Download Class 4english Math

Published by
National Curriculum and Textbook Board
69-70, Motijheel Commercial Area, Dhaka-1000.
[All rights reserved by the publisher]
ial Edition
First Print (Bangla version) : October 2005
First Print (English Version) : February 2007
Computer Graphics
Laser Scan Limited
15/2 Topkhana Road, BMA Bhaban.
Cover & Illustrations:
Md. Abdul Momen Milton
Picture Coloured by
Ahmed Ullah
Printing Supervision
Hafija Sultana
Design
National Curriculum and Textbook Board
Website version developed by Mars Solutions Limited
Preface
For improving the existing quality of Primary Education in Bangladesh, National Curriculum
and Textbook Board (NCTB) in collaboration with PEDP-2 initiated an extensive program for
development of curriculum and teaching learning materials in 2002. In the light of this
program the curriculum, textbooks and other teaching learning materials of Primary levels
have been prepared, revised and evaluated.
The textbook entitled, 'Elementary Mathemetics' has been prepared on the basis of
attainable competencies for the students of Class Four. The subject matter of the textbook is
derived from the basic issues of the religion familiar to the children through their family
practices. This will facilitate our young learners to know how they can make best use of this
religious knowledge & values in their day-to-day life.
The contents of the book are analyzed and explained in such a manner with practical
examples, illustrations and system of planned activities, that students are inspired to study the
subject with a keen interest.
This book is originally published in Bangla. From the year 2007 NCTB is publishing the English
version of the textbook. English is the language of choice in today's globalized world. To
facilitate the verbal and written communication skills of our future citizens and suitably
prepare them for international competition, we decided to translate the original Bangla
textbooks into English. It's pleasant to note that the number of English medium schools in
Bangladesh is increasing very fast. In this context NCTB decided to publish all the textbooks
of Primary level in English. This was a big endeavour for us. Despite our all efforts the first
edition may not be totally error free. However, in the future editions we shall try to remove all
errors and discrepancies.
Finally, I would like to express my heartfelt thanks and gratitude to those who have made
their valuable contributions in writing, editing, evaluating and translating this book. I
sincerely hope that the book will be useful to those for whom it has been prepared.
Prof. Md. Mostafa Kamaluddin
Chairman
National Curriculum and Textbook Board
Dhaka
Chapter-One
Number
Read and write numbers in figures and. in words (The
first one is done for you)
Number
Thousands
Hundreds Tens Ones
(in
Ten
Thousands
figures) thousands
5021
990
4672
7903
6550
3989
8095
8704
9008
9999
10000
5
0
2
1
Number
(in words)
Five thousand
twenty one
2
Number
Make groups of tens, hundreds and thousands (Two
examples are given)
Elementary Mathematics
3
Count and Read
tens
hun.
hun.
hun.
thou.
thou.
hun.
hun.
thou.
thou.
hun.
tens
tens
tens
ten thou.
thou.
thou.
thou.
thou.
hun.
hun.
thou.
7 thousand 4
hundred 8 tens 9
7489
hun.
hun.
tens tens tens tens tens tens tens tens
thou.
thou.
thou.
thou.
thou.
ten thou.
thou.
thou.
ten thou.
thou.
thou.
ten thou.
hun.
hun.
hun.
hun
ten thou.
.
hun
.
hun
10 thousands
10000
thou.
ten thou.
.
1 thousand 5
hundred 3 tens 7
1537
tens
40 thousand 6
hundred 1 ten 5
40615
4
Number
Count and read
ten thou.
ten thou.
ten thou.
ten thou.
ten thou.
57 thousand 3
hundred 2
thou.
thou.
thou.
thou.
thou.
thou.
thou.
57302
hun.
hun.
ten thou.
hun.
ten thou.
ten thou.
ten thou.
ten thou.
73 thousand 2
hundred 4 tens
ten thou.
ten thou.
hun
ten thou.
hun
tens
ten thou.
thou.
thou.
tens
tens
ten thou.
thou.
73240
tens
ten thou.
ten thou.
100 thousand
1 lac
ten thou.
ten thou.
ten thou.
ten thou.
ten thou.
100000
Elementary Mathematics
5
Read
Number
(in
figures)
Lacs
Crores
Ten
Lacs
Thousands
Lacs
Ten
Thousthousands ands
Hund- Tens Ones
reds
10001
1
0
0
0
1
10009
1
0
0
0
9
50716
5
0
7
1
6
99999
9
9
9
9
9
100000
1
0
0
0
0
0
780612
7
8
0
6
1
2
4509325
4
5
0
9
3
2
5
9999999
9
9
9
9
9
9
9
0
0
0
0
0
0
0
10000000
1
Let us look:
(1) 1 Ajut (10000) = 10 thousand
(2) 1 Nijut = 10 lac
(3) Generally we read 10 thousand instead of 1 Ajut
(4) We read 10 lac instead of 1 Nijut
Number
(in words)
Ten
thousand
one
Ten
thousand
nine
Fifty
thousand
seven
hundred
sixteen
Ninety nine
thousand
nine
hundred
ninety nine
One lac
Seven lac
eighty
thousand
six hundred
twelve
Forty five
lac nine
thousand
three
hundred
twenty five
Ninety nine
lac ninety
nine
thousand
nine
hundred
ninety nine
One crore
6
Number
Read and write (in figures and words) [One is done for you] :
Number
(in
figures)
crores
Nijuts
907865
Lacs
9
Ajuts
Th
0
H
7
8
T
6
O
5
Number
(in words)
Nine lac
seven
thousand
eight
hundred
sixty-five
12934
78050
420199
2877602
8909009
10532040
Read and write in figures [One is done for you] :
Number
(in words)
Five lac ninety thousand
seven hundred four
Twenty five thousand
nine hundred sixty eight
Six lac seventy five
thousand two hundred
thirty
Fourteen lac four
thousand forty eight
Seventy six lac ten
thousand ninety four
Ninetynine lac nine
thousand ninety
One crore thirty lac fifty
thousand seven
crores Nijuts
Lacs Ajuts
5
9
Th H T O Number
(in
figures)
0
7
0 4
590704
Elementary Mathematics
7
Read the numbers by using comma
By using comma
Numbers
(in
figures)
Numbers (in
words)
Cro Lacs Thousand s
res
Cr. Ni L Aj Th
H T
O
45798
4
5,
7
9
8
97060
9
7,
0
6
0
6,
4
2,
1
0
9
642109
8908005
8
9,
0
8,
0
0
5
16543217 1,
6
5,
4
3,
2
1
7
Forty five
thousand seven
hundred ninety
eight
Ninety seven
thousand sixty
Six lac forty-two
thousand one
hundred nine
Eighty nine lac
eight thousand
five
One crore sixty
five lac forty three
thousand two
hundred seventeen
Let us look
•
•
•
•
Starting from right to left, the first comma is put after three digits
to indicate thousand.
After two digits of the first comma the second comma is put to
indicate lac.
After another two digits the second comma the third comma is
put to indicate crore.
10 hundred =1 thousand, 100 thousand = 1 lac and 100 lac = 1
crore.
8
Number
Writing numbers by using comma
Numbers (in words)
Numbers in
(figures)
By using comma
Cr L Th H T O
0, 00, 00, 0 0 0
Fifty-five thousand nine
55, 9 7
hundred seventy seven
Ninety-nine thousand eight
99, 8 6
hundred sixty-five
Eight lac thirty-eight
8, 38, 6 2
thousand-six hundred twentyfour
Eighty five lac five thousand
85, 05, 0 5
fifty five
Two crore, nine lac twelve
2, 09, 12, 7 0
thousand seven hundred nine
7
55977
5
99865
4
838624
5
8505055
9
20912709
Let us look:
•
Only one comma is used for writing a figure in thousand; two
commas are used for writing a figure in lac and three commas are
used for writing a figure in crore.
•
Starting from the right, the digits of ones tens and hundreds are
written first.
•
The first comma has been put after the hundred digit, then, the
thousand digits are written.
•
The second comma has been put after the thousand digits, then,
the lac digits are written.
•
The third comma has been put after the lac digits, then, the crore
digit is written.
•
O (zero) is put as and when required.
Elementary Mathematics
9
Read numbers using commas and write in words (One is
done for you) :
Numbers
(in figures)
8205690
By using
commas
82,05,690
Numbers (in words)
Eighty two lac five
thousand six hundred
ninety
68704
942059
4935200
34406070
Using commas, read and write numbers in figures (One is
done for you) :
Numbers (in words)
Fifty-seven lac forty-five
thousand eight hundred sixteen
Ninety thousand nine
hundred ninety
Six lac seventy-five
thousand four hundred eighty
six
Eighty eight lac three
thousand nine hundred two
Five crore ninety-one lac
four thousand five hundred
forty
By using
comma
Numbers (in digits)
57,45,816
5745816
10
Number
Place Values
Look at the diagram
and write the number
(One is done for you)
write the place values
(One is done for you)
85 6 7
7 ones
Th
H
T
O
7385
H
T
O
4 3 9 9
Th
H
T
O
9 0 4 8
Th
H
T
O
88 8 8
Th
H
T
O
7
6 tems
=
5 thosuand =
60
500
8 thosuand = 8000
7 9 0 2
Th
=
Elementary Mathematics
11
Place Values
4 2 7 1 5 3
Ni
L
Aj
Th
H
T
O
1 Ones =
3
5 Tens =
1 Hun =
7 Thus =
50
100
7000
2 Aj =
4 Lacs =
20000
400000
1 Ones =
3 Tens =
0 Hun =
7 Thus =
0 Ajs =
1
30
00
7000
0
5 Lacs =
6 Nis =
500000
6000000
0 Ones =
0 Ten =
2 Hrs =
6 Thus =
4 Ajs =
0 Lac =
8 Nis =
0
0
200
6000
400000
0
8000000
4 Ones =
1 Ten =
5 Huns =
8 Thus =
3 Ajs =
2 Lac =
6 Nis =
4
10
500
8000
30000
200000
6000000
6 5 0 7 0 3 1
Ni
L
Aj
Th
H
T
O
8 0 4 6 2 0 0
Ni
L
Aj
Th
H
T
O
6 2 3 8 5 1 4
Ni
L
Aj
Th
H
T
O
1 0 0 0 0 000
Ni
L
Aj
Th
H
T
O
0 Ones =
0 Tens =
0 Thun =
0 Thu =
0 Ajs =
0 L=
1 Nis =
C=
0
0
0
0
0
0
1000000
0000000
12
Number
Place Values
write the place values
Look at the diagram and
write the numbers
(One is done for you)
(One is done for you)
7302850
N
L
Aj
Th
H
T
O
0 Ones =
0
5 Tens =
50
8 Hun =
800
2 Thous = 2000
0 Ajs =
0
3 lac
= 300000
7047013
7 Ni
2439601
N
L
Aj
Th
H
T
O
5806702
N
L
Aj
Th
H
T
O
8004650
N
L
Aj
Th
H
T
O
9909995
N
L
Aj
Th
H
T
O
=7000000
Elementary Mathematics
13
Place Values
11111111
Cr.
Ni
Crore Nijuts
L
Aj
Lacs Ajuts
Th
H
T
O
Thousands Hundreds Tens Ones Numbers
(in
words)
1
One
1
0
Ten
1
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
0
One Lac
1
0
0
0
0
0
0
Ten Lac
0
0
0
0
0
0
0
One
Hundred
One
Thousand
Ten
Thousand
One
1
Place values are read starting from the right side:
10 ones are 1 ten
10 tens are 1 hundred
10 hundred are 1 thousand
10 thousand are 1 Ajut
10 Ajut or 100 thousand are 1 lac
10 lac are 1 Nijut
10 Nijut or 100 lac are 1 crore.
Crore
14
Number
Find the place value and add them (Two are done for you)
8925367
The place value of 7 →
The place value of 6 →
The place value of 3 →
The place value of 5 →
The place value of 2 →
The place value of 9 →
The place value of 8 →
1620598
The place value of 8 →
The place value of 9 →
The place value of 5 →
The place value of 0 →
The place value of 2 →
The place value of 6 →
The place value of 1 →
3805792
The place value of 9 →
The place value of 8 →
The place value of 5 →
The place value of 7 →
The place value of 0 →
The place value of 3 →
The place value of 2 →
6532871
The place value of 1 →
The place value of 2 →
The place value of 3 →
The place value of 5 →
The place value of 6 →
The place value of 7 →
The place value of 8 →
7
60
300
5000
20000
900000
8000000
8925367
6819024
The place value of 9 →
The place value of 8 →
The place value of 0 →
The place value of 1 →
The place value of 6 →
The place value of 4 →
The place value of 2 →
7586923
The place value of 3 →
The place value of 2 →
The place value of 9 →
The place value of 6 →
The place value of 8 →
The place value of 5 →
The place value of 7 →
9863054
The place value of 3 →
The place value of 4 →
The place value of 0 →
The place value of 6 →
The place value of 5 →
The place value of 9 →
The place value of 8 →
8536974
The place value of 9 →
The place value of 8 →
The place value of 7 →
The place value of 6 →
The place value of 5 →
The place value of 4 →
The place value of 3 →
9000
800000
0
10000
6000000
4
20
6819024
Elementary Mathematics
Comparison of Numbers
Find the greater or smaller numbers and use symbols to show them
7865, 8743
7865 = 7 thousand 8 hundred 6 tens
5
8743 = 8 thousand 7 hundred 4 tens
3
Of the two four-digit
numbers, the one with the
∴7865 is smaller, 8743 is greater
greater digit in the
or, 7865 <8743; smaller sign <
thousands place is greater
than the other.
Rule for reading : 7865 is smaller
than 8743.
Again it can be written in the
following way
8743> 7865; greater sign >
Rule for reading : 8743 is greater
than 7865.
9754, 9745
9754 = 9 thousand 7 hundred 5 tens
4
9745 = 9 thousand 7 hundred 4 tens 5
∴ 9754 is greater, 9745 is smaller
or, 9754 > 9745
If the digits of the
thousands place of two four
digit numbers are the same,
look at the digits of
hundreds tens and ones to
find which number is
greater than the other.
