisible by 2,3,4,6,8,12 and 24 so all these are factors of 24
• MULTIPLES: Multiples of any numbers are the numbers which are exactly divisible by the number.
• For e.g. Multiples of 4 are 4x1=4, 4x2=8, 4x3=12
REMEMBER
• 1 is the smallest factor of every number. • A factor of any number is always less than or equal to that number. • Every number is a multiple of 1. • Multiples are infinite while factors are limited. • Every number is a multiple of itself.
PRIME AND COMPOSITE NUMBER
PRIME NUMBER: A number greater than 1 and having only two factors, is known as prime number.
E.g. 2,3,5,7
NOTE: There is no even prime number except 2. COMPOSITE NUMBER: A number greater than 1 and having more
than two factors, is known as composite number. e.g. 4,6,8,10
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TWIN PRIMES: Two prime numbers with a difference of 2 are called TWIN PRIMES.E.g.3and 5 (difference=2)
PRIME FACTORIZATION
DEFINITION: When every factor of a number is a prime number, the
factorisation of that number is known as prime factorisation of the
number.
Example: 18 = 2 x 9
= 2 x 3 x 3 2 and 3 are the prime factors of 18
HCF (HIGHEST COMMON FACTOR)
DEFINITON: The greatest number that divides the numbers exactly is
called the HCF of the number
Example: Let us take two composite numbers 18 and 24
Factors of 18 = 1,2,3,6,9,18
Factors of 24 = 1,2,3,4,6,8,12,24
Common factors = 1,2,3,6
6 is the highest common factor so HCF = 6
TEST FOR DIVISIBILITY
DIVISIBILITY RULES FOR 2, 3,4,5,6,9,10
2 -A number is exactly divisible by 2,if the last digit is 0,2,4,6 or 8. e.g.
30, 648, 122
3 - A number is exactly divisible by 3 if the sum of its digits is exactly
divisible by 3.e.g. 231 2+3+1=6 and 6/2=3.
5 -A number is exactly divisible by 5 if the last digit is 0 or 5. e.g. 305,
500
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4 - A number is exactly divisible by 4 if the last two digits are divisible
by 4 for e.g. 24
8 - A number is exactly divisible by 8 if the last three digits are divisible
by 8 e.g. 888
6 - A number is exactly divisible by 6, if it is exactly divisible by both 2
and 3. e.g. 48 48/2=24 48/3= 16
9 - A number is exactly divisible by 9 if the sum of its digits is exactly
divisible by 9 e.g. 153 1+5+3=9 and 9/9=1
10 - A number is exactly divisible by10 if the last digit is 0.e.g. 20, 30
METHOD FOR FINDING HCF
HCF by Prime factorisation method
Example-Find the HCF of 8 and 4 by prime factorisation method.
Methods of finding HCF
• HCF by Prime factorisation method
• Example-Find the HCF of 8 and 4 by prime factorisation method
Solution
8
2 X 4 4
2 X 2 X 2 2 X 2
8= 2X2X2
4= 2X2
HCF=2X2=4
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HCF BY DIVISION METHOD.
EXAMPLE: Find the HCF of 20 and 24 by DIVISION.
Solution -
2 20 2 24
2 10 2 12
5 5 2 6
1 3 3
1
20 = 2×2×5 24= 2×2×2×3
Common factor = 2×2
H C F= 2×2 = 4
HCF BY LONG DIVISION METHOD
STEP 1: Divide the greater number by the smaller number and find the
remainder.
STEP 2: Divide the smaller number or the divisor by the remainder.
STEP 3: Continue till you reach the last divisor. It is the HCF.
HCF BY LONG DIVISION METHOD
EXAMPLE: Find the HCF of 20 and 24 by long division method.
SOLUTION: 1
20 ) 24 (1
-20
4 ) 20 (5
-20
0
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Since last divider is 4, thus Hcf =4
LCM(LEAST COMMOM MULTIPLE) DEFINITION: The least multiple
,which is common for two or more numbers is their LCM.
