Mathematics Grade 4: Volume 1 9798619545043, 9781678013646
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Copyright © 2020, February 29, by Hassan A. Shoukr All rights reserved. This book or any portion thereof may not be reproduced or used in any manner whatsoever without the express written permission of the author except for the use of brief quotations in a book review. ASIN: B0859N575V ISBN: 979-8-61954-504-3 ISBN: 978-1-67801-364-6 ISBN: 979-8-64037-042-3
ﺑﺎﻟﮭﯿﺌﺔ اﻟﻌﺎﻣﺔ ﻟﻠﻜﺘﺎب13262 \ 2000 رﻗﻢ اﻹﯾﺪاع 3079ﺗﺮﺧﯿﺺ وزارة اﻟﺘﺮﺑﯿﺔ و اﻟﺘﻌﻠﯿﻢ ﺑﻤﺼﺮ رﻗﻢ
Dear parent, teacher, and student, The series of Mathematics Grade for primary stage is based on several principles: 1-
Mathematics is the queen of other sciences, i.e. all science has a certain amount of mathematics.
2-
Therefore, it must be accessible to all students and with different abilities, attitudes and inclinations...
Based on the previous principles, we did all of the following: 1) The mathematics grade series for primary
stage explains in details each topic of the book in simple, new and easy ways with solved examples followed by graduated exercises. Our strategy is analysis then construction, we analyze the mathematics topic into its initial blocks (small lessons); then we build every two blocks then every three and so on until the topic is finally built in a pyramid form from all its apparent aspects. 2) The mathematics grade series is built on using
the colors well to help the reader to understand quickly every part.
3) In addition to, the dictionary of new terms and
words in the course at the end of the book. We did it in English-Arabic 4) In the end of the book, there are five self-tests
as a bank of problems to measure the abilities of self-analysis and self-assessment. 5) After self-Test, there are ten Exam Style
Papers to ensure the capabilities of using the different skills in the books. Mathematics Grade 4 Volume 1 has the topics: large numbers and their operations, divisibility of numbers, factorizations of numbers(HCF and LCM), angles (their types, how can you construct an angle?, and how can you measure an angle?), triangles(their types and how can you construct a triangle?), and how can you get the perimetre and area of squares and rectangles? We are pleased to know your opinion and observations about the book with our correspondence Hassan A. Shoukr
Contents Subject How Much Do You Remember?
Arithmetic I-Large numbers Millions Place Value Applications ♦ Different types of problems
II-Operations On Large Numbers Addition Subtraction Multiplication ♦ Multiplying by 2-digit number ♦ Multiplying by 3-digit number Long Division ♦ Parts of Division ♦ Dividing by 1-digit number ♦ Dividing by 2-digit number Applications ♦ Different types of problems ♦ Word problems
III-Divisibility Definition Divisibility by 2 Divisibility by 3 Divisibility by 5 Divisibility by 6 Divisibility by 10 Prime numbers Applications ♦ Different types of problems
Page 1 6 7 7 11 13 13 22 22 24 26 26 27 29 29 30 32 35 35 45 55 55 56 57 58 59 60 61 63 63