46059, 47182
If the digits of the Ajuts
46059 = 4 Ajut 6 thousand 0
place of two five-digit
hundred 5 tens 9
numbers are the same, then
47182 = 4 Ajut 7 thousand 1
it is necessary to compare
hundred 8 tens 2
them by looking at the
∴ 46059 is smaller, 47182 is greater digits of thousands,
hundreds tens and ones.
or, 46059 < 47182
100000, 99999
If there are two numbers
100000 = 1 lac
one having six-digits and
99999 = 9 Ajut 9 thousand 9
the other having five digits,
hundred 9 tens 9
the number having six∴ 100000 is greater, 99999 is smaller digits will always be
greater.
or, 100000 > 99999
15
16
Number
Comparison of Numbers
Find the greater or smaller number and use symbols to show them
(two are done for you)
8539, 8607
8539 smaller, 8607 greater
8539 <8607
6789, 7654
7654 greater, 6789 smaller
7654 > 6789
4385, 4702
_____ smaller, ______ greater
7825 , 78 52
______ greater , ______smaller
9628, 9569
_______ greater, _______ smaller
35706, 43082
_______ smaller, _______greater
56789, 56817
_______ greater, _______smaller
85398, 85389
_______ smaller, _______greater
78029, 79103
_______ gre ater, _______smaller
99999, 88888
_______ greater, _______smaller
Elementary Mathematics
Arranging Numbers in Sequence
Arranging numbers in ascending and descending orders
4309, 8142, 7214, 8124, 6037, 7218
Here, 4309 =4 thousand 3 hundred 0 tens 9
8142 =8 thousand 1 hundred 4 tens 2
7214 =7 thousand 2 hundred 1 ten 4
8124 =8 thousand 1 hundred 2 tens 4
6037 =6 thousand 0 hundred 3 tens 7
7218 =7 thousand 2 hundred 1 tens 8
Looking at the digits of thousands, hundreds tens and ones we can
find the numbers in descending order
8142, 8124, 7218, 7214, 6037, 4309
or, 8142>8124>7218>7214>6037>4309
Again by arranging in ascending order we can find
4309, 6037, 7214, 7218, 8124, 8142
or, 4309 <6037<7214<7218<8124<8142
65904, 891254, 98536, 890397, 7706253, 6814300, 98563
Here, 65904= 65 thousand 9 hundred 0 ten 4
891254= 8 lac 91 thousand 2 hundred 5 tens 4
98536= 98 thousand 5 hundred 3 tens 6
890397 = 8 Lac 90 Thousand 3 hundred 9 tens 7
7706253= 77 lac 6 thousand 2 hundred 5 tens 3
6814300= 68 lac 14 thousand 3 hundred
98563= 98 thousand 5 hundred 6 tens 3
Looking at the digits of lacs, thousands, hundreds, tens and units
we can find the descending order.
7706253, 6814300, 891254, 890397, 98563. 98536, 65904
or, 7706253> 6814300>891254>890397>98563>98536>65904
Again, by arranging in ascending order we can find
65904, 98536, 98563, 890397, 891254, 6814300, 7706253,
Or, 65904<98536<98563<890397<891254<6814300<7706253
17
18
Number
Arranging Numbers in Order
Arrange in ascending/descending (one is done for you)
Number
65032, 8973, 26940,
53278, 80149,
84256, 9856
8265, 9702, 8814,
7523, 9537, 7532
65068, 72498,
66253, 72085,
54987, 39875,
55210
543210, 35721,
68057, 54321,
759002, 752083
889531, 902671,
888530, 725094,
875320, 923510,
732189
992570, 8548071,
8877510, 999999,
8888888, 632571,
8509724
5746800, 3587260,
889006, 4370582,
5526897, 5742810,
64085913
Descending
84256, 80149,
65032, 53278,
26940, 9856, 8973.
84256>80149>65032
>53278>23940>9856
<8973.
Ascending
8973, 9856, 26940,
53278, 65032,
80149, 84256.
8973<9856<26940
<53278<65032
<80149<84256
Elementary Mathematics
19
Making the Greatest and the Smallest Numbers
Making the greatest and smallest numbers by using one digit once only
Given digits : 5, 2, 7, 9
Given digits : 6, 0, 8, 5, 3
9>7>5>2
2<5<7<9
8>6>5>3>0
0<3<5<6<8
∴ Greatest number : 9752
Smallest number : 2579
∴ Greatest number : 86530
Smallest number : 30568
Given digits : 4, 5, 0, 8, 7, 2
Given digits : 3, 6, 7, 4, 5, 8
8>7>5>4>2>0
0<2<4<5<7<8
∴ Greatest number : 875420
Smallest number : 204578
8>7>6>5>4>3
3<4<5<6<7<8
∴ Greatest number : 876543
Smallest number : 345678
The greatest and smallest numbers with one or more digits
Numbers having given
Greatest numbers
Smallest numbers
digits
Number of one digit
9
1
Number of two digit
99
10
Number of three digit 999
100
Number of four digit
9999
1000
Number of five digit
99999
10000
Number of six digit
999999
100000
Let us look:
• The given digits are first written in order of the greatest to the smallest
by using symbols and then the greatest and the smallest number are
made.
• When there is a zero (0) in the given digits, 0 is written just after the
second smallest number. Afterwards smallest number is formed.
• 9 is written once or more to make the greatest numbers of one or more
digits.
• A required number of 0s (zeros) are written on the right of 1 to make
the smallest numbers of several digits.
20
Number
Roman Numbers
This is the picture of a wall clock.
The time indicating numbers are
written in Roman digits on the dial
of the clock. Generally we see
Roman number symbols written on
the door of classroom.
Number
Number
(in figure) (in words)
1
2
3
4
5
6
one
two
three
four
five
six
Number
(Roman
system)
I
II
III
IV
V
VI
Number
Number
(in figure) (in words)
7
8
9
10
11
12
seven
eight
nine
ten
eleven
twelve
Number
(Roman
system)
VII
VIII
IX
X
XI
XII
Let us look:
• I is used to indicate 1 in Roman system; V is used for 5; X is used for
10.
• Here, numbers from 1 to 12 are written using the Roman numbers I, V
and X
• When I is put on the right of I, V and X, the values are increased by 1.
such as, II=1+1=2; III=2+1=3; VI=5+1=6; XI=10+1=11
• When I is put on the left of V and X, the values are decreased by 1.
such as, IV=5-1=4; IX=10-1=9; VI=5+1=6; XI=10+1=11
Look at the roman digits and write the numbers:
II= ________, IV = _______, X = ________
VII = ______, XII = ______
Write the numbers in Roman digits:
3= ______,
5 = _______, 8 = ________
11 = ______, 12 = ______
Elementary Mathematics
21
Exercise-1
1.
2.
3.
4.
5.
6.
Read and write the following numbers by using commas:
25734, 47099, 880539, 5704602, 805505,
8888888, 100000000.
Write the following numbers in figures:
Forty-five thousand seven hundred twelve; sixty-eight thousand
nine hundred three; six lac fifty thousand three hundred nine;
twenty-seven lac four thousand fifty-nine; Eighty eight lac seven
hundred eight; Ninety-nine lac nine thousand nine hundred
ninety; one crore.
Fill in the blanks:
in 5908726
(a) The place value of 9 is ______
(b) The place value of 8 is ______
(c) The place value of 7 is ______
(d) The place value of 6 is ______
(e) The place value of 5 is ______
(f) The place value of 0 is ______
(g) The place value of 2 is ______
Find the place value of each 5 in the number 555555.
Find the greater or smaller number in the following pairs and use
symbols to show which one is greater/smaller.
8326, 8362; 7506, 8021; 6798, 6789;
85320, 76203, 58249, 58492.
Arrange the following groups of numbers in ascending/
descending order.
(a) 4321, 3270, 4902, 3357, 4287, 5027.
(b) 8506, 5786, 6210, 5768, 6218, 8560.
(c) 72570, 72607, 83259, 60892, 60829, 83592.
(d) 90062, 83001, 90257, 78253, 78235, 83251, 79805
22
7.
8.
9.
10.
11.
12.
13.
14.
Number
Without repeating any digit, arrange the following groups of
numbers to make the greatest and smallest numbers possible.
(a) 3, 9, 8
(b) 9, 0, 7
(c) 4, 3, 8, 5
(d) 6, 0, 3, 8
Write the greatest and smallest numbers having 5 digits.
Write the greatest number using the digits 5, 7, 3, 2, 8 and find
the place value of 8, 7 and 3 in each number.
Write the numbers that come immediately after the greatest
numbers with two digits and with three digits.
Write the numbers that come immediately before the greatest
numbers with six digits and with four digits.
Write the following numbers in the Roman system:
2, 5, 8, 9, 11, 12
Convert the following Roman digits into numbers:
IV, VII, IX, X, VI, XII.
Match the numbers with the Roman numerals by drawing lines:
6
9
4
11
8
10
5
7
IX
V
VIII
IX
VI
IV
VII
X
Elementary Mathematics
23
Chapter-two
Addition and Subtraction
Example 1 : Add: 2403, 3214, 1052, 2310
Solution : 2403
3214
2 thousand 4 hundred 0 ten 3
1052
3 thousand 2 hundred 1 ten 4
1 thousand 0 hundred 5 tens 2
+2310
2 thousand 3 hundred 1 ten 0
8979
8 thousand 9 hundred 7 tens 9
= 8979
Answer: 8979.
Example 2 : Add: 3257, 4062, 2103, 5130,1436
Solution : 3257
3 thousand 2 hundred 5 tens 7
4062
4 thousand 0 hundred 6 tens 2
2103
2 thousand 1 hundred 0 tens 3
5130
5 thousand 1 hundred 3 tens 0
+1436
1 thousand 4 hundred 3 tens 6
15988
15 thousand 8 hundred 17 tens 18
=15 thousan 8 hundred 18
Answer: 15988.
tens 8
=15 thousand 9 hundred 8tens 8
=15988
Let us Look :
• The numbers are arranged in the order of ones under ones, tens
under tens hundreds under hundreds etc.
• The addition starts from the ones.
• When the sum of the digits of one place is a two-digit number,
only the right hand digit is put in the place and the left-hand
digit is carried to the left-hand side column and is added with the
numbers of this column.
24
Addition and Subtraction
Example 3 : Add: 25007, 40018, 14023, 3625,16241
Solution : 25007
Aj
Th
Hund.
Tens
40018
14023
+1
+2
+1
3625
+ 16241
0
0
5
2
98914
4
1
1
9
Answer: 98914.
1
2
2
4
1
0
0
6
2
9
0
4
3
6
8
Ones
7
8
3
5
1
4
Let us look:
• The sum of the digits of units is 24. 4 is put in the sum line under
the unit position. 2 is Carried to the left Column and is written on
the top under tens.
• In the same way the digits of the tens are added and carried 1 is
written on the top under hundreds
• The addition process of hundreds, thousands and ten-thousand has
been performed in the same way.
Example 4 : Write the number along side and add:
2304, 1620, 40521, 12032, 21350
/ / / /
/ / / /
/ / / / /
/ / / / /
/ / / / /
Solution : 2 3 0 4 + 16 2 0 + 4 0 5 2 1 + 12 0 3 2 + 213 5 0 = 77827
Answer : 77827.
Let us look:
•
The numbers are written side by side by putting a plus sign Â+Ê
between them.
At the end of the row of the numbers symbol Â=Ê is put and then
the sum is written.
•
The digits in different places are added with the same place digits
and (/) mark is put on the digits.
•
The carrying number after addition is added with the left digit.
Comments : Learners will do the addition by using the traditional
way. They will not be asked to give any explanation.
Elementary Mathematics
Add (one solution is given for you) :
32079
2572
42140
18204
1039
15379
5625
3804
16802
20531
1356
20531
14362
4921
90801
25
24607
13852
15034
30791
16248
Fill in the boxes :
2160 + 1532 + 1425 + 2054 =
35046 + 1725 + 2908 + 26871 + 10425 =
15270 + 26085 + 14326 + 13572 + 30628 =
Example 5 : There were 3528 students in a school. 1675 more students got
admitted into that school at the beginning of the year. Find the total number
of the students in the school now.
Solution : The number of students was
: 3528
Number admitted : 1675
∴ Total number of students : 5203
Answer : 5203 Students.
Example 6 : 2590 lichis were picked from a tree. 3255 lichis were picked
from another tree. Again 2834 lichis were picked from a third tree. How
many lichis were picked in total from the three trees?
Solution :
The number of lichis picked from 3 trees
2590
3255
2834
Number of lichis picked
8679
from 3 trees
∴ Total number of lichis = 8679
Answer: 8679 lichis.
26
Number
Example 7 : Subtract 6142 from 8574.
Solution : 8574
8 thousand 5 hundred 7 tens 4
-6142 − 6 thousand 1 hundred 4 tens 2
2432
2 thousand 4 hundred 3 tens 2
Subtraction from
subtrahend
Difference
=2432
Example 8 : Subtract 7583 from 94632
Solution :
Ten
Thous
9
+1
−
8
94632
-7583
87049
Thousands Hundreds Tens
Ones
4+10
7
7
2+10
3
9
6
5+1
0
3+10
8+1
4
Here, 94632 is the number for subtraction from, 7583 is the subtrahend or
number to be subtracted and 87049 is the difference or the result of
the subtraction.
Answer : 87049.
Example 9 : Subtract 20847 from 63205
Ten
Thousands Hundreds Tens Ones
Solution : 63205
-20847
Thous
6
3
2+10
0+10
5+10
42358
+1
+1
+1
Answer : 42358.
-2
0
8
4
7
4
2
3
5
8
Example 10 : Subtract 16729 from 52314 by putting them side by side.
/ / / / /
/ / / / /
Solution : 5 2314 − 167 29 = 35585
Here, 52314 is the number for subtraction from
16729 is the number to be subtracted and (subtrahend)
35585 is the difference.
Answer : 35585.
(Number for Subtraction from−
Number for Subtraction from=
subtrahend =Difference
Difference + Subtrahend
(Number for Subtraction from)- Difference =
(Number to beSubtracted/subtrahend
Let us look:
• A minus sign (−) is put between number for subtraction from and number to be
subtracted. Symbol (=) is put at the end of these two numbers.
• This (/) marks has been put above the digits of units tens, hundreds etc for
checking whether the digits are added or not.
Elementary Mathematics
27
Subtract (One is done for you):
96025
6254
-38276
-3079
57749
Fill in the boxes (One is done for you):
4635− 1586 =
3049
92507 − 73058 =
84306
-57283
26531
-8607
4503 − 3821 =
86013 − 58320 =
Example 11: The number for subtraction from is 8506 and the number to be
subtracted is 3789.
Calculate the difference.
Solution : Number for subtraction from : 8506
Number to be subtracted
: 3789
Difference
: 4717
Answer : 4717.
Example 12 : The sum of two numbers is 7209. One number is 915. Find
the other number.
Solution : Sum of two numbers : 7209
One number
: -915
∴ Other number : 6294
Answer: 6294.
Example 13: The sum of three numbers is 57304. Two of these numbers are
13528 and 4952. Find the third one.