For example let us take two numbers 2 and 6.
Multiples of 2 – 2,4,6,8,10,12,14,16
Multiples of 6 - 6, 12,18
Common multiples – 6,12 but 6<12
Hence 6 is the LCM of 2 and 6.
METHODS OF FINDING LCM
1. SHORT DIVISION METHOD 2. PRIME FACTORISATION METHOD 3. COMMON DIVISION METHOD
EXAMPLES
EXAMPLE: Find the LCM of 8 and 10 by Short division method.
Solution:
2 8, 10
2 4, 5
2 2, 5
5 1, 5
1, 1
LCM = 2×2×2×5= 40
RELATION BETWEEN LCM AND HCF
PRODUCT OF TWO NUMBERS = LCM X HCF
LCM = PRODUCT OF TWO NUMBERS / HCF
HCF = PRODUCT OF TWO NUMBERS /LCM
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EXAMPLE: Find the LCM of two numbers whose product is 120 and
their HCF= 2.
Solution: LCM = PRODUCT OF TWO NUMBERS / HCF
=120/2 =60 so LCM =60
EXAMPLE: Find the HCF of two numbers whose product is 375 and
their LCM =75
Solution: HCF = PRODUCT OF TWO NUMBERS /LCM
= 375/75
=5
So HCF = 5
WORD PROBLEMS
EXAMPLE- A certain number of fruits can be arranged in groups of
3,4,6,8 with no fruits left behind. Find the number of fruits?
SOLUTION- Number of fruits = LCM of 3,4,6,8
3 = 3x1 4 = 2x2 6 =2x3 8 =2x2x2
LCM = 2X2X2X3 = 24
So the number of fruits is 24
EXAMPLE- Vaidik wants to plant 8 onions and 12 cabbages in his
garden. What is the greatest number of rows possible if each row has
every kind of vegetable and all the row has same number of vegetable ?
SOLUTION- Greatest no of possible rows = HCF of 8 and 12
8 = 2 x 4 12 = 2 x 6
= 2 x 2 x2 = 2 x 2 x 3
HCF = 2 X 2 = 4
So the greatest number of possible rows is 4 .
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Unit -7
FRACTIONS
• Definitions of fraction ,like fraction, unlike fraction, proper
fraction, improper fraction, unit fraction ,mixed fraction
• Comparison of like fractions ,addition and subtraction of like
fractions
• More about fractions ,relationship of fraction with division,
converting an improper fraction into a mixed fraction. Converting
a mixed fraction into an improper fraction
• equivalent fractions, reduction of fraction to lowest terms
,fractions of whole number
• comparison of unlike fractions ,addition, subtraction
,multiplication , reciprocal division of fractions and their word
problems.
DEFINITIONS
• Fraction – a fraction is a part of a whole. For example 1/2 is a
fraction which means 1 out of total 2 parts.
• Like factions are those fractions which have the same denominator
for e.g.-1/8 and 2/8
• Unlike factions are those fractions which do not have the same
denominator for e.g.-1/8 and 2/5
• Proper fraction - any fraction having its numerator less than the
denominator is called proper fraction . Example - 1/3 , 6/7 , 3/11
,12/13 are proper fraction.
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• Improper fraction - any fraction having its numerator greater than
or equal to the denominator is called improper fraction . Example -
5/3 , 9/7 , 14/11 ,13/13 are improper fraction.
• Unit fraction- when a proper fraction has 1 as its numerator, it is
called a unit fraction.example- 1/3, 1/4 ,1/7 are unit fraction
• Mixed fraction - whenever a whole number is combined with a
proper fraction , we get a mixed fraction
• Example – 3+1/4 = 3 1/4 is a mixed fraction
COMPARISON OF LIKE FRACTIONS
• While comparing like fractions , just compare the values of the
numerators
• 5/12 < 7/12
• EXAMPLE- Arrange the following fraction in ascending order:
3/4 , 1/4, 7/4
• Solution - compare the numerators 1<3<7
so, 1/4< 3/4< 7/4 ascending order.