IV-Factorization Factors Highest common factor ( H.C.F ) Multiples Least or Lowest common multiple ( L.C.M )
73 73 78 81 82
Geometry I-Angles Measuring angles Types of angles Drawing angles
II-Triangles Sum of measures of angles of triangles Types of triangles ♦ According to the measures of angles ♦ According to the lengths of sides Drawing a triangle ♦ By one angle and two sides ♦ By one side and two angles
III-Squares and Rectangles Perimetre ♦ Perimetre of square ♦ Perimetre of rectangle Area ♦ Units ♦ Area of square ♦ Area of rectangle
88 88 91 92 95 95 98 98 101 104 104 107 111 111 111 113 117 117 120 121
Tests and Exams I-Self tests II-Model Exams Mathematical Terms
125 155 169
How much Do You Remember? Table Work 1×0 =
2×0 =
3×0 =
4×0 =
1×1 =
2×1 =
3×1 =
4×1 =
1×2 =
2×2 =
3×2 =
4×2 =
1×3 =
2×3 =
3×3 =
4×3 =
1×4 =
2×4 =
3×4 =
4×4 =
1×5 =
2×5 =
3×5 =
4×5 =
1×6 =
2×6 =
3×6 =
4×6 =
1×7 =
2×7 =
3×7 =
4×7 =
1×8 =
2×8 =
3×8 =
4×8 =
1×9 =
2×9 =
3×9 =
4×9 =
1 × 10 =
2 × 10 =
3 × 10 =
4 × 10 =
1 × 11 =
2 × 11 =
3 × 11 =
4 × 11 =
1 × 12 =
2 × 12 =
3 × 12 =
4 × 12 =
5×0 =
6×0 =
7×0 =
8×0 =
5×1 =
6×1 =
7×1 =
8×1 =
5×2 =
6×2 =
7×2 =
8×2 =
5×3 =
6×3 =
7×3 =
8×3 =
5×4 =
6×4 =
7×4 =
8×4 =
5×5 =
6×5 =
7×5 =
8×5 =
5×6 =
6×6 =
7×6 =
8×6 =
5×7 =
6×7 =
7×7 =
8×7 =
5×8 =
6×8 =
7×8 =
8×8 =
5×9 =
6×9 =
7×9 =
8×9 =
5 × 10 =
6 × 10 =
7 × 10 =
8 × 10 =
5 × 11 =
6 × 11 =
7 × 11 =
8 × 11 =
5 × 12 =
6 × 12 =
7 × 12 =
8 × 12 =
9×0 =
10 × 0 =
11 × 0 =
12 × 0 =
9×1 =
10 × 1 =
11 × 1 =
12 × 1 =
9×2 =
10 × 2 =
11 × 2 =
12 × 2 =
9×3 =
10 × 3 =
11 × 3 =
12 × 3 =
9×4 =
10 × 4 =
11 × 4 =
12 × 4 =
9×5 =
10 × 5 =
11 × 5 =
12 × 5 =
9×6 =
10 × 6 =
11 × 6 =
12 × 6 =
9×7 =
10 × 7 =
11 × 7 =
12 × 7 =
9×8 =
10 × 8 =
11 × 8 =
12 × 8 =
9×9 =
10 × 9 =
11 × 9 =
12 × 9 =
9 × 10 =
10 × 10 =
11 × 10 =
12 × 10 =
9 × 11 =
10 × 11 =
11 × 11 =
12 × 11 =
9 × 12 =
10 × 12 =
11 × 12 =
12 × 12 =
Easy Revision Tests Test 1
Test 2
Test 3
Test 4
1) 6 + 5 =
8+3=
9+4
=
7+6=
2) 13 6 =
15
17
=
14 9 =
3) 8 × 5 =
7×9=
4×8
=
5×7=
4) 16 ÷ 4 =
25 ÷ 5 =
18 ÷ 3
=
27 ÷ 9 =
5) 30 × 10 =
20 × 4 =
10 ×50 =
40 × 6 =
6) 500 × 6 =
300 ÷ 3 =
200 ÷ 2 =
100÷10 =
7) 35 × 0 =
342 ÷ 8 =
212 × 4 =
32300÷10 =
8) 469 ÷ 7 =
576 × 5 =
576 × 5 =
575 × 5 =
9) 786 ÷ 2 =
780 ÷10 =
0 ÷1384 =
1384 ÷8 =
10)4800×2=
5001 × 3 =
561 ÷ 3 =
832 × 8 =
Test 7
Test 8
Test 5
7=
Test 6
8
1) 9 + 3 =
7+5=
8+4=
5+9=
2) 16
12
14
11 4 =
7=
9=
6=
3) 6 × 9 =
9×5=
7×4=
3×8=
4) 21 ÷ 3 =
36 ÷ 4 =
35 ÷ 7 =
40 ÷ 8 =
5) 40 × 10 =
50 × 4 =
5 × 100 =
100 × 2 =
6) 50 ÷ 10 =
400 ÷ 10 =
100 ÷10 =
30 ÷ 3 =
7) 180 + 3 =
246 + 8 =
420
523 4 =
8) 330 205= 323 9) 48 × 20 =
136=
73 × 7 =
3=
136 + 13 =
145 16 =
36 × 2 =
56 × 2 =
10)168 ÷ 2 =
165 ÷ 1 =
Test 9
252 ÷ 4 =
Test 10
260 ÷ 4 =
Test 11
Test 12
1) 8 + 5 =
7+8=
9+6=
7+9=
2) 13 7 =
16
15
14
3) 3 × 9 =
8×7=
5 × 12 =
6×8=
4) 36 ÷ 6 =
49 ÷ 7 =
56 ÷ 8 =
63 ÷ 7 =
5) 20 × 10 =
3 × 100 =
40 × 100 =
100 × 20 =
6) 60 ÷ 10 =
4000÷100=
200 ÷10 =
300 ÷ 3 =
7) 42 × 5 =
123 × 7 =
74 + 22 =
82 + 8 =
8) 720 ÷ 8 =
560 ÷ 2 =
1107
621
1 1 − = 2 4
1 1 − = 2 3
1 1 + = 4 4
1 2
1 +9 = 3
2-
9)
10) + 8 = Test 13
9=
Test 14