Solution : Two numbers :
13528
+ 4952
18480
Sum of three numbers : 57304
Sum of two numbers : -18480
∴ Third number
: 38824
Answer: 38824.
28
Number
Example 14 : Mojid has Tk. 40895. Kaniz has Tk. 5389 less than Majid.
Fatema has Tk. 987 more than Kaniz. How much money does Fatema have?
Solution : Mojid has Tk. 40895
Tk. 5389
∴ Kaniz has Tk. 35506
Again, Kaniz has Tk. 35506
+ 987
Fatema has Tk. 36493
Answer : Fatema has Tk. 36493.
Example 15 : The sum of the ages of the father and the son is 125 years.
The sonÊs age was 36 years 10 years ago. What will be the age of the father
after 8 years?
Solution : SonÊs age before 10 years 36 years
+ 10 years
SonÊs present age
46 years
Sum of the present age of father and son
125 years
SonÊs present age
- 46 years
∴ FatherÊs present age
FatherÊs present age
∴ FatherÊs age after 8 years
79 years
+8 years
87 years
Answer : 87 years.
79 years
Elementary Mathematics
29
Exercise-2
1.
Add.
(a)
6402
2571
3089
796
(b)
4586
2190
5034
1708
(c)
1526
7085
2600
4954
3197
(d)
4059
5321
8647
2905
3574
(e)
23570
46089
5907
2653
3792
(f)
30791
28056
13807
26085
(g)
12653
17206
24089
32570
10234
(h)
25046
17280
13705
20451
21603
2.
Fill in the boxes:
(a) 2357 + 2506 + 723 + 406 =
(b) 3240 + 1528 + 4039 +1247 =
(c) 2470 + 1305 + 2167 + 3280 + 1075 =
(d) 13264 + 20315 +13204 + 15023 + 398 =
(e) 12304 + 15360 + 20471 + 19052 + 31052 =
3.
Subtract :
(a)
8504
726
(b)
7524
2830
(c)
92503
8607
(d)
61382
9307
(e)
91605
32738
(f)
60327
14569
(g)
51608
12532
(h)
72530
22075
4.
Fill in the blanks :
(a) 6325 896
=
(c) 7052 1695 =
(e) 96021 57234 =
(b) 7104 1368 =
(d) 85213 7359 =
(f) 63205 24307 =
30
Number
5. Subtract and fill in the boxes :
(a) 9403 Here, Number for subtraction from :
4387
Number to be subtracted : :
Difference :
(b)
73251
5374
subtrahend :
Difference :
Number for subtraction from :
(c) 6214− 2578 = Here, Difference
:
Number for subtraction from :
subtrahend
:
(d) 5302− 1805 =
Here,
subtrahend
:
Number for subtraction from :
Difference
:
6.
7.
8.
9.
10.
11.
12.
Here,
2310 people live in a village. 4096 People live in another village and
985 people live in yet another village. Find the number of people
living in three villages.
The sum of two numbers is 83502. One number is 2408. Find the
other number.
In one sum the difference is 5240 and the subtrahend is 759. Find the
number for subtraction from.
Subtract the greatest five-digit number from the smallest Six-digit
number.
The difference between two numbers is 4215. If the number for
subtraction from is 8350. What will be the subtrahend.
3825 mangoes were picked from a tree. 876 mangoes less were
picked from another tree. How many mangoes were picked from the
two trees.
There were 4876 students in a school. 512 students left the school at
the beginning of the year and 1954 students got admitted into the
school. How many students were there afterwards?
Elementary Mathematics
13.
31
The sum of three numbers is 73052. Two numbers of them are 12504
and 38957. Find the third number.
14.
982 is subtracted from a number. If 899 is added to this difference,
the sum becomes 7895. Find the number?
15.
Write the five-digit greatest and smallest numbers by using 3, 5, 7, 0
and 9 each for once only. Now find the difference between these two
numbers.
16.
The sum of a number, when added to 999 becomes 1,00,000. What is
the number?
17.
The sum of the present ages of the father and the daughter is 107
years. The daughterÊs age was 27 years 12 years ago. What would be
the age of the father after 10 years.
18.
Sumi was asked to write four thousand six hundred nine. She wrote
46009. How much less or more did she write ?
19.
Mr.Farid sold paddy for Tk. 4605 and mustard for Tk. 2947. He paid
Tk 3500 to his son and Tk1750 to his daughter. He deposited the rest
of the amount to bank. What amount did he deposit in the bank?
20.
Latifa Begum has Tk. 50890. She paid Tk. 5775 to her son, Tk. 3050
to her elder daughter and 2846 to her younger daughter. She also
repaid a loan of Tk. 1790 and deposited the rest of the amount to
bank. What amount did she deposit in the bank?
32
Addition and Subtraction
Chapter 3
Multiplication
35
×
9 =
315
Multiplicand Multiplier Product
Example 1 : Multiply : 357
× 29
Solution :
357
× 29
3213
7140
10353
Answer: 10353
3 5 7
Explanation :
357
× 29
3213
7140
10353
357×9
357 ×2 tens
2
7
5 tens
3 hundred
Now, 357 × 9
and
357 × 2 tens
Hence, 357 ×9
357 ×2 tens
357×29
7 ×9 = 63
5 tens ×9 = 45 tens
3 hundred × 9 = 27 hundred
9
9
2 tens
= 63
= 450
= 2700
3213
7 ×2 tens = 14 tens
= 140
5 tens ×2 tens = 50 × 2 tens = 100 tens
= 1000
3 hundred × 2 tens = 300 × 2 tens =600 tens = 6000
7140
= 3213
= 7140
= 10353
Elementary Mathematics
Example 2 : Multiply :
33
532
× 37
532
× 37
3724
15960
19684
Answer : 19684
532
× 37
3724
15960
19684
Solution :
Example 3 : Multiply :
Solution : 513
×127
3591
10260
51300
65151
532 × 7
532×3 tens
513
× 127
513
×127
3591
10260
51300
65151
513 × 7
513×2 tens
513×1 hundred
Answer : 65151
Example 4: Multiply
409
× 235
2045
12270
81800
96115
Answer : 96115
Solution :
409
× 235
409
× 235
2045
12270
81800
96115
409 × 5
409×3 tens
409 ×2 hundred
34
Multiplication
Example 5: Multiply
236 by 150
Solution : 236
150
11800
23600
35400
236 5 tens
236 1 hundred
Explanation:
236 5 tens
236 1 hundred
∴ 236 150
= 1180 tens
= 236 hundred
= 11800
= 23600
= 35400
Answer : 35400
Let us look : There is zero (0) in the ones place of the multiplier. For
this multiplication is not operated in the ones place. Hence, putting 0 in
the ones place the operation of tens is made in the first step. The
product is written from one digit left. In the second step, o is put in the
ones and tens place and the multiplication of hundreds is written from
two digits left.
Example 6: Multiply 235 by 102
Solution :
235
102
470
23500
23970
236 2
235 1 hundred
Answer : 23970
Let us look: There is zero (0) in the tens place. For this reason
multiplication of tens is not shown here. Hence, 0 is put in the ones
and tens place of the product. The multiplication of hundreds is written
from 2 digits left.
Elementary Mathematics
35
Example 7 : Multiply : 1452
32
Solution : 1452
32
2904
43560
46464
Example 8 : Multiply : 3526
23
Solution : 3526
23
10578
70520
81098
Answer : 46464
Answer : 81098
Example 9 : Multiply : 384
× 100
Solution :
384
100
38400
Example 10 : Multiply : 1000
100
Solution : 1000
100
100000
Answer : 100000
Answer : 38400
Example 11 : Multiply using simple method:
(a) 245 99 (b) 51 98
Solution : (a) 245 99
= 245 (100-1)
= (245 100)-(245 1)
= 24500 - 245
= 24255
Answer : 24255.
(b) 51 98
= 51 (100-2)
= (51 100)-512)
= 5100 - 102
= 4998
Answer : 4998.
36
Multiplication
Commutative law of Multiplication
Example 12 : Multiply (a)
73
256
(b)
256
73
Solution:
(a) 73
256
438
3650
14600
18688
Answer : 18688
(b) 256
73
768
17920
18688
Answer: 18688
Let us look: The product of multiplication remains the same if the
multiplicand and multiplier are interchanged.
Example 13 : Shakil has Tk. 324/ Robin has money 27 times of what
Shakil has. How much money does Robin have?
Solution :
Shakil has Tk. 324
Robin has Tk. (32427)
Now,
324
27
2268
6480
8748
Answer : Robin has Tk. 8748
Example 14 : 1 kg of mangoes costs Tk. 65. How much does 30 kg of
mangoes cost?
Solution : 1 kg of mango costs Tk. 65
∴ 30 kg of mangoes cost Tk. (6530)
Now,
65
30
1950
Answer : Tk. 1950
Elementary Mathematics
37
Exercise-3
Calculate the product (questions 1-6) :
1.
4.
391
2.
378
69
48
478
5. 358
63
97
3.
522
56
6.
5826
17
Multiply (Question 7-12)
7.
437 by 60
10. 402 by 160
8.
231 by 103
9.
509 by 125
11. 350 by 130
12.
6529 by 15
Calculate the product by using simple method (Questions 13-18):
13.
293 99
14. 912 99
15. 863 90
16.
275 90
17. 99 999
18. 99 990
19. A book has 156 pages. Each page contains 132 words. How many
word does this book contain?
20. Each bag contains 85 kg of rice. How much rice do 980 bags of
this type contain?
21. One duckling costs Tk. 46. How much would 205 duckling cost?
22. One labour earns Tk. 65 per day. How much will he earn in 4
months 5 days? (30 days=1 month).
23. Reba sold 185 chicken from her poultry she received Tk. 75 for
each chicken. How much did she receive altogether?
24. A bundle of five-taka notes contains Tk 500. How much money
would be in 35 bundles?
38
Multiplication
Chapter 4
Division
Example 1 : Divide 964 by 4.
Solution : 4 ) 96 4 ( 241
8
16
16
4
4
0
Answer : Quotient 241
Explanation : 964 = 9 hundred 6 tens 4
4 ) 9 hundred 6 tens 4 ( 2 hundred 4 tens 1
8 hundred
16 tens
(since, 1 hundred 6 tens=16 tens)
16 tens
4
4
0
Let us look : Remainder is zero (0) in the sum It means that 964 is
divisible by 4.
Example 2 : Divide 932 by 6
Solution : 6 ) 932 (155 Explanation: 932 : 9 hundred 3 tens 2
6
6 ) 9 hundred 3 tens 2 ( 1 hundred 5 tens 5
33
6 hundred
30
33 tens [since, 3 hundred 3 tens =33 tens]
32
30 tens
30
32 [since, 3 tens 2 = 32]
2
30
2
Answer : Quotient 155, Remainder 2.
Let us look: Remainder is 2 in the sum. Hence the dividend is not
divisible by the divisor.
Elementary Mathematics
39
Example 3 : Divide 2842 by 14
Solution: 14) 2842 (203 Explanation: 2842 = 2 th 8 huns 4 tens 2
28
= 28 huns 4 tens 2
42
14) 28 huns 4 tens 2 (2 huns 0 tens 3
42
28 huns
0
4 tens
0
42
[since, 4 tens 2 = 42]
42
0
Answer : Quotient 203
Example 4 : Divide 6257 by 17.
Solution: 17 ) 6257( 368
51
115
102
137
136
1
Answer : Quotient 368, Remainder 1.
Example 5 : Divide 2409 by 48.
Solution: 48 ) 2409 ( 50
240
9
0
9
Answer : Quotient 50, Remainder 9.
Let us look : In the second step, 9 from dividend is carried and
written. Now. 9 is smaller than divisor 48. Hence one zero (0) is put in
the ones place of the quotient. There is no other digit in the dividend to
divide. So the process of division is ended here and 9 is found to be the
remainder.
∴ 2409 is not divisible by 48.
40
Division
Relation between multiplication and division
Let us look : Multip-× Multiplier = Product
licand
9
×
7
63 ÷ 9 = 7
= 63
63 ÷ 7 = 9
Multiplicand × Multiplier
= Product
Product ÷ Multiplicand = Multiplier
Product ÷ Multiplier
= Multiplicand
In case of divisibility:
Dividend ÷ Divisor = Quotient
63
÷
9
= 7
Dividend ÷ Divisor=Quotient
Divisor×Quotient =Dividend
Dividend ÷Quotient=Divisor
63 ÷ 7 = 9
9×7 = 63
So it can be said that Division is an opposite process of Multiplication in
the case of divisi bility.
Dividend = Divisor × Quotient + Remainder
Let us look again:
Divisor = (Dividend -Remainder)÷ Quotient)
Devisor Dividend Quotient
Quotient = (Dividend -Remainder) ÷ Divisor
9 )
67
63
4
( 7 Here 67 = 9 × 7+4
Remainder
Example 6 : In a division the divisor is 45,quotient 23 and remainder is 29.
Find the dividend.
Solution : We know, Dividend = Divisor× Quotient + Remainder
Here, Divisor × Quotient = 45 ×23 = 1035
∴ Dividend = 1035 + 29 = 1064
Answer : 1064
Elementary Mathematics
41
Example 7 : In a division the divisor is 1335, quotient 22 and the remainder
is 15. Find the dividend.
Solution : We know, Divisor = (Dividend Remainder) ÷ quotient
∴ Divisor = (1335 15) ÷ 22
= 1320 ÷ 22 = 60
Answer : 60
Example 8 : 25 kg rice cost tk 475. How much does 1 kg rice cost.
Solution : 25 ) 475 (19
25
225
225
0
Answer : tk. 19
Example 9 : 1 kg flour costs tk 17. How much flour can be bought for tk
408?
Solution : 17 ) 408 (24
34
68
68
0
Answer : 24 kg
Example 10 : Divide the smallest four digit number by 64
Solution : Smallest four digit number = 1000
64 ) 1000 (15
64
360
320
40
Answer : Quotient 15, Remainder 40.
42
Multiplication
Example 11 : Divide the greatest five digit number by 64
Solution : Greatest five digit number = 99999
64 ) 99999 (1562
64
359
320
399
384
159
128
31
Answer : Quotient 1562, Remainder 31.
Dividing by 10 or 100
Example 12 : Divide 624 by 10
Solution :
10 ) 624 (62
60
24
20
4
Answer : Quotient 62, Remainder 4.
Let us look: The divisor contains a zero (0) at the end. So if a comma (,) is
set in a place one digit left from the end of the dividend, we can have 62, 4.
Here the number left to comma is the quotient and the right-hand number is
the remainder.