• EXAMPLE- Arrange the following fraction in descending order:
6/14 , 9/14, 13/14
• Solution - compare the numerators 13>9>6
so, 13/14>9/14> 6/14 descending order.
ADDITION AND SUBTRACTION OF LIKE FRACTIONS
• EXAMPLE- Add 3/7 + 1/7
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• Solution - Add the numerators.
Write down the same denominator.
3/7+1/7 = 3+1
EXAMPLE- Subtract 5/9 – 1/9
Solution - Subtract the numerators.
write down the same denominator
5/9- 1/9 = 5-1
CONVERTING A MIXED NUMBER INTO AN IMPROPER
FRACTION
• EXAMPLE – Convert 3 1/2 into an improper fraction.
• Solution –3 =3X2+1/2 = 7/2
• EXAMPLE – Convert 13/2 into a mixed fraction.
• Solution - step 1 - Divide 13 by 2
• 2)13(6
• --12
• -----------
• 01
• Step 2 - now write the fraction as Q X R/D
• Ans - 6 X 1/2
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EQUIVALENT FRACTION
• DEFINITION – There are two different ways of describing the
same amount like 1/2 = 2/4 =3/6.These fractions are called
equivalent fractions
• To find equivalent fractions of a fraction just multiply the
numerator and the denominator of the fraction by the same
number.
QUESTIONS RELATED TO EQUIVALENT FRACTIONS
EXAMPLE- Find the equivalent fractions of 3/4
SOLUTION- MULTIPLY BY 2 3/4 X 2/2 = 6/8
MULTIPLY BY 3 3/4 X 3/3 = 9/12
So, 3/4 = 6/8 = 9/12
3/4 , 6/8 ,and 9/12 are equivalent fractions.
EXAMPLE – Check whether 2/6 and 3/9 are equivalent or not.
SOLUTION – 2 Cross multiply the fractions 6 x
3 = 18
2 x 9 = 18 since the value of both the
products are equal , the fractions are equivalent
Reduction of a fraction to its lowest term
• A fraction can be reduced to its lowest term by dividing the
numerator and denominator by their HCF.
• EXAMPLE- Reduce 2/4 to its lowest term.
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• SOLUTION – HCF of 2 and 4 is 2
2/2
------- =1/2
4/2
Therefore 2/4 = 1/2
Fraction of a whole number
• Division of the given whole number by the denominator of the
given unit fraction is called the fraction of that number.
• Example 1– There are 9 balls . 1/3 of them are grey . How many
balls are grey ? How to find 1/3 of 9?
• Solution – 1/3 of 9
= 1/3 x 9
= 9/3
=3
Example 2 – Find 3/8 of a day in hours?
Solution- 1 day = 24 hours
3/8 of a day
= 3/8 of 24 hours
Step 1 Divide hours 24/8 = 3
Step 2 Multiply the number obtained in step 1 by the numerator of the
fraction 3 x3 = 9 hours
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Ans. There are 9 hours in 3/8 of a day.
Comparison of unlike fractions
• EXAMPLE- Compare 3/4 and 2/3
• Solution – Step 1- Find the LCM of denominators that is of 4 and 3
. LCM of 4 and 3 is 12
Step 2- Now, make the denominators of both the fractions 12
3 = 3 x 3 = 9
2 = 2 x 4 = 8
Compare 9/12 > 8/12
So, 3/4 > 2/3
ADDITION OF FRACTIONS
• EXAMPLE – Add 1/3 + 2/5
• Solution – Step 1- Find the LCM of 3 and 5
LCM of 3 and 5 is 15
Step 2 – Make the denominators of both the fractions 15.