6=
8=
1 = 4
8=
23 =
1 1 + = 5 5
1 = 2
3-
Test 15
Test 16
1) 12 + 9 =
14 + 7 =
19 + 4 =
17 + 6 =
2) 23 16 =
25
37
34
3) 4 × 12 =
9×8=
7×6=
5 × 11 =
4) 81 ÷ 9 =
42 ÷ 6 =
96 ÷ 8 =
64 ÷ 8 =
5) 30 × 100 =
5 × 1000 =
20 × 50 =
500 × 10 =
6) 200 ÷ 10 =
500 ÷ 50 =
6000 ÷10 =
4000 ÷ 400 =
7) 68 × 4 =
70 × 5 =
61 + 30 =
117 + 27 =
8) 694 ÷ 2 =
936 ÷ 2 =
1287
936
17 =
18 =
33 =
19 =
39 =
9)
2 1 + = 3 3
2 1 + = 4 4
5 3 − = 8 8
3 3 − = 5 5
3 8
8 2 − = 9 9
1 1 + = 5 5
1 1 + = 4 4
10) −
1 = 8
Test 17
Test 18
Test 19
Test 20
1) 19 + 13 =
17 + 15 =
28 + 14 =
25 + 19 =
2) 36
32
41
34 16 =
19 =
17 =
24 =
3) 8 × 8 =
12 + 9 =
11× 11 =
7 × 12 =
4) 72 ÷ 6 =
132 ÷ 2 =
72 ÷ 8 =
54 ÷ 9 =
5) 20 × 30 =
40 × 50 =
200 × 50 =
500 × 20 =
6) 5000÷500 = 4000÷2000=
6000÷300 =
8000 ÷ 40 =
7) 61 × 30 =
1745 + 8=
168 + 45 =
2140
2335 18 =
117 × 3 =
8) 1107 ÷ 27 = 621 ÷ 3 = 9)
5 3 = − 12 12 3 8
5 8
10) + =
438=
17 9 = − 20 20
4 4 1 + + = 9 9 9
1 1 5 + + = 8 8 8
4 3 + = 5 5
1 1 1 −1 = 3 3
5 4 1 +1 = 9 9
If you have the queen Mathematics you will be a king. Has s an A . S ho u k r
Large Numbers Hundred Thousands and Millions Well, dear pupil comes to understand how to reading and writing hundred thousands and millions. Firstly: Millions means 7, 8 and 9-digit numbers Can you complete as the pattern? Millions Thousands H
TU
Millions have 6-zeroes
HTUHTU 1 0 0 0 0 0 can be read as
100-thousand
5 1 0 0 0 0 can be read as 8 9 4 0 0 0 can be read as 4 5 0 0 0 0 can be read as 1 2 4 0 0 0 0 can be read as 3 5 0 5 0 0 0 can be read as 7 7 5 0 0 0 0 can be read as 4 0 5 6 0 0 0 can be read as 9 0 0 8 0 0 0 can be read as
1-million and 240-thousand
1 0 0 4 5 0 0 0 can be read as
10-million and 45-thousand
1 5 6 0 4 0 0 0 can be read as 2 3 0 9 0 0 0 0 can be read as 5 6 0 0 8 0 0 0 can be read as 1 0 0 0 0 7 0 0 0 can be read as
100-million and 7-thousand
2 5 4 5 0 0 0 0 0 can be read as 7 0 1 0 0 6 0 0 0 can be read as ♦
6-digit numbers
Example 1: i)
Write in letters 547935
Write in digits ii) 405-thousand forty-three
and
Solution Divide the number into U, T, H, Th, , M
547005 547-thousand and five
405-thousand and forty-three 405-thousand forty-three
+
405 000 43 405 043
The number in digits is 405043
Exercise 1: A-Write in letters i)
255417
ii)
605001
iii)
200450
iv)
202005
v)
530114
vi)
700006
vii)
700001
viii)
855064
ix)
500306
x)
950001
xi)
641000
xii)
549004
xiii)
650002
xiv)
650001
xv)
126000
B-Write in digits xiii)
214-thousands and fifty
xiv)
300-thousands, 625-hundreds
xv)
904-thousands, 50-hundreds and fifty-two
xvi)
451-thousands and 1425-units
xvii) 242-thousands and ninety-one xviii) 500-thousands and 9-hundreds xix)
420-thousands, 141-hundreds and two
xx)
215-thousands and 41-hundreds
xxi)
102-thousands, 2-tens and one
xxii) 225-thousands xxiii) 600-thousands, 42-hundreds and five xxiv) 12458-hundreds xxv) 1254-units ♦
7-digit numbers
Example 2: Write in letters i) 3548935
Write in digits ii) 6-million, 45-thousand and 4-hundred Solution
Divide the number into U, T, H, Th, , M
3548935 3-million, 548-thousand, 9hundred and thirty-five