This process is easier for finding quotient and remainder of a sum for
dividing by 10.
Example 13: Divide 4257 by 100.
Solution: The divisor 100 contains two zeroes (0) at the end. So putting a
comma ( , ) in the place two digits left to the end of dividend we can have
42, 57.
Answer : Quotient 42, Remainder 57.
Elementary Mathematics
43
Example 14: Divide 4700 by 100.
Solution: Here the divisor is 100. Putting a comma (,) in a place two digits
left to the end of dividend, we can have 47,00.
Answer : Quotient 47, Remainder 0.
Let us look: When dividing a number by 10 or 100 by using a simple
method, the number of zero in the divisor should be counted first. Next a
comma (,) is to put in a place two digits left to the end of the dividend. The
number left to the comma is the quotient and that to the right is the
remainder.
Exercise- 4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Calculate the quotient and the remainder:
(a) 525 ÷5
(b) 462 ÷6
(c) 637 ÷7
(d) 835 ÷9
(e) 4248 ÷4 (f) 5676÷6
(g) 2436÷7 (h) 7530 ÷8
(i) 7665÷15 (j) 6336 ÷24 (k) 5488÷37 (l) 8001÷65
Calculate quotient and remainder by using simple method.
(a) 650 ÷10
(b) 740 ÷10 (c) 815 ÷10 (d) 5462 ÷100
(e) 6300 ÷100
(f) 7035 ÷100
(a) Dividend is 3640, Divisor is 70, Find quotient.
(b) Dividend is 3414 Divisor is 43, Remainder is 17, Find quotient.
(c) Dividend is 7363, Quotient is 49, Remainder is 13 Find divisor,
(d) Dividend is 3579, Divisor IS 47, Find Remainder.
(e) Divisor is 72, Quotient 123, Remainder 6, Find dividend.
25 dozens of ball pens cost Tk. 950. What is the price of 1 dozen ball pen?
The price of 85 kg rice is Tk. 1445. What is the price of 1 kg rice?
A tape of 1716 metre length is cut to 78 pieces. How much is the
length of one piece?
Divide the smallest four digit number by 16.
Divide the greatest four digit number by 78.
Make the greatest number, using digits 2,3,5,7 once and divide it by
the greatest two-digit numbers.
Make the smallest number using digits 1,4,0,8 once and divide it by
the smallest two digit number.
Distribute Tk. 7642 among 52 persons equally. How much will
everyone get and how much will remain?
Tk. 9702 is to be distributed among 63 persons equally. How much
will everyone get?
44
Multiplication
Chapter-5
Simple Problems
(Related to Addition, Subtraction, Multiplication and Division)
Example 1 : 5 pencils cost tk. 60. How much is the price of 9 pencils
of this type?
Solution : 5 pencils cost tk. 60
∴ 1 pencil costs tk. (60 ÷ 5)
= tk. 12
∴ The price of 9 pencils is tk. (12×9)
= tk. 108
Answer : Tk. 108
Example 2 : 216 banana trees are planted in 9 rows in a garden. How
many banana trees can be planted in 15 rows of such type?
Solution : 216 banana trees are in 9 rows
∴ In 1 row the number of trees = (216 ÷9)
= 24
∴ In 15 rows the number of trees = (24×15)
= 360
Answer : 360 trees
Example 3: There are 165 mangoes in a basket. There are 12 baskets
of managoes. Swapon got 280 and Robin got 210 mangoes from these
mangoes. The rest of the mangoes were distributed equally among
other 5 persons. How many mangoes did each person get?
Solution : 1 basket contains 165 mangoes.
∴12 baskets contain (165 × 12) mangoes
= 1980 Mangoes
Elementary Mathematics
45
Swapon and Robin got (280 + 210) mangoes
= 490 mangoes
The rest of the mangoes are (1980-490)
= 1490 mangoes
Now 5 persons got 1490 mangoes
∴ 1 person got (1490 ÷ 5) mangoes
= 298 mangoes
Answer : 298 mangoes
Example 4: Rupa and Moni have Tk. 875 altogether. Moni has Tk.
125 more than Rupa. What amount do Rupa and Moni have?
Solution : Moni has Tk. 125 more than Rupa. If Tk. 125 is deducted
from the total amount, they will have the equal amount.
Now Tk. (875-125)=Tk. 750
∴ The amount of Rupa = (750 ÷ 2) = 375
∴ The amount of Moni = (375 + 125) = 500
Answer : Rupa : Tk. 375 and Moni Tk. 500
Example 5 : The sum of ages of a mother and a daughter is 55 years.
The mothers age is 4 times as much as the daughterÊs. What are the
ages of the mother and the daughter?
Solution :
DaughterÊs age = 1 time of daughterÊs age
MotherÊs age = 4 times of daughterÊs age
∴The total of mother and daughterÊs age=5 times of daughterÊs age.
∴ DaughterÊs age is (55 ÷ 5) years
= 11 years
∴ MotherÊs age = (11 × 4) years
= 44 years
Answer : MotherÊs age 44 years, daughterÊs age 11 years.
46
Simple Problems
Exercise -5
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
There are 75 oranges in one basket. How many oranges are there
in 8 baskets of the same type?
169 coconuts were distributed equally among 14 persons. How
many did each person get? How many coconuts were left?
8 farmers receive tk 560 as wages by working in a paddy field.
How much is the wage of one farmer per day?
The daily wage for one labour is Tk. 154. How much is the wage
of 28 labourers altogether?
MinaÊs present age is 25 years. Rina is 6 years Junior to her.
RinaÊs age is 11 years more than MithuÊs. Find out the total ages
of Mina, Rina and Mithu.
Kala baught fish for Tk. 145 and vegetables for Tk. 75. He paid
for it with a 500 taka note. How much amount will the
shopkeeper refund to him?
Sohel bought 15 kg of rice at the rate of Tk. 18 each, one hilsha
fish for Tk. 150 and 3 kg of potato at the rate of Tk. 12 from the
market. How much did he spend for buying these things?
The monthly salary of a man is Tk. 7500. He spends Tk. 7275 in
each month. How much will be his savings after 10 months?
The weight of 10 bags of sands is equal to the weight of 8 bags
of cement. One bag of cement weight 50 kg. How much is the
weight of one bag of sands?
There were 575 students in a primary school. 216 students were
admitted at the beginning of the year, but 35 students left the
school. How many students are altogether now in that school?
The product of two numbers is 912. When one number is
multiplied by 5, it become 95. What is the other number?
The quotient of two numbers is 25. When one number is
multiplied by 4, it become 96. What is the other numbers?
The sum of the ages of a father and his son is 68 years. The
fatherÊs age is 3 times as much as his sonÊs. Find the present ages
of the father and the son. What will be their ages after 5 years?
Elementary Mathematics
14.
15.
16.
17.
18.
19.
20.
21.
47
Minu has Tk. 75 more than Chinu and Tk. 35 less than Rabu.
Minu has Tk. 215. How much does each of Chinu and Rabu
have? How much do the three persons have altogether?
The divisor is 4 times as much as the remainder and the quotient
6 times of the divisor. If the Remainder is 3 what will be the
dividend?
The price of 4 chickens and 3 ducks is Tk. 639 altogether. The
price of a duck is Tk. 85. How much is the price of a chicken?
Some guavas were distributed equally among 45 children. The
children got 14 guavas each and 25 guavas remained. How many
guavas were there altogether?
310 lichis were distributed among 50 children. The boys got 5
lichis each and each girl got 2 more than each boy. Find the
number of boys and girls.
Mr. Abdur Rahim bought 25 kg of rice and Pulse for tk 126, oil
for tk 215 and other commodities for tk 105. Each kg of rice
costs tk 18. He gave the seller two 500-taka notes. How much
money should he get back?
Moni and Lipi have tk 936 altogether. Moni has tk 208 less than
Lipi. How much money does each of them have?
14 times of a number when added with 75 it becomes 859. What
is the number?
22. There are 105 members in a society 15 new members joined them.
Each of the members paid tk 75 for a picnic. How much money
was collected as subscription?
48
Simple Problems
Chapter Six
Simplification
Example 1 : Simplify : 154 − 64 + 27 + 59 − 76 − 84
Solution :
154 − 64 + 27 + 59 − 76 − 84
=
90 + 27 + 59 − 76 − 84
=
117 + 59 − 76 − 84
=
176 − 76 − 84
=
100 − 84
=
16
Answer : 16.
Let us look : (a) The expression has only the addition and subtraction
activities.
(b) The simplification is made from the left hand side by
adding and subtracting numbers step by step.
154 − 64 = 90 ; 90 + 27 = 117 ; 117 + 59 = 176;
176 − 76 = 100 ; 100 − 84 = 16
Alternative solution : 154 − 64 + 27 + 59 − 76 − 84
=
(154 + 27 +59) − (64 + 76 + 84)
=
240 − 224
=
16
Let us look : (a) The positive numbers are added together :
154 + 27 + 59 = 240
(b) The negative numbers are added together;
64 + 76 + 84 = 224
(c) The second sum is subtracted from the first sum;
240 − 224 = 16
(d) Brackets () are used for taking the numbers of the same
sign altogether. The alternative process is easier for
solving the simplification.
Elementary Mathematics
49
Example 2 : Simplify : 225 45 60 + 35 + 68 123
Solution : 225 45 60 + 35 + 68 123
=
(225 + 35 + 68) - (45 + 60 + 123)
=
328 - 228
=
100
Answer : 100.
Example 3 : Simplify : 27− 28 + 110 − 120 54 + 23 + 105
Solution : 27 − 28 + 110 − 120 − 54 + 23 + 105
=
(27 + 110 + 23 + 105) −(28+120+54)
=
265 − 202
=
63
Answer : 63
Example 4: Simplify: 2 × 3 × 4 × 5 × 6 × 7
Solution : 2 × 3 × 4 × 5 × 6 × 7
=
6×4×5×6×7
=
24 × 5 × 6 × 7
=
120 × 6 × 7
=
720 × 7
=
5040
Answer : 5040.
Let us look : (a) The problem contains only multiplication activities.
(b) The solution is made by multiplying the numbers
gradually from the left-hand side.
50
Simplification
Simple 5: Simplify : 45 × 4 - 28 × 3 - 15 ×5 + 32 × 2
Solution : 45 × 4 - 28 × 3 - 15 ×5 + 32 × 2
=
180 - 84 - 75 + 64
=
(180 + 64) - (84 + 75)
=
244 - 159
=
85
Answer : 85
Let us look :
(a) The problem has three types of operations. They are addition,
subtraction and multiplication.
(b) At first the operation of multiplication is done. Then the
positive numbers are added together and after it the negative
numbers are added together.
(c) The second sum is subtracted from the first sum.
First of all, the operation of multiplication is to be done when
the problem contains the operations of addition, subtraction and
multiplication.
Example 6 : Simplify : 22 × 22 - 3 × 25 × 2 + 4 × 16 - 160 - 5 × 19
Solution : 22 × 22 - 3 × 25 × 2 + 4 × 16 - 160 - 5 × 19
=
484 - 75 × 2 + 64 - 160 - 95
=
484 - 150 + 64 - 160 - 95
=
(484 + 64) - (150 + 160 + 95)
=
548 - 405
=
143
Answer : 143.
Elementary Mathematics
51
Exercise - 6
Simplify :
1.
75 + 36 - 32 - 39 + 25
2.
772 - 528 + 188 - 222 - 85
3.
85 - 215 + 529 + 50 - 249
4.
455 + 25 - 36 - 157 + 105 - 111 - 82
5.
502 + 112 - 325 - 40 - 101 + 52 - 98
6.
38 - 55 + 278 + 29 - 115 - 8 - 147
7.
2×3×6×8×9
8.
4×8×3×5×7
9.
6×2×8×7×9
10.
5 × 12 - 6× 9 + 18 × 11 - 37 × 2 - 40 × 3
11.
4 × 9 × 21 + 7 - 41× 3 - 32 × 12 - 16 × 16
12.
3 × 5 × 7 - 43 × 4 +6 × 11 × 2 - 13 × 5+3 × 2+4
13.
7 × 9 × 8 - 3 × 6 × 9 + 42- 25 × 8 + 23 × 4 - 5 × 5 × 11 + 3
14.
103 × 10 +7 - 25 × 6 × 5 - 5 × 5 × 9 + 3 × 2 - 13 × 5
15.
25 - 4× 2 + 11 × 5 - 6 × 7 × 3 + 2 × 4 × 8
52
Simplification
Chapter Seven
Measures and Multiples
There are 12 lichis for one pupil.
Each receives one lichi
12 × 1 = 12
There are 12 lichis for 2 pupils.
They receive 6 lichis each;
6 × 2 =12
There are 12 lichis for 3 pupils.
They receive 4 lichis each;
4 ×3 = 12
There are 12 lichis for 4 pupils.
They receive 3 lichis each,
3 × 4 = 12
Elementary Mathematics
53
There are 12 lichis for 6 pupils.
They receive 2 lichis each.
2 x 6 =12
There are 12 lichis for 12 pupils.
They receive 1 lichis each;
1 x 12 = 12
Let us look:
12 lichis can be shared among 1, 2, 3, 4, 6 or 12 pupils equally.
12 is divisible by 1, 2, 3, 4, 6 and 12.
Each of 1, 2, 3, 4, 6 and 12 is the multiplier or factor of 12.
12 lichis cannot be shared equally among 5 pupils.
5 is not the multiplier or factor of 12.
The numbers by which a larger number can be divided are the
measures or factors of that number. The measures are also named
as factors.
54
Measures and Multiples
4 = 1× 4
= 2 ×2
8 = 1× 8
= 2× 4
4 is divisible by 1, 2 and 4.
∴ Each of these numbers is the factor of 4.
4 is not divisible by 3. So 3 is not any
measure or factor of 4.
∴ All the factors of 4 are : 1, 2, 4
8 is divisible by 1, 2, 4 and 8.
∴ Each of these numbers is the factor of 8.
8 is not divisible by 3, 5, 6 or 7.
Hence none of these numbers is the factor
of 8.
∴ All the factors of 12 are : 1, 2, 4, 8
12 = 1 × 12
= 2× 6
=3× 4
12 is divisible by 1, 2, 3, 4, 6 and 12.
∴ Each of these numbers is the factor of
12.
12 is not divisible by 5, 7, 8, 9, 10 or 11.
Hence none of these numbers is the factor
of 12.
∴ all the factors of 12 are : 1, 2, 3, 4, 6, 12.