1 1 X 5/3 X 5 = 5/15
2 2 X 3/5 X 3 = 6/15
Step 3 – Now , add the two like fractions 5/15+ 6/15= 5 + 6/15 =
11/15
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SUBTRACTION OF FRACTIONS
• EXAMPLE – Subtract and write in simplest form- 4 X 1/2 – 2 X
1/3
• SOLUTION – Step 1 - convert into improper fraction
9/2 – 7/3
Step 2- find the LCM of 2 & 3 which is = 6
Step 3 – convert into like fractions.
and
Step 4 - 27/6 – 14/6 = 27 - 14 = 13/6 = 2
MULTIPLICATION OF FRACTIONS
• MULTIPLICATION IS A REPEATED ADDITION
• EXAMPLE- Multiply 9 x 2/3
• SOLUTION- .Step 1- Rewrite the whole number as fraction 9 = 9/1
Step 2- Multiply the numerators =( 9 x 2)/1x3
Step 3 Simplify the fraction. 18/3 = 6 so 9 x 2/3 = 6
Multiplying a fraction by another fraction
EXAMPLE - Multiply 2/3 x 3/4
SOLUTION –( 2x3)/(3x4)=6/12=1/2
IMP NOTE – MULTIPLICATION OF A FRACTION BY ZERO ALWAYS
RESULTS IN ZERO 3/5 X 0 = 0
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MULTIPLICATIVE INVERSE OF A FRACTION
• Multiplicative inverse is also known as reciprocal .
• To find the reciprocal of a fraction , just interchange the values of
the numerator and denominator.
EXAMPLE – Find the M.I.of (i) 1/8 (ii) 2/3
SOLUTION – (i) The M.I. of 1/8 is 8
(ii) The M.I. of 2/3 is 3/2
IMP NOTE -The reciprocal of 1 is 1
The reciprocal of 0 is not defined.
If one fraction is reciprocal of the other fraction , their product will
always be 1. for example 1/6 x 6 = 1/6 x 6/1 = 1
DIVISION OF FRACTIONS
• Division of a whole number by a fraction.
• Example: Divide 8 by 1/3
• Division of a fraction by a whole number
• 8x3=24.
• WORD PROBLEMS
• Q1) A basket has 48 balls.3/4 of the balls are blue. How many balls
are blue ?
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• Sol – Total no. of balls = 48
• Blue balls = 3/4
• No of blue balls = 3/4 of 48
• = 3/4 x 48 ( 48/4 =12, now multiply 12 by 3 )
• = 36
• Q2) Sunil painted 1/4 of a wall. Sona painted 2/5 of the same
wall.
• How much of the wall did they painted together ?
• Sol – Sunil painted wall = 1/4
• Sona painted wall = 2/5
• The wall they painted together = 1/4 + 2/5
• LCM of 4 and 5 is 20
• 1/4x5/5=5/20
• 2/5x4/4=8/20
• 5/20 + 8/20 = 13/20
• The wall they painted together=13/20
WORD PROBLEMS
• Q3) A ruler is 1 foot long. Vedic broke 1/8 of this ruler. How much
length of ruler is left ?
Solution – Length of ruler = 1 foot
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Broken part of ruler = 1/8
Length of ruler left = 1 – 1/8 = 1/1 -1/8
=( 8-1)/8 = 7/8
Length of ruler left =7/8
WORD PROBLEMS
• Q4) 8/9 kg of sweets is to be equally distributed among 16
students. How much will each student get ?
• Solution – Quantity of sweets = 8/9 kg
Total students = 16
Each student get = 8/9 16
= 8/9 x 1/16
= 1/18 ( 8 divided by 16 = 1/2 ,then 1/2x 1/9 =
1/18)
so each student will get 1/18 kg of sweets.
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UNIT-8
DECIMAL NUMBERS
• CONCEPT OF DECIMAL
• VARIOUS TYPES OF DECIMALS.
• The point between the two numbers is called the decimal point.
Tenth means 1/10 and 1 one = 10 tenth
The number 3/10 is written as 0.3 Such a number is called a decimal
fraction. The dot between 0and 3 is called the decimal point.