6-million, hundred 6-million 45-thousand 4-hundred
45-thousand
and
6 000 000 045 000 400 6 045 400
The number in digits is 6045400
4-
Note: When writing the number in letters, if one of M, Th, or more are zeroes. Then you mustn t write 0-million or 0thousand or Exercise 2: A-Write in letters i)
2535412
ii)
1245801
iii)
8004520
iv)
1120065
v)
4000154
vi)
7100036
vii)
7000001
viii)
5120464
ix)
7004356
x)
4562001
xi)
1204501
xii)
9100254
B-Write in digits xiii)
2-million, 4-thousand and 4
xiv)
5-million, 14-thousand, 5-hundred and fifty-two
xv)
2-million, 452-thousand and forty-one
xvi)
1-million, 5-thousand and 9-hundred
xvii) 6-million, 40-thousand, 1-hundred and two xviii) 5-million xix)
2-million and 55-thousand
xx)
3-million and sixty-five
xxi)
1-million, 42-hundred and five
xxii) 9-million, 450-thousand and two ♦
8-digit numbers
Example 3: i)
Write in letters 23518932
Write in digits ii) 61-millions, 4-thousand and forty
Solution Divide the number into U, T, H, Th, , M
61-million, 45-thousand and forty
23518932 23-million, 518-thousand, 9hundred and thirty-two
61-million 4-thousand forty
61 000 000 040 000 40 61 040 040
The number in digits is 61 040 340
Exercise 3: A-Write in letters i)
32532412
ii)
91245801
iii)
80074520
iv)
21120165
v)
41000154
vi)
71600036
vii)
17000024
viii)
51220464
ix)
75004356
x)
84558001
xi)
12034501
xii)
19100254
xiii)
8000001
xiv)
90000023
xv)
85000002
B-Write in digits xiii)
26-million, 104-thousand and 1
xiv)
35-million, 140-thousand, 8-hundred and fifty-two
xv)
82-million, 42-thousand and forty-three
xvi)
19-million, 51-thousand and 90-hundred
xvii) 56-million, 4-thousand, 10-hundred and four xviii) 15-million and two xix)
45-million and five
xx)
100-thousand
xxi)
45-million, 5-hundred and seventy-two
xxii) 11-million and forty-one xxiii) 56-million and 56-thousand
♦
9-digit numbers *
Example 4: Write in letters i) 130547935
Write in digits ii) 206-million, thousand and three
405forty-
Solution Divide the number into U, T, H, Th, , M
130547935 130-million, 547-thousand, 9-hundred and thirty-five
206-million , 405-thousand and fortythree 206-million 405-thousand forty-three
206 000 000 405 000 43 206 405 043
The number in digits is 206 405 043
Exercise 4: A-Write in letters xvi)
252535417
xvii) 601245001
xviii) 204800450
xix)
201112005
xx)
xxi)
534000114
787100006
xxii) 700000001
xxiii) 855120064
xxiv) 587004306
xxv) 954560001
xxvi) 641204000
xxvii) 549100004
xxviii)56000001
xxix) 400000002
xxx) 522000020
B-Write in digits xxvi) 221-million, 14-thousand and fifty xxvii) 554-million, 94-thousand, 50-hundred and fifty-two xxviii) 221-million, 42-thousand and ninety-one •
Milliard has 9-zeroes for instance 2000 000 000 can be read as 2-milliard and so on. Also milliard is called billion
• •
Trillion has 12-zeroes for example 5000 000 000 000 can be read as 5-trillion and so on Quadrillion has 15-zeroes for example 3000 000 000 000 000 is 3-quadrillion