14 is divisible by 1, 2, 7 and 14;
14 = 1 × 14
= 2× 7
Each of this numbers is the factor of 14
14 is not divisible by 3, 4, 5, 6, 8, 9, 10,
11, 12,13
Hence none of these numbers is the factor of 14
17=1 × 17
∴ All the factors of 14 are : 1, 2, 7, 14
17 is divisible by 1 and 17;
but 17 is not divisible by any other
numbers smaller than 17.
∴ The factors of 17 are : 1 and 17.
Let us look :
Every number is the factor of its own.
1 can be the factor of any number.
Elementary Mathematics
55
The factor of 1 is 1 alone.
Any factor of a number must be equal to or smaller than the
number itself.
Any number other than 1 has several factors. These factors
except the numbers itself are equal to or smaller than the half of
this number.
The numbers 2 , 3, 5, 7, 11, 13, 17 etc. have only two factors.
They are 1 and the number itself.
The number should be written in the form of the multiple of two
numbers of all possibilities. Then it is possible to find all the
factors of the numbers.
The smallest factor of any number is 1 and the greatest one is the
number itself.
Determination of measures
Example 1 : Find all the factors of 32.
Solution : 32 = 1 × 32
= 2 × 16
=4×8
Answer : The possible factors of 32 are : 1, 2, 4, 8, 16, 32.
Example 2 : Find the possible factors of 49
Solution : 49 = 1 × 49
= 7 ×7
Answer : The possible factors of 49 : 1, 7, 49.
Example 3 : Find the possible factors of 60
Solution : 60
=
1 × 60
=
2 × 30
=
3 × 20
=
4 × 15
=
5 × 12
=
6 × 10
Answer : The possible factors of 60 : 1, 2, 3, 4, 5, 6, 10, 12, 15, 20,30,
60.
56
Measures and Multiples
Example 4 : Find the possible factors of 175.
Solution : 175 =
1 × 175
=
5 × 35
=
7 × 25
Answer : The possible factors of 175 : 1, 5, 7, 25, 35, 175.
Example 5 : Find the possible factors of 200.
=
1 × 200
=
2 × 100
=
4 × 50
=
5 × 40
=
8 × 25
=
10 × 20
Answer : 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200.
Solution : 200
Prime numbers, Composite numbers
The numbers which have no factors other than itself and 1, are prime
numbers. The prime numbers are greater than 1.
The numbers which have at least one factor other than itself and 1, are
composite numbers. The composite numbers are greater than 1.
Numbers
2
3
4
5
6
7
8
9
10
11
Whether the number has
any factor other than and
itself
No
No
Yes, eg 2
No
Yes, eg 2
No
Yes, eg 2
Yes eg 3
Yes, eg 2
No
Category
Prime number
Prime number
Composite number
Prime number
Composite number
Prime number
Composite number
Composite number
Composite number
Prime number
Elementary Mathematics
Numbers
12
13
14
15
16
17
18
19
20
57
Whether the number has
any factor other than and
itself
Yes, eg 2
No
Yes, eg 2
Yes, eg 3
Yes, eg 2
No
Yes, eg 2
No
Yes, eg 2
Category
Composite number
Prime number
Composite number
Composite number
Composite number
Prime number
Composite number
Prime number
Composite number
Let us look:
The prime numbers up to 20 are: 2, 3, 5, 7, 11, 13, 17, 19
The smallest prime number is 2.
The prime numbers other than 2 are odd. But all the odd
numbers are not prime.
Composite numbers have at least 3 factors.
1 is neither prime nor composite.
Verifying the divisibility
(a) Divisible by 2
2 × 0 = 0, 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, 2 × 4 = 8,
2 × 5 = 10, 2 ×6 = 12, 2 × 7 = 14, 2 × 8 = 16, 2 ×9 = 18,
Hence, when any number is multiplied by 2, the product number will
have 0, 2, 4, 6 or 8 in the ones place. So it can be said that numbers
having 0, 2, 4, 6 or 8 digits in the ones place would be divisible by 2.
These numbers are known to us as even number.
(b) Divisible by 3
Let us look,
10
=
100 =
1000 =
3×3+1
3 × 33 + 1
3 × 333+1
58
Simplification
so 261
= 200 + 60 + 1
= 2 × 100 + 6 ×10 + 1
= (3× 66 +2) + (3 ×18 + 6) +1
= (3× 66 + 3 18) + (2 + 6+1)
The first term inside the brackets in the right-hand side is divisible
by 3 (Quotient : 66 + 18). So the divisibility of 261 by 3 would be
possible if the other term 2+6+1 inside the brackets is divisible by 3.
The sum of this term is the sum of the digits contained in the
number 261. In this situation 2+6+1=9 and it is divisible by 3,
consequently, 261 is divisible by 3.
If the sum of the digits of a number is divisible by 3, the number
will also be divisible by 3.
418 is not divisible by 3. It is because 4+1+8=13 and it is not
divisible by 3. 672 is divisible by 3. It is because 6+7+2=15 and it is
divisible by 3.
(c) Divisible by 5
0 × 5 = 0, 1 × 5 = 5, 2 × 5 = 10, 3 × 5 = 15, 4 × 5 = 20,
5 × 5 = 25, 6 × 5 = 30, 7 × 5 = 35, 8 × 5 = 40, 9 × 5 = 45,
If any number is multiplied by 5, the product of multiplication
will contain 0 or 5 in its ones place. Hence if the digit of the ones
of a number contains 0 or 5, the number must be divisible by 5.
Example 6 : Is 87 a prime or a composite number?
Solution : 8 +7 = 15, it is divisible by 3
∴ 87 is a composite number.
Answer : 87 is a composite number.
Example 7 : Is 745 a prime or a composite number?
Solution : The digit in the unit of this number is 5.
∴ The number is divisible by 3.
Answer : 745 is a composite number.
Elementary Mathematics
59
Example 8 : Is 43 a prime or a composite number?
Solution : 43 is an odd number. Hence, it is not divisible by 2.
4+3 = 7 and it is not divisible by 3. So 43 is not divisible by 3.
Again the digit of ones in 43 is not 0 or 5.Hence 43 is not divisible by 5.
7 is the next prime number after 5. But 7×7=49 and it is greater than
43. So it is not needed to verify the divisibility of 43 by a prime
number greater than 7.
∴ 43 is a prime number.
Answer : 43 is a prime number.
Example 9 : Is 89 a prime or a composite number?
Solution : 89 is an odd number. Hence, it is not divisible by 2.
8+9 = 17 and it is and divisible by 3. So 89 is not divisible by 3.
Again, the digit of ones place in 89 is not 0 or 5. Hence 89 is not
divisible by 5.
Now let us divide the number
7) 89 ( 12
directly and observe,
7
89 is not divisible by 7.
19
14
5
11 is the next prime number after 7. But 11×11=121 which is greater
than 89. Hence, it is not needed to verify the divisibility of 89 by a
prime number greater than 7.
89 is a prime numbers.
Answer : 89 is a prime number.
Let us look:
To identify a prime or a composite number it is enough to verify
its divisibility by such prime numbers whose squares are smaller
than the number itself.
Prime numbers up to 20 have been identified previously. Now the
prime numbers between above 20 and up to 100 are as follows:
23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
60
Simplification
Prime factors
The factors of 12 are 1, 2, 3, 4, 6 and 12
2 and 3 are the prime factors among them. 12 may be expressed as
the product of two factors using any way. In each way 12 is
expressed in terms of these two prime factors 2 and 3. Such as,
12 =2 × 6 = 2 × 2 × 3
12 = 3 × 4 = 3 × 2 × 2 = 2 × 2 × 3
2 × 2 × 3 is called the prime factor analysis of 12.
The analysis of factors for 36 may be performed at least in four ways:
36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3
36 = 3 × 12 = 3 × 2 × 6 = 3 × 2 × 2 × 3 = 2 × 2 × 3 × 3
36 = 4 × 9 = 2 × 2 × 3 × 3
36 = 6 × 6 = 2 × 3 × 2 × 3 = 2 × 2 × 3 × 3
Whatever may be the way, the analysis of factors for 36 always
produces the same result. The result is 2 × 2 × 3× 3
12 = 2 × 6
2 3
36 = 2 × 18
2×9
3×3
12 = 3 × 4
2 2
36 = 3 × 12
2×6
2×3
36 = 4 × 9
2× 2 3×3
36 = 6 × 6
2×3 2×3
Elementary Mathematics
61
Express the following numbers into prime factors (one example is given
here:)
Numbers
Analysis
Prime factor analysis
5 × 5× 7
5 × 35
175
5× 7
120
135
273
343
472
Common Measures, Greatest Common Measures
All measures (factors) of 12 are 1, 2, 3, 4, 6 and 12.
All measures (factors of 18 are 1, 2, 3, 6, 9 and 18.
Therefore, the numbers 1, 2, 3 and 6 are the measure of both 12 and 18.
Hence, these are the common measures of 12 and 18. The largest of the
common measures is 6 and we call 6 the greatest common measure or
G.C.M of 12 and 18
[Alternatively we call GCM as the Highest Common Factor (HCF)
The greatest measure that lies among the common measures/factors of
two numbers is the Greatest Common Measure (GCM)
4
12
Measures/Factors of 12
1
2
3
6
9
18
Measures/factors 18
62
Simplification
Example 10 : Find the GCM of 24 and 36.
Solution : All measures of 24 : 1, 2, 3, 4, 6, 8, 12, 24.
All measures of 36 : 1, 2, 3, 4, 6, 9, 12, 18, 36.
∴ Common measures of 24 and 36 : 1, 2, 3, 4, 6, 12
∴ Greatest common measure of 24 and 36 = 12
Example 11 : Find the GCM of 15 and 22.
Solution : All measures of 15 : 1, 3, 5, 15.
All measures of 22 : 1, 2, 11, 22.
∴ The only common measure of 15 and 22 : 1.
∴ Greatest common measure of 15 and 22 = 1
The GCM of two numbers may be 1 in some cases
Example 12 : Find the GCM of 16 and 64.
Solution : Here the greater number is divisible by the smaller number.
16 ) 64( 4
64
0
Hence, smaller number is the determined GCM
Answer : GCM of 16 and 64 =16.
The GCM of two numbers may in some cases be the smaller number itself
Finding out GCM by using prime factors :
Example 13 : Find the GCM of 225 and 455
5)225(45
Solution : 225=5 × 45
20
=5×5×9
25
= 5×5×3×3
25
= 3 ×3×5× 5
0
455 = 5×91
5 ) 455( 91
= 5× 7×13
45
5
5
0
Among the prime factors of 225 and 455
There is only a common factor it is 5.
The GCM/HCF=5.
Elementary Mathematics
63
Example 14 : Find the GCM of 12, 30 and 36.
Solution : 36 is divisible by 12. Hence, the GCM of 12 and 30 will be the
required GCM.
Now,
12 = 2 × 6=2×2×3
30 = 2 ×15=2×3×5
Two prime factor analysis of 12 and 30 contains a common part. It is 2×3.
∴ The required GCM = 2×3=6
Example 15 : Find the GCM of 45, 60 and 75.
Solution :
45 = 3 ×15=3×3×5
60 = 2 × 30=2×2×15=2×2×3×5
75 = 3 × 25 = 3×5×5
The prime factor analysis of 45, 60 and 75 contains a common part. It is
3×5.
The required GCM = 3×5=15
Multiples
When a number is divisible by another number, the first number is called
the multiple of the other number. It is because, if the other number is
multiplied by the quotient the product becomes the first number.
Such as, 21 = 3×7 and as such 21 is the multiple of 3. Again, it is the
multiple of 7 too.
Hence, any number is the multiple of it factors/measures.
One multiple of a number can be derived by multiplying it by any
number.
Such as , 7 ×1=7, 7× 2=7, 7×3=21, 7× 4=28, 7× 5=35 etc.
Every product is the multiple of 7.
Let us look:
If a number is divisible by another numbers the first number becomes
the multiple of both the divider and the quotient.
Every number is the multiple of 1 and the number itself.
Every number has infinite multiples.
Every number itself is the smallest multiple of it.
64
Simplification
Example 16: Find the ten multiples of 8.
Solution : 8×1=8
8×2=16
8×4=32
8×3=24
8×6=48
8×5=40
8×8=64
8 ×7=56
8×10=80
8×9=72
First ten multiples of 8 : 8, 16, 24, 32, 40, 48, 56, 64, 72, 80.
Common multiples, Lowest common multiple
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 etc.
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 etc.
Let us look : 24, 48, etc are the multiples of both 6 and 8. Hence both
24 and 48 are the common multiples of 6 and 8. 24 is the lowest one
among the common multiples.
The lowest common multiples of 6 and 8 = 24.
In short, lowest common multiple is called LCM.
Example 17 : Find the LCM of 10 and 30.
Solution : 30 is the multiple of 10, because 30=3×10
Again, the lowest multiple of 30 is 30 itself.
1 CM of 10 and 30 = 30
In some cases the LCM of two numbers may be the larger number
Finding LCM by Prime factorization method.
Example 18: Find the LCM of 18 and 24
Solution : 18 = 2×9=2×3×3
24 = 2×12=2×2×6=2×2×2×3
In the two groups of factors, 2 is found maximum three times and 3 is
found maximum twice.
The LCM=2×2×2×3×3 =8×9=72
Elementary Mathematics
65
Example 19 : Find the LCM of 35 and 50.
Solution : 35 = 5 ×7
50 = 2×25=2×5×5
In the two groups of factors 2 is found maximum once, 5 found twice and 7
once.
∴ The required LCM=2 ×5 ×5 ×7=10×35
=350
Example 20 : Find the LCM of 12, 18 and 24.
Solution : 24 = 2 ×12 is the multiple of 12. Hence, the LCM of 18 and 24
will be the required LCM.
LCM of18 and 24=72 (Example 18)
∴ The LCM=72
Example 21 : Find the LCM of 16, 24 and 56.
Solution : 16 = 2 ×8=2×2× 4=2 ×2×2 ×2
24 = 2 ×12=2×2×6=2 ×2 ×2×3
36=2 ×18=2×2× 9=2× 2×3×3
In the three groups of factors 2 is found maximum four times, 3 is found
maximum twice.
∴ The required LCM=2× 2×2×2 ×3 ×3=16×9
=144
66
Simplification
Exercie-7
1.
Find the possible measures of the following :
(a) 16
(b) 30
(c) 42
(d) 45
(f) 105
(g) 144
(h) 189
(i) 216
(e) 324
(j) 234
2.
Identify and write which numbers of the following are prime or
composite:
(a) 28 (b) 31 (c) 47 (d) 87 (e) 91
3.