Imp note – If there is no whole number, we write 0 in the place of whole
number before the decimal. Example: 0.7 for 7/10
1/100 means one hundredth
1/1000 means one thousandth
5/100 = Five hundredth in decimal form can be written as 0.05
4/1000 = decimal form can be written as 0.004
1 tenth = 10 hundredths & 1 hundredth = 10 thousandths
4.735 on a place value chart can be written as follows
4.735 = ones tenths hundredths thousandths
4 7 3 5
EXAMPLE – Expand the following
(1) 4.21 (ii) 0.786 (iii) 32 .01
Solution-(i) 4.21 = 4 + 2/10 +1/100
(ii) 0.786 = 7/10 + 8/100 + 6/1000
(iii) 32.01 = 30 +2 + 1/100
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EXAMPLE- Express each one as a decimal number.
25 +3/10 +7/100
1 + 3/10 + 1/100 + 3/1000
2 + 8/100 + 3/1000
Solution- (i) 25 +3/10 +7/100 = 25.37
1 + 3/10 + 1/100 + 3/1000 = 1.313
2 + 8/100 + 3/1000 = 2.083
EQUIVALENT DECIMALS - Decimals like 0.2, 0.20, 0.200 are
called equivalent decimals.
LIKE DECIMALS – Decimal that have the same number of decimal
places are called like decimals. for e g 0.2, 3.3 , 8.8 have same number
of decimal places that is 1
UNLIKE DECIMALS – Decimal that have the different number of
decimal places are called unlike decimals. for e.g. 0.25, 3.3 , 8.786 have
different number of decimal places
EXAMPLE- Change the following unlike decimal into like decimals.
3.6 , 8.123, 9.28
Solution – Like decimal – 3.600 , 8.123, 9.280
EXAMPLE- Compare the decimals using >, = or <
(a) 7.36 7.48 (b) 24.37 24.370
Solution (a) 7.36 7.48
(b) 24.37 24.370
EXAMPLE- Rearrange 5.12 , 5.21 , 5.3 , 5 in ascending order
Solution: Ascending order - 5 , 5.12 , 5.21 , 5.3
EXAMPLE- Rearrange 7.62 , 70.003, 7.6 , 70.21 in descending order
Solution- Descending order – 70.21 , 70.003 , 7.62, 7.6
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Example- Add 3.18 + 4.9
Solution- Convert the decimals into like decimals and now add the
numbers vertically by placing the decimal one below the other
3.18
+ 4.90
Example – Subtract 67.5 – 66.84
Solution- Convert into like decimals and subtract. Borrow if required
67.50
- 66.84
Example- Multiply 0.8 x 0.7
Solution – 0.8 (decimal place = 1)
0.7 (decimal place = 1)
8 x 7 = 56 (ignoring decimal points)
1 decimal place + 1 decimal place = 2 decimal places so, 0.8 x
0.7 = 0.56
Remember- Place the decimal point , counting from the extreme right in
the product
Multiplying decimal by 10 , 100 ,1000
Let us take a decimal number 8.7623
10 x 8.7623 = 87.623 (Multiplication by 10 results in the shifting of
decimal point one place to its right)
100 x 8.7623 = 876.23 (Multiplication by 100 results in the shifting of
decimal point two places to its right)
1000 x 8.7623 = 8762.3 (Multiplication by 1000 results in the shifting of
decimal point three places to its right)
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Division of a decimal is carried out in the same way as we divide a
whole number . But be careful to place a decimal point at the right place
in the quotient . Always put a decimal point directly above the decimal
point of the dividend
Example – Divide 28.60 by 4
Dividing a decimal by another decimal
Example- Divide 0.72 by 6
Solution – Step 1 – 0.6 x 10 = 6
Step 2 - 0.72 x 10 = 7.2
so, 0.72 /0.6 = 7.2 /6
The decimal point moves,2 places and 3 places to the right when
multiplied by 10, 100 & 1000 respectively. In case of division , the
decimal point moves to the left as many places as number of zeroes in
the divisor.