Analyse the following numbers into prime factors :
(a) 60
(b) 72
(c) 93
(d) 112
(e) 135
(f) 222
(g) 255 (h) 320
(i) 484
(j) 512
4.
Find which numbers are divisible by 2, 3 or 5 and write them
separately:
(a) 69
(b) 80
(c) 95
(d) 165
(e) 255
(f) 360
(g) 435
(h) 555 (i) 717
(j) 777
5.
Write five multiples for each of the following numbers:
(a) 6 (b) 9
(c) 12
(d) 16
(e) 21
(f) 28 (g) 35
(h) 42
(k) 44
(j) 50
6.
Find the possible common measures of the following pairs of
numbers.
(a) 18, 42 (b)30, 75
(c) 36, 45
(d) 24. 64 (e) 48, 56
(f) 60, 100
7.
Find the GCM:
(a) 16, 24
(b) 28, 42
(e) 12, 18, 30
(f) 16, 24, 40
8.
Find the LCM:
(a) 12, 16
(e) 36, 45
(i) 12, 18, 24
(b) 18, 24
(f) 42, 56
(j) 20, 25, 30
(c) 35, 49
(g) 51, 63, 99
(c) 20, 50
(g) 46, 69
(k) 32, 48, 72
(d) 45, 60
(h)( 48. 60, 72
(d) 15, 35
(h) 38, 57
Elementary Mathematics
67
Chapter-Eight
Mathematical Symbols
Number Symbols :
All numbers are written by using ten symbols. These are 0, 1, 2, 3, 4, 5, 6,
7, 8 and 9. Hence, they are called number symbols. The symbols which
are used for writing numbers are called digits.
Operational symbols : + (plus sign),sign), (Division sign) are the operational symbols.
Relation Symbols : = (Equalization sign), > (Greater sign / Larger sign),
< (Smaller sign or small sign). These two signs indicate the relation
between two numbers. Such as7 × 12 = 84, 27 + 11 > 38-7
×
36 ÷
Opposite Relation symbols
=/ (Not equal ) 16 =/ 27
> (Not greater ) 23 > 32
< (Not smaller ) 32 < 23
×
Such as : 6 ×
35-17 > 15 + 12
40 + 5 < 60 -35
72 ÷
×2
Mathematical sentence ( Statement )
The ten examples mentioned above are mathematical sentences or
statements. Mathematical statements may be true or may be false. The
above mentioned statements are true. Examples of false statement:
10 ÷ 2 = 6
2×13> 25 + 11
27 ×
68
Mathematical Symbols
Open Sentence
+ 5 = 12
It is also a type of mathematical sentence. Here symbol is used to express
an indefinite number. We cannot say whether a sentence is true or false until a
definite number is set in . This type of sentence is called an open sentence.
When a definite number is put in the symbol of an open sentence, we get
a mathematical statement. This statement may be true or false such as:
7 + 5 = 12 True statement
6 + 5 = 12 False statement
9 + 5 = 12 False statement
True statement will only be found when 7 is put in
symbol of
+ 5 = 12
Number terms
A number term is formed when some numbers are attached with operational
sign. Such as :
_
26 ÷ 2 + 3 × 7 15
Bracket sign
Brackets are used to indicate which operation is to be performed first: Such
_
as : 37 ( 13 + 8 )
In this mathematical term, addition operation is to be performed first and
then the sum to be subtracted from 37.
Example 1: Show that (37-13) _ 8 = 37 _ (13 + 8)
_
_
Solution : Left-hand side = (37 13) 8
_
= 24 8
= 16
_
_
Right-had side = 37 (13 + 8) = 37 21 = 16
∴ Left-had side = Right-hand side
Example 2 : Use number symbol and opposite relation symbol with the
following statements.
(a) Seven hundred eighty-nine is not greater than the nine hundred nine.
(b) Four thousand four and forty Thousend four are not equal.
Solution : (a) Seven hundred eighty-nine =789
Nine hundred nine = 909
Answer : 789 > 909
(b) Four thousand four = 404
Answer : 4004 = 40004
Elementary Mathematics
69
Example 3 : Make a true statement by putting relation symbol in the
(blank) space.
(a) 137 _ 69 47 + 14
(b) 16 ×12 24×8
(c) (72÷9)×4 (72× 4)÷9
(d) (225 _ 169) _ 33 347 _ (211 + 56)
(e) (43 + 26)×7 43×7 + 26 ×7
Solution (a) : 137 _ 69 = 68
47 + 14 = 61
∴ 68 > 61
Answer : 137 _ 69 > 47 + 14
Solution (b) : 16 ×12 = 192
Again, 24×8 = 192
∴ 16 ×12 = 24 ×8
Answer : 16 ×12 = 24 ×8
Solution (c) : (72÷9) ×4 = 8×4 = 32
Again (72×4 ) ÷9 = 288 ÷9 = 32
∴ 72÷9 ×4 = (72×4)÷ 9
Answer : 72 ÷9×4 = (72 ×4)÷9
Solution (d) : (225 _ 169) _ 33 = 56 _ 33 = 23
Again 347 _ (211 + 56) = 347 _ 267 = 80
Now, 23 < 80
∴ (225 _ 169) _ 33 < 347 _ (211 + 56)
Answer : (225 _ 169) _ 93 < 347 _ (211 + 56)
Solution (e) : (43 + 26)×7 = 69×7 = 483
Again 43×7 + 26 ×7 = 301 + 182 = 483
∴ (43 + 26) ×7 = 43 ×7 + 26× 7
Answer : (43 + 26)×7 = 43×7 + 26×7
70
Mathematical Symbols
Example 4: Write the questions by using symbols and find out the unknown
figures.
(a) When 17 is subtracted from a number the difference becomes 12. What is
the number?
(b) When a number is multiplied by 9 the product becomes 108. What is the
number?
Solution : (a) If symbol is taken for the unknown number, the question
becomes:
If −17 = 12; What is equal to?
Subtraction is the reciprocal operation of addition. Hence the number in space would be the sum of 12 and 17.
∴ = 12 + 17 = 29
Answer : 29
(b) If is taken for the unknown number, the question becomes.
If × 9 = 108, What is equal to?
Division is the reciprocal operation of multiplication. Hence the number in
would be the quotient of 108 and 9.
= 108 ÷ 9 = 12
Answer : 12
Exercise-8
1.
(a)
(b)
(c)
(d)
(e)
2.
(a)
(b)
(c)
(d)
(e)
Use symbols to express the following sums:
One hundred twenty-seven plus three hundreds three
Four hundred sixty-five minus two hundred twenty-nine
Eighty-eight multiplied by seven
Two hundred eighty-nine divided by seventeen
Difference between seventy-two and thirty-eight multiplied by eleven.
Use symbols to express the following statements:
The difference between three hundred fifty-seven and two hundred
twelve is greater than fifty
The quotient of three hundred sixty and eighteen is not equal to
twenty-one.
The six times of the sum of twenty-one and sixteen is equal to the
sum of the six times of twenty one and six times of sixteen.
23 times of 23 is greater than five hundred.
If the quotient of one hundred forty-four and twelve is multiplied by
sixteen, the product becomes one hundred ninety-two.
Elementary Mathematics
3.
4.
5.
6.
7.
(a)
(b)
(c)
(d)
(e)
Estimate which of the following statements are true or false.
_
_
(a) 76 47
14 = 195 ÷ 13
(b) 537 + 482 = 689 + 320
(c) 59 + 47 > 63 + 39
(d) (380 ÷19) ×17 =/ (397÷17)×19
(e) (187 + 121) + 97 < 187 + (121 + 97)
Fill in the boxes ( ) with any of the relation symbols of =, > and
< , so that the statement formed is True.
_
(a) 49 + 28 103
37
(b) 23×13 × 21× 24
_
_
_
(c) (189 27)
69 189
(27 + 69)
(d) (13×11) ×7 13×(11× 7)
(e) (343 ÷49)×6 343÷(7×7)
Fill in the boxes ( ) with such an operational sign that the
statement becomes true.
(a) 71 35 = 9 4
(b) 36 24 = 18 48
(c) 7 5 = 105 3
(d) 87 38 = 7 7
(e) 324 18 = 19 18
Put numbers in the following open sentence so that the statements
formed become true.
_
_
(a) 7 = 32
17
(b) 6 × = 27 × 2
(c) ÷11 = 6 + 7
(d) 72 ÷ = 24÷3
_
_
(e) 17 11 = 39
Express the following questions by using symbols and find out the
unknown numbers.
When a number is added with 23, the sum becomes 191. What is
the number?
When thirty-nine is subtracted from a number, the difference
becomes 86. What is the numbers?
The product becomes 169 when a number is multiplied by 13.
What is the numbers?
The quotient becomes 15 when a number is divided by 19. What is
the number?
When 8 is added to a number, the sum becomes equal to the
difference between 17 and 9. What is the number?
71
72
Mathematical Symbols
Chapter-Nine
Simple Fraction
Two thirds
Or, two divided by three
2
3
Five eighths
Or, five divided by eight
5
8
Three fourths
Or, three divided by four
3
4
Seven twelfths
Or, seven divided by twelve
7
12
Let us look:
Numerator A line is drawn. The number of divisions has been
2
written under the line as denominator and the
3
denominator number of shaded divisions has been written
above the line as numerator. In the following fractions:
2 5 3 7
, , ,
3 8 4 12
etc. Numerators are smaller than denominators.
Elementary Mathematics
73
4
4
3
4
2
4
1
4
1
2
2
3
3
5
5
5
2 3 4
is equal to 1.
, , and
2 3 4
5
Here, Numerator =Denominator.
Let us look : Each one of
Equivalent Fractions
1
2
1
2
2
4
3
6
4
8
5
10
6
12
2
4
3
6
4
8
5
10
6
12
74
Simple Fraction
1× 2 2 1 1× 3 3
= ; =
=
2 2× 2 4 2 2×3 6
1× 4 4 1 1× 5 5
1
=
= ; =
=
2 2 × 4 8 2 2 × 5 10
1× 6 6
1
=
=
2 2 × 6 12
1 2 3 4 5 6
= = = = =
Let us look :
2 4 6 8 10 12
1 2 3 4 5 6
∴
, , , , ,
etc are equivalent fractions.
2 4 6 8 10 12
3 3× 2 6 3 3× 3 9
=
= ; =
=
5 5 × 2 10 5 5 × 3 15
3 3 × 4 12 3 3 × 5 15
=
= ; =
=
5 5 × 4 20 5 5 × 5 25
3 6 9 12 15
etc are equivalent fractio ns.
∴ , , , ,
5 10 15 20 25
1
=
If the numerator and the denominator of a fraction are multiplied
by the same number the new fraction will be equivalent.
Let us look again : ∴
9 9÷3 3
=
=
15 15 ÷ 3 5
12 ÷ 4 3
=
20 ÷ 4 5
15 15 ÷ 5 3
=
=
25 25 ÷ 5 5
12
20
=
If the numerator and the denominator of a fraction are divided by
the same number, the fraction becomes the equivalent of the
previous fraction.
Elementary Mathematics
75
The smallest form of fraction
24 24 ÷ 2 12
=
=
72 72 ÷ 2 36
Here, the numerator and the denominator of the fractions are divided
by the common factor 2. This division and omitting 2 as common
factor of the numerator and the denominator is the same.
12 12 ÷ 2 6
=
=
36 36 ÷ 2 18
6÷2 3
6
=
=
18 18 ÷ 2 9
3 3÷3 1
And
=
=
9 9 ÷3 3
Again.
1
12 6 3
, , and are smaller than the
3
36 18 9
denominators of the original fractions.
Let us look : The denominators of
∴
24
72
=
1× 2 × 2 × 2 × 3
1
= is the smallest form of the original
1× 2 × 2 × 2 × 3 × 3 3
fraction.
Let us look : If we divide the denominator and the numerator by the
greatest Common factor of them, we can have the smallest form of the
24 24 ÷ 24 1
=
=
fraction :
72 72 ÷ 24 3
26
in the smallest form.
Example 1: Express
34
26 1× 2 ×13 13
Solution :
=
=
34 1× 2 ×17 17
There is no factor of 13 an 17 other than 1.
26
13
∴
is the smallest form of
.
34
17
Let us look : 1 is the highest common factor of the numerator and the
denominator of a fraction.
76
Simple Fraction
Example 2 : Express the following fractions in the smallest form
54
48
(b)
(a)
81
96
2
54 1× 2 × 3 × 3 × 3
=
(a) solution : =
81 1× 3 × 3 × 3 × 3
3
∴The smallest form of
Answer :
54
81
is
2
3
2
3
48 48 ÷ 4 12
12 12 ÷12 1
=
=
Again
=
=
96 96 ÷ 4 24
24 24 ÷12 2
1
48
∴ is the smallest form of
2
96
(b) solution :
Answer :
1
2
We can have the smallest form of a fraction, when the numerator and
the denominator are divided by its common factors step by step.
Fractions of equal denominators
9
15
and
have the same denominators. These are the fractions of
32
32
3
equal denominators. 7 and
are not the fractions of equal
8
5
denominators. These can be expressed with the equal denominator
form
7 7 × 5 35
=
=
8 8 × 5 40
3 3 × 8 24
=
=
5 5 × 8 40
Again,
Denominator is made 40 in both the fractions.
7 7 ×10 70
=
=
Denominator is made 80 in both the fractions.
8 10 × 8 80
3 3 ×16 48
=
=
×
5
16
80
5
Elementary Mathematics
77
The fractions may be converted into equal denominator form by
making the denominators in the form of common multiples. Generally
the LCM of the denominators are used to make them equal
denominator form.
5
3
Example 3 : Convert and into fractions having 48 as
6
4
denominator.
5 × 8 40
=
6 × 8 48
3 3×12 36
48 ÷ 4 =12 ∴ =
=
4 4 ×12 48
40
36
and
Answer :
Solution : 48 ÷ 6 =8 ∴
48
5
6
=
48
Example 4 : Express
5
12
and
7
8
as the fractions having equal
denominators.
Solution : The LCM of the denominators of 12 and 8 is 24 .
5 × 2 10
5
24 ÷ 12 =2 ∴ =
=
12 12 × 2 24
7 7 × 3 21
24 ÷ 8 =3 ∴ =
=
8 8 × 3 24
10
21
and
Answer :
24
24
Let us look: The LCM of the denominators is divided by each
denominator. The numerator and the denominator of each fraction are
multiplied by the quotient. In this way the fractions of equal
denominators are made.
Fill in the blanks after making the smallest equal denominators (One is
done for you)
(a)
(c)
3
5
2
7
7
8
5
6
=
=
=
=
21
35
10
35
(b)
(d)
5
=
6
4
=
9
3
=
4
5
=
6
78
Simple Fraction
2
3
(e)
=
4
=
7
5
=
8
5
=
14
(d)
5
=
8
3
=
16
Example 5 : Put the right numbers in the box:
(a)
5
6
=
30
Solution: a) Here, 30÷6=5
5 5 × 5 25
∴ =
=
6 6 × 5 30
Answer :
5 25
=
6 30
(b)
24
6
=
11
b) Here, 24 ÷6=4
6 × 4 24
6
∴ =
=
11 11× 4 44
Answer :
6 24
=
11 44
Comparison between fractions
4
8
5
8
5
8
is greater than
Again,
4
8
4
,
8
in other words 5 >
8
5
8
4
8
is smaller than , In other words
4
8
<
5
8
Elementary Mathematics
79
2
9
2
9
5
9
5
9
is smaller than ; in other words;
Again,
5
9
is greater than
2
;
9
2
9
<
5
9
In other words;
Let as look : The denominators of
But numerator 5 > numerator 2 ∴
5
9
>
5
2
and are
9
9
5 2
>
9 9
7
∴ 6<
11 11
6
= 6 parts out of 11
11
7
= 7 parts out of 11
11
6<7
10
=10 parts out of 17
17
6
=6 parts out of 17
17
10>6 ∴
2
9
the same
6
10
>
17 17
If the denominators are the same, the fraction with
greater numerator is greater.
3
Example 6 : Which one of 7 and
is greater? Show it by using
10
10
mathematical symbol.
7
= 7 parts out of 10
Solution :
Q 7>3
Answer
10
3
= 3 parts
10
7
3
∴ >
10 10
7
3
:
>
10 10
out of 10
80
Simple Fraction
Example 7: which one of
5
16
and
9
16
is smaller?
Solution : The denominators of the two fractions are 16,
The numerators of these fractions are 5 and 9.
5
9
<
Q 5<9∴
16
Answer :
5
16
16
is smaller.
3
11
3
5
Let us look: The numerators of
Q 11 > 5
3
11
<
3
8
3
5
and
3
11
are the same.
3
5
3
16
Q 8 < 16 ∴
3
8
>
3
16
When the numerators are same, the fractions having a
smaller denominator is greater.
Example 8 : Which one of the fractions 7 and 7 is greater?
18
15
Show it by using mathematical symbol
Solution : The numerators of the two fractions are the same,
Denominators are 18 and 15
7
7
>
18 > 15 ∴
15
Answer :
7
15
>
18
7
18
Elementary Mathematics
Example 9 : which one of
81
3
15
and
4
5
is greater? Use the symbol.
Solution : The denominators of the two fractions are 15 and 5. The
LCM of them are 15.
3 ×1 3
3
=
15 ÷ 15 = 1
∴ =
15 15 ×1 15
4 4 × 3 12
=
15 ÷ 5 = 3
∴ =
5 5 × 3 15
3
3
12
4
> , In other words, >
12 > 3
∴
15 15
5 15
4 3
Answer :
>
5 15
Let us look : The two fractions are converted into the fractions of
equal denominators. Then their numerators are compared to identify
greater one.
Example 10 : Arrange the following fractions in order of the smallest
7 13 11
to the highest values : ,
,
8 16 24
Solution : The denominators of the fractions are 8, 16 and 24. The
LCM of these numbers is 48
7 7 × 6 42
=
48 ÷ 8 = 6
∴ =
8 8 × 6 48
13 13 × 3 39
=
48 ÷ 16 = 3
∴ =
16 16 × 3 48
11 11× 2 22
48 ÷ 24 = 2
∴
=
=
24 24 × 2 48
22 39 42
∴ < <
Q 22 < 39<42
48 48 48
11 13 7
Or,
< <
24 16 8
Answer : By arranging in order from the smallest to the highest;
11 13 7
, , .
24 16 8
82
Simple Fraction
Addition of Fraction
(a)
3
7
1
7
4
7
We can see in the picture:
(b)
5
9
2
9
3 1
+
7 7
1
9
=
4
7
=
3 +1
7
8
9
5 + 2 +1
5 2 1
We can see in the picture: + + = 8 =
9 9 9 9
9
Let us look :
The denominator of the sum of some fractions of equal denominator is
the common denominator of the fractions and the numerator is the sum
of the numerators of the fractions.
7 2
3
Example 11 : Add + +
13 13 13
7 2 3 + 7 + 2 12
3
Solution : + + =
=
13 13 13
13
13
12
Answer :
13
Example 12 : Add :
5
2
+
11 22
2× 2 4 2 5
4 5 4+5 9
2
=
= ∴ + = + =
=
11 11× 2 22 11 22 22 22
22 22
9
22
Solution :
Answer :
Let us look : These fractions are not of equal denominators. First of all
these fractions are converted into equal denominator fraction. Then
they are added.
Elementary Mathematics
1 1 5
Example 13 : Add: + +
8 6 12
Solution : The denominators of the fractions are 8, 6 and 12. The LCM
of these denominators is 24.
1 1× 3 3 (
=
=
Q 24 ÷ 8 = 3)
8 8 × 3 24
1 1× 4 4 (Q 24 ÷ 6 = 4)
=
=
6 6 × 4 24
5 5 × 2 10 (
=
=
Q 24 ÷12 = 2)
12 12 × 2 24
1 1 5 3 4 10 3 + 4 + 10 17
+ + = + + =
=
8 6 12 24 24 24
24
24
17
Answer :
24
2 3 1
Example 14 : Add + +
5 10 15
Solution : The denominators of the fractions are 5, 10 and 15 and the
LCM of these is 30
2 2 × 6 12 (
=
=
Q 30 ÷ 5 = 6)
5 5 × 6 30
3 3× 3 9
=
=
10 10 × 3 30
1 1× 2 2
=
=
15
×
2
30
15
(Q 30 ÷10 = 3)
(Q 30 ÷15 = 2)
2 3 1 12 9 2 12 + 9 + 2 23
∴ + + = + + =
=
30
5 10 15 30 30 30
30
Answer:
23
30
83
84
Simple Fraction
Subtraction of fraction
6
7
4
2
7
Let us look at the picture :
7
4 2 6−4
6
−
= =
7
7 7
7
It is found that the denominator of the difference between the two
fractions is the common denominator. On the other hand, the
numerator of the difference between the fractions is the difference
between the of the two numerators.
11
13
8
13
3
13
Let us look at the picture :
3
8
11 − 3
11
=
=
13 13
13
13
Example 15 : Subtract :
7
9
Solution :
−
7
9
5 7−5 2
=
=
9
9
9
−
5
9
Answer : 2
9
Example 16 : Find the difference :
Solution :
Answer :
3 7 −3 4
7
−
=
=
25 25
25
25
4
25
7
25
−
3
25
Elementary Mathematics
85
1
8
Example 17 : Subtract :
from
5
25
8 1
Solution : − , the denominators are 25 and 5
25 5
The LCM of the denominators is 25
8
25
=
8 ×1 8
=
25 ×1 25
[Q 25÷25=1 ]
1 1× 5 5
[Q 25÷5=5 ]
=
=
5
5 × 5 25
8 5 8−5 3
8 1
− = − =
=
25 5 25 25
25 25
3
Answer :
25
5 1
−
Example 18 : Find the difference
12 4
Solution : Here the denominators are 12 and 4. The LCM of these
denominators is 12
5 5 ×1 5
[Q 12 ÷12=1]
∴
=
=
12 12 ×1 12
3
1 1× 3
∴ =
=
[Q 12 ÷4=3]
4 4 × 3 12
1
5 3 5−3 2 1
5 1
∴ − = − =
= =
12 4 12 12
12 12 6
6
1
Answer :
6
Let us look : The fractions are not of equal denominators. These are
converted into the fractions of equal denominators and subtraction is
made after it. Afterwards the difference is made into smallest form.
3
Example 19 : Find the value of 14
3 4 3
3
[Converting 1 and into equal
Solution : 1- = −
4 4 4
4
denominator fractions]
4−3 1
=
−
4
4
1
Answer :
4
86
Simple Fraction
Fill in the blanks with the right numbers :
(One is done for you )
1
3
2
1
2
(a)
(b)
+
=
+
=
5
6
5
5
3
(d)
17
28
−
3
=
14
(e)
15
22
(g)
5
6
+
1
=
4
(h)
3
4
−
−
3
=
11
(c)
(f)
17
24
+
1
=
12
7
−
2
5
15
=
3
=
10
Simplification of fractions
1 2
1
Example 20 : Simplify + −
9
3 9
Solution : The denominators are 9, 3, 9 and the LCM of these
denominators is 9
1×1 1
1
[Q 9 ÷9 =1 ]
=
=
9
9 ×1 9
1× 3 3
1
=
=
[Q 9 ÷3 =3 ]
3
3× 3 9
2 ×1 2
2
=
=
9
9 ×1 9
1 1 2 1 3 2 1+ 3 − 2 4 − 2 2
+ − = + − =
=
=
9 3 9 9 9 9
9
9
9
2
Answer :
9
Let us look : The fractions are not of equal denominators. These are
converted into equal denominator form and afterwards the
simplification is made.
3 5
7 28
Solution : The denominators are 14, 7, 28 and its LCM is 28
3 5 18 12 5 18 + 12 − 5 30 − 5
9
+ − = + − =
=
14 7 28 28 28 28
28
28
25
=
28
25
Answer :
28
Example 21 : Simplify :
9
14
+
Elementary Mathematics
87
Exercise-9 (A)
1.
2.
Fill in the blanks :
(a) The numerator is 7, the denominator is 10 and the fraction is
(b) The numerator is 5, the denominator is 25 and the fraction is
(c) The numerator is 9, the denominator is 15 the fraction is
(d) The numerator is 24, the denominator is 72 and the fractions is
Determine the equivalent fractions of the following (three for each
fraction).
(a)
3.
5
6
(b)
4
7
(c)
3
8
Convert the following fractions into the smallest form and then fill in
the blanks :
25
=
40
34
=
(d)
85
54
=
72
38
(e)
=
76
(a)
15
=
45
24
(f)
=
78
(b)
(c)
Convert the following fractions in the smallest equal denominator from
from 4 - 9 :
4.
7.
7
1
13
5
5 , 6 and 7
and
5.
and
6.
6 7
10
15
24
6
8
1 4
7
5 3 and 8 9. 3 1 3 and 9
and
8. ,
,
, ,
21
7 14
30
16
15 45
8 2 4
Write appropriate number in the
10.
13.
4
=
5 20
8 24
=
15
7 49
11. =
8
8
=
14.
81 9
space :
3
=
21 42
49 7
=
15.
12
12.
Which one of the following fractions is greater or smaller? Use symbols.
16. (a)
3 3
9 7
, (b)
,
5 4
14 70
(c)
17 7
,
30 15
(d)
3 7
,
11 33
88
Simple Fraction
Add : (Question 17-21)
5 1
2 4
18.
19.
17.
+
+
9 6
15 75
9 3 1
3 2 1
+ +
+ +
20.
21.
28 14 3
12 9 6
1 1 3
+ +
2 6 12
Subtract : (Question 22-26)
3 2
1 2
1 3
7 9
4
23. −
24. −
25.
26. 1−
22.
−
−
12 9
3 11
4 17
13 26
5
Simplify : (Question 27 – 34)
1 1 1
1 3 1
3 1 2
5 1 2
27.
28. + −
29. − +
+ +
30.
+ −
6 3 4
6 4 8
5 2 3
14 2 7
2 4 3
2 1 4
2 5 1
5 5 5
32.
33. + − 34. − +
+ +
31. + +
5 25 10
15 3 15
9 6 3
8 6 12
Problems related to fractions
Example 1 : Adib had some litchis. He gave 3 parts of it to his sister
8
Rumi. How may lichis are left to him?
3
lichis
8
3
8 3
Adib had with him (1- ) part = ( - ) part
8
8 8
8-3
5
=
=
part
8
8
Solution :
Rumi got
5
part
8
Example 2 : In a garden there are 73 part of m,ango terrs and 72 part of jackfruit
trees. Rest of the part there are black-berry trees. How many part of the gardeen
have black-berry trees?
Answer :
Solution : 3 + 2
7 7
3+2 5
=
=
7
7
89
Elementary Mathematics
5 part of the garden have mango and jackfurit trees.
7
Rest of the garden,(1 −
7
5
7
) part
5
=( − ) part
7 7
2
5 =( 7 − 5 ) part =
7
7
part
Answer : 2 part of the garden have black-berry trees.
7
Example 3: A man handed over
1
8
of his property to his wife, 1 to his
2
son and 1 to his daughter. How much property was with him
4
afterwards?
Solution : The man gave his daughter
1
8
, son
1
2
and daughter
1
4
Total property handed over :
⎛1 1 1⎞
⎜⎜ + + ⎟⎟
⎝8 2 4⎠
=
1+ 4 + 2
7
=
8
8
The rest property was = (1=
Answer :
1
8
7
8
8−7
8
)
=
1
8
of his property was with him
Let us look :
The property of the man is supposed to be 1.
1 = 8 (8 parts out of 8 ie whole)
8
Exercise-9 (b)
1.
2.
1
3
and the rest of the property belongs to Jolly. How much of that
property does Jolly own?
2
1
of a bamboo is under mud,
of it is in water and the rest of it
5
2
is above the water. How much is above the water?
Jeba is the owner of
1
2
of the property, Runa is the owner of
90
3.
4.
5.
6.
7.
Simple Fraction
A farmer has cultivated jute in
1
portion of his land, paddy in
13
1
3
portion and wheat in
portion. What portion of land is under
2
26
cultivation?
3
4
A man gave
part of his money to his wife, and
part as
5
15
donation. Then how much money was with him?
7
3
portion of his vacation at home, portion at
Robin has spent
16
16
his maternal uncle’s house and the rest of the time he was in a
scout camp. How many days were in his vacation he spent in scout
camp?
1
1
Mr. Afjal had some money. He gave of it to his wife and
3
15
of it as donation. He gave the rest of the amount to his son and
daughter. How much part of his money he gave to his Son and
Doughter?
2
Shiplu bought a story book with of his pocket money and one
3
1
icecream with . How much Shiplu spent of his pocket money?
6
1
Mr. Rofiq planted teak trees in 2 part of his garden, sal trees in
4
4
part and planted Jarul trees in the rest of his garden. What part
of his garden was used to plant Jarul trees?
1
1
portion of a bamboo is under mud, portion is in water and the
9.
5
2
rest of portion is above water. How much portion of the bamboo is
above the water ?