Dividing a decimal by 10,100,1000
Example – Let us take a decimal number 276.4
Then 276.4 divided by 10 = 27.64
276.4 divided by 100 = 2.764
276.4 divided by 1000 = 0.2764
Q1- Neetu bought 200.2 g of sweets. Varsha bought 150.55 g of sweets.
How much sweets did they buy in all ?
Sol – Neetu bought sweets = 200.20g
Varsha bought sweets = 150.55 Total
sweets They buy 350.75 g sweets in all.
Q 2- Tom weighs 62.08 kg. Tim weighs 41.8 kg .How much more does
Tom weigh than Tim ?
Sol - Tom weighs = 62.08 kg
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Tim weighs = - 41.80 kg
Difference = 20.28kg
Q3) A bottle contains 2.55 litre of water . How much will 9 such bottles
contain ?
Solution – A bottle contains = 2.55 l of water
So, 9 such bottles will contain = 2.55 x 9 l of water
2.55
x 9
22.95 9 bottles will contain 22.95 l of
water
Q4) 4 balls weigh 2.24 kg. Find the weight of each ball.
Sol - Weight of 4 balls = 2.24 kg
So weight of each ball = 2.24/4
So weight of each ball = 0.56 kg
• Tenth means 1/10 and 1 one = 10 tenth
• The number 3/10 is written as 0.3 Such a number is called a decimal
fraction. The dot between 0and 3 is called the decimal point.
• Imp note – If there is no whole number, we write 0 in the place of
whole number before the decimal. Example: 0.7 for 7/10
• 1/100 means one hundredth
• 1/1000 means one thousandth
• 5/100 = Five hundredth in decimal form can be written as 0.05
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• 4/1000 = decimal form can be written as 0.004
• 1 tenth = 10 hundredths & 1 hundredth = 10 thousandths
• 4.735 on a place value chart can be written as follows
• 4.735 = ones tenths hundredths thousandths
4 7 3 5
• EXAMPLE – Expand the following
• 4.21 (ii) 0.786 (iii) 32 .01
Solution-(i) 4.21 = 4 + 2/10 +1/100
(ii) 0.786 = 7/10 + 8/100 + 6/1000
(iii) 32.01 = 30 +2 + 1/100
EXAMPLE- Express each one as a decimal number.
• 25 +3/10 +7/100
• 1 + 3/10 + 1/100 + 3/1000
• 2 + 8/100 + 3/1000
Solution- (i) 25 +3/10 +7/100 = 25.37
• 1 + 3/10 + 1/100 + 3/1000 = 1.313
• 2 + 8/100 + 3/1000 = 2.083
EQUIVALENT DECIMALS - Decimals like 0.2, 0.20, 0.200 are
called equivalent decimals.
LIKE DECIMALS – Decimal that have the same number of decimal
places are called like decimals. for e g 0.2, 3.3 , 8.8 have same
number of decimal places that is 1
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UNLIKE DECIMALS – Decimal that have the different number of
decimal places are called unlike decimals. for e.g. 0.25, 3.3 , 8.786
have different number of decimal places
• EXAMPLE- Change the following unlike decimal into like
decimals.
3.6 , 8.123, 9.28
Solution – Like decimal – 3.600 , 8.123, 9.280
EXAMPLE- Compare the decimals using >, = or <
(a) 7.36 7.48 (b) 24.37 24.370
• Solution (a) 7.36 7.48
• (b) 24.37 24.370
• EXAMPLE- Rearrange 5.12 , 5.21 , 5.3 , 5 in ascending order
• Solution: Ascending order - 5 , 5.12 , 5.21 , 5.3
• EXAMPLE- Rearrange 7.62 , 70.003, 7.6 , 70.21 in descending order
• Solution- Descending order – 70.21 , 70.003 , 7.62, 7.6