10. Ghoni Mia profits some money by culturing fish in his pond.
3
He repaid a loan with of his money and 3 of his money he bought
7
14
food for fish. How much money of profits left with him ?
8.
11.
Girls those who took part in the SSC examination from a girls high school.
1
7
17
of them got A+,
got A and
got A- . Rest of them failed.
24
12
48
How many girls failed of the total ?
Chapter - Ten
Decimal Fraction
Let us look at the shaded box below:
1
= .1 (Decimal one)
10
2
Two tenth =
= .2 (Decimal two)
10
3
Three tenth = = .3 (Decimal three)
10
4
Four tenth = = .4 (Decimal three)
10
5
Five tenth = = .5 (Decimal three)
10
6
Six tenth = = .6 (Decimal three)
10
7
Seven tenth = = .7 (Decimal three)
10
8
Eight tenth = = .8 (Decimal three)
10
9
Nine tenth = = .9 (Decimal three)
10
One tenth =
Let us look: Each one of the above figures is a fraction.
Its denominator is 10
Let us look at the following shaded boxes:
One hundredth =
1
100
Five hundredth =
5
100
92
Desimal Fraction
1
100
5
=
100
6
=
100
One hundredth
=
Five hundredth
Six hundredth
= .01 (Decimal zero one)
= .05 (Decimal zero five)
= .06 (Decimal zero six)
……………………………………....................
Nine hundredth
=
Ten hundredth
=
Eleven hundredth
=
Sixteen hundredth
=
Twenty hundredth
=
9
100
10
100
11
100
16
100
20
100
= .09 (Decimal zero nine)
= .10 (Decimal one zero)
= .11 (Decimal one one)
= .16 (Decimal one six)
= .20 (Decimal two zero)
…………………………………………………..
Sixty hundredth
=
Ninety nine hundredth
=
Let us look: .01 =
.99 =
1
,
100
60
100
99
100
.03 =
= .60 (Decimal six zero)
= .99 (Decimal nine nine)
3
,
100
.05 =
5
,
100
.16 =
16
,
100
.19 =
19
,
100
99
100
Each one of the above is a fraction and it has 100 as a denominator.
93
Elementary Mathematics
Example 1: Fill in the blanks (Two are done for you):
(a)
7
10
=
18
100
=
9
10
=
25
100
=
5
10
=
40
100
=
3
10
=
75
100
=
1
10
=
2
100
=
.7
(b)
.18
Shade twelve squares:
The Fractions which have 10 or multiples of 10 as denominators are
expressed shortly by putting one decimal point. These are decimal
fractions.
Mixed Decimal Fraction:
In 2.3, 2 is a whole number and .3 is a decimal number. Hence, 2.3 is a
mixed decimal fraction.
Let us look: A whole number and a decimal fraction exist in a mixed
decimal fraction.
94
Desimal Fraction
The system of reading a decimal fraction
To read a decimal fraction, first say the word ‘decimal’ and then read
the right-hand side digits. If the fraction is a mixed one, in other words
when there is a whole number and a fraction, the whole number is read
in the conventional way as we read normal numbers.
Decimal fraction (In figures)
.01
.10
.99
99.09
80.08
Decimal fractions (In words)
Decimal zero one
Decimal one zero
Decimal nine nine
Ninety nine decimal zero nine
Eighty decimal zero eight
Let us look:
You can’t read .10 as decimal ten.
You can’t read .99 as decimal ninety nine.
Write the following decimal numbers in words (The first one is
done for you):
Decimal fraction (In figures) Decimal fractions (In words)
8.46
Eight decimal four six
12.429
205.001
555.505
Write the following decimal numbers in figures which are written
in words:
Decimal fraction (In words)
Decimal fractions (In figures)
Fifteen decimal two seven
15.27
Seven decimal zero three
7.03
Seventy two decimal nine eight
72.98
Three hundred decimal zero three
300.03
Taka five and Paisa five
TK 5.05
95
Elementary Mathematics
Write the following decimal numbers in figures which are written in
words:
(First one is done for you):
Decimal fraction (In words)
Decimal fractions (In figures)
Ten decimal zero three
10.03
Fifteen decimal three two one
One hundred and three
decimal zero one three
One hundred two decimal nine nine
Taka ten and paisa five
Place value:
0,1,2,3,4,5,6,7,8 and 9 are the ten digits or symbols. We use these
symbols for writing numbers. Hence, our number system is ten based.
This base counting starts from the unit.
Let us look at the following diagram:
Thousands
Hundreds
•
Tens
times
10
•
times
100
•
times
1000
Decimal point
•
Decimal fraction portion
Tenths
•
Hundredths
•
Thousandths
•
ones
.1
Whole number portion
.01
.001
The place values of the left hand digits of unit position have increased
10 times; on the other hand, the right hand digits values of the unit
position have decreased by one tenth. The left hand places are named
as tens, hundreds, etc. The right hand side places are called one tenth,
one hundredth, one thousandth etc. The whole number position and the
decimal fraction position can be identified by putting a point (.) at the
right of the unit place. This point is called the decimal point. The
values of one tenth, one hundredth, one thousandth etc are expressed in
the form of .1,.01,.001 etc respectively.
Place value: 36.94
Thous
ands
1000
Hundreds
Tens
Units
One tenth
One
hundredth
One
thousandth
100
10
1
1
or .1
10
1
or .01
100
1
or
1000
.001
3
6
9
4
96
Desimal Fraction
Here,
figures
In words
Place value is three tens
Place value is six units
Place value is nine tenth
Place value is four hundredth
3
In number
or
or
or
or
30
6
.9
.04
6 .9 4
Write the place values (the first one is done for you):
In figure 18.05
Place value of
1
is
1 ten
or
Place value of
8
is
8 units
or
Place value of
0
is
0 tenth
or
Place value of
5
is
5 hundredth or
(a) 28.64, (b) 42.50, (c) 64.64, (d) 8.04, (e) 0.03
Converting simple fraction into decimal fraction:
Again,
10
8
.0
.05
1
1
1
5
5
= x1= x =
= .5
2
2
2
5
10
1 1
1
2
2
= x1= x =
= .2
5 5
5
2
10
1
1
1
25
25
= x1= x
=
= .25
4
4
4
25
100
1 1
1
20
20
= x1= x
=
= .20
5 5
5
20
100
9
9
9
2
18
=
x1=
x =
= .18
50 50
50
2
100
Let us look: The numerator and the denominator are multiplied by the
same number in order to make denominator 10 or 100.
1
5
= .2,
1
5
= .20, So, .20 = .2
In other words, ‘0’ can be omitted from the end of a decimal number. It
does not affect the value.
97
Elementary Mathematics
Convert the simple fractions into decimals (First one is done for
you):
3
20
7
20
9
10
37
50
3
40
=
3
20
x1=
3
20
x
5
5
=
15
100
=.15
=
=
=
=
Converting decimal fractions into simple fractions :
3
1
3
6
5
1
=
(b) .5 =
=
Example 2:
(a) .06 =
100
50
10
50
2
9
(c) .45 =
45
100
2
38
=
9
20
(d) .76 =
76
100
19
=
=
38
50
19
25
25
20
50
Let us look: Here, the number after the decimal point has been taken
as the numerator. Under the numerator 1 is written in the denominator.
Equal number of zeroes against each number of digits after the decimal
point is written after 1.
Convert the following decimal fraction into simple fraction (First
one is done for you):
.26 =
13
=
26
100
13
50
50
.4 =
.25 =
.64 =
.04 =
.48 =
.75 =
98
Simple Fraction
Comparison of decimal fractions
2
10
= .2
.2
.1
.3
1
10
3
10
= .1,
= .3
Let us look at the figure :
2
1
>
10 10
or .2 > .1,
Similarly, .4 > .3
3
10
>
2
10
or .3 > .2
.5 > .4 etc.
Let us look at the following diagrams,
24
100
= .24
Here, .38 > .24
38
100
= .38
99
Elementary Mathematics
Example 3: Find the smaller or greater of the following decimals:
.53 and .56
Solution:
.53 =
53
100
53
100
56
,
100
<
and .56 =
.56
100
so, .53 < .56
Let us look: If the tenth decimal is greater, the fraction will be a
greater one. Again, if the tenth decimals of the fractions are the same,
the greater the hundredth decimal the greater is the fraction.
Compare (First one is done for you):
.34 and .43
.50 and .05
.35 and .53
.39 and .09
.57 and .59
.65 and .68
Here, .43 > .34
Here, … … …
Here, … … …
Here, … … …
Here, … … …
Here, … … …
Because,
Because,
Because,
Because,
Because,
Because,
10th decimal 4 > 10th decimal 3
… …
… …
… …
… …
… …
… …
… …
… …
… …
… …
Exercise – 10 (a)
1. Write in words:
(a) 5.73
(f) 9.01
(b) 8.09
(c) 8.58
(d) 25.25
(e) 9.9
2. Write in figures:
(a) Seven decimal zero nine
(c) Five decimal five five
(b) Eight decimal zero one
(d) Nine decimal nine nine
3. Write the place value of each digit in the following numbers:
(a) 4.34
(e) 13.03
(b) 15.05
(f) 20.02
(c) 18.26
(g) 8.89
(d) 17.52
100
Desimal Fraction
1. Convert the following simple fractions into decimal fractions:
(a)
9
10
(b)
1
10
(c)
8
10
(d)
7
10
(e)
3
20
(g)
9
20
(h)
4
25
(i)
9
25
(j)
13
50
(k)
21
50
(f)
7
20
2. Convert the following decimal fractions into simple ones:
(a) .3
(b) .7
(g) .07 (h) .99
(c) .9
(d) .13
(i) .89
(e) .27
(j) .98
(f) .39
(k) .01
3. Convert the following decimal fractions into the smallest
form of simple fractions:
(a) .4
(f) .40
(b) .04
(g) .50
(c) .5
(h) .60
(d) .20
(i) .75
(e) .25
(j) .80
4. Compare the following pairs of decimal fractions by using >/
< symbols:
(a) .38, .46 (b) .58, .72 (c) .09, .90 (d) .30, .03
(e) .86, .87 (f) .64, .60 (g) .5, .25 (h) .37, .32
101
Elementary Mathematics
Addition and Subtraction of decimal fractions
Example 4:
Add: 72.36 and 34.21
Solution: 72.36
34.21
106.57
Explanation:
Answer: 106.57
Hundreds
Tens
1
7
3
0
Ones Tenths Hundredths
2
4
6
3
2
5
6
1
7
Let us look: Hundredth decimal is added to hundredth decimal and the
sum is written under hundredth decimal. Tenth decimal is added to the
tenth decimal and the sum is written under the tenth decimal. This
process has also been followed in case of whole numbers.
Example 5:
Add: 4.9 and 3.55
Explanation:
4.90
3.55
8.45
Hundreds
Tens
Ones Tenths Hundredth
4+1
3
8
Answer: 8.45
9
5
4
s
0
5
5
Let us look: 9 tenths and 5 tenths are equal to 14 tenths. 4 tenths is written
under this tenths place and 1 is carried to the next units place and added to
figures 4.
Add (First one is done for you):
5.8
.21
6.01
1.40
3.07
.04
.53
.98
.18
1.4
.03
3.03
4.83
.53
2.30
102
Desimal Fraction
Example 6:
Subtract: 5.7 from 7.3
Solution: 7.3 Explanation: 7 ones 3 tenths = 6 ones 13 tenths
- 5.7
5 ones 7 tenths = 5 ones 7 tenths
1.6
1 ones 6 tenths
Alternatively,
Hundreds Tens Ones Tenths Hundredths
3+10
7
+
7
51
1
6
Answer:
1.6
Let us look: 7 tenths can not be subtracted from 3 tenths. Hence, 10
tenths are added to 3 tenths to make 13 tenths. To balance both
numbers, (10 tenths = 1 unit). 1 unit is added to 5 units of the lower
row. In this way 6 is subtracted from 7 units.
Example 7:
Subtract: 2.54 from 3.41
Solution:
3.41
- 2.54
.87
Answer: 0.87
Explanation:
3 units
2 units
3 units
2 units
2 units
2 units
0 unit
4 tenths
5 tenths
3 tenths
5 tenths
13 tenths
5 tenths
8 tenths
1 hundredth
4 hundredths
11 hundredths
4 hundredths
11 hundredths
4 hundredths
7 hundredths
103
Elementary Mathematics
Alternatively,
Hundreds Tens Units Ones Hundredths
3
4+10
1+10
5+1
4
2+1
0
8
7
Subtract (the first one is done for you):
8.4
6.9
1.5
9.43
5.18
12.75
8.97
0.62
.49
5.05
0.56
7.42
1.39
Exercise 10 (b)
1. Add:
(a) 0.5
0.8
(e) 5.02
6.39
2. Subtract:
(a) 5.4
2.9
(e) 12.1
2.46
(b)
(f)
0.7
3.05
(c)
2.36
4.63
(d) 3.64
5.87
4.76
2.77
(g)
4.03
6.38
(h) 9.09
6.66
(b) 7.44
3.19
(c) 5.58
.07
(f)
(g)
12.1
.01
5.01
3.15
(d) 8.64
2.46
104
Desimal Fraction
Multiplication of decimal fraction
Example 8: Multiply: 3.2 × 4
Solution:
Let us Add,
3.2
3.2
3.2
3.2
12.8
3.2
× 4
12.8
Answer: 12.8
Explanation:
Tens Ones Tenths Hundredths
3
2
× 4
1
2
8
Let us look: At first 2 tenths is multiplied by 4 and the result became 8
tenths. Afterwards 3 ones is multiplied by 4 and it became 12 units. 12
units comprises 1 ten and 2 ones.
Example 9: Multiply: 12.25 × 16
Solution:
12.25
× 16
73.50
122.50
196.00
Answer: 196.00
Multiply (First one is done for
you):
3.42
.4
× 4
× 8
13.68
Example 10: 5 × 8.3
Solution:
5
× 8.3
1.5
40.0
41.5
Answer: 41.5
Explanation: First of all, the
multiplication
operation is done by the ones
of multiplies 6.
Afterward,
the multiplicand is multiplied
by 1 ten of the multiplier. the
result is written in one digit
left.
12.12
× 6
0.03
× 12
× 5
8.3
× 5
41.5
Answer: 41.5
Again, 8.3
4.04
× 15
. ..
. ..
...
. ..
Picture of dirrerent weights
325045
Saturday
7
9
9
Elementary math
145
146
Geometry
Elementary math
147
148
Geometry
5999
15.66451, 66451
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
8
3
3
5
10
4
1
15
48
ones
ones
ones
ones
ones
ones
ones