xxix) 301-million, 500-thousand and 9-hundred xxx) 600-million, 420-thousand, 141-hundred and two xxxi) 215-million and 41-hundred xxxii) 102-millions, 2-tens and one xxxiii) 255-million and 25-thousand xxxiv) 100-million, 6-thousand, 42-hundred and five xxxv) 12458-hundred xxxvi) 5641-thousand xxxvii)200-million, 5-thousand and one
Place Value Well, dear pupil comes also to understand how to determine the place value of a digit in a number Example 5: Determine the ten-millions Determine the place value of digit in each of the following: 8 in each of the following: i) 395468201 ii) 142800354 Solution M
Th H T U
HTU HTU 3 9 5 4 6 8 2 0 1 Thus, The ten-millions digit is 9
M
Th
H T U
HTU HTU 1 4 2 8 0 0 3 5 4 Thus, 8 is hundred-thousands digit or 800 000
Exercise 5: A- Determine the hundred-thousand digit in each of the following: i)
1245803
ii)
12014571
iii)
14879602
iv)
22314
v)
789
vi)
1205481
vii)
200154701
viii)
12457125
ix)
4005
x)
54200065
xi)
421336901
xii)
754210021
B-Circle the unit-million digit in each of the following: xiii)
12458103
xiv)
xvi)
22314
xvii) 7089
xviii) 1205081
xix)
20915471
xx)
xxi)
xxii) 12450040
1214571 13417125
xxiii) 453219804
xv)
14819602 4005
xxiv) 41200035
C- Determine the place value of the bold blue digit in each of the following: xxv) 12458103
xxvi) 1214571
xxvii) 14819602
xxviii) 22314
xxix) 7089
xxx) 1205081
xxxi) 20915471
xxxii) 13417125
xxxiii) 4005
xxxiv) 45128001
xxxv) 124536
xxxvi) 4512873
D- Determine the place value of the digit 5 in each of the following: xxxvii)12458103
xxxviii)1214571
xxxix) 54819602
xl)
xli)
xlii)
1205081 12456870
25314
70895
xliii) 215002
xliv) 2514360
xlv)
xlvi) 215405
xlvii) 4215980
xlviii) 5321467
Applications Different Types of Problems Well, dear pupil, come also to understand how to solve the following types of problems. ♦
Come to understand the following type
Example 5: Complete using , or = : a)
2135468 10245783
b)
12457831 12435786
Solution 7-digits
Now, 12457831 .. >.. 12435786 Exercise 5: Complete using or = : i) 12450243 iii) 123004451
6245365 245360247
ii) 42151460
42120457
iv) 48702145
122450445
v) 824500369
854211216
vi) 92045783
vii)621503465
621640214
viii) 678542013
ix) 452003698
452103654
x) 548721001
xi) 3-million
120-thousand
92321454 548721001
xii)3-million
20-thousand 21450-units
xiii)2-thousand
214-hundred
xiv)214-ten
xv)2-thousand
32154-units
xvi)21-hundred
♦
9321445
321-ten
Also come to understand the following type
Example 6: Complete the missing digits: i) 124587702 > 1245
0214 ii) 1245
14 < 124505487
Solution You note, the number of digits are the same, so that we compare between the place values
124587702 > 1245
0214
Because of the sign >, thus we put any digit less than 8 say, 0 or 7
You note, the number of digits are different, so that we put any digit from 0, ,9
1245 14 < 124505487 Now, 1245 1 14 < 124505487
Now, 124587702 > 1245 1 0214 Exercise 6 Complete the missing digits: i) iii)
1245
243 > 6245365
12300
451 = 123004451 iv) 487
v) 824500369 < 854 vii) 621
ii) 42151460 < 421
11216
0457
2145 > 122450445
vi) 92321474 = 923214
03465 = 621503465 viii) 678542
4
13 > 9321445
ix)
52003698 > 452103654 x) 548721001 < 548721
xi) 12565482 = 1256 xiii) 65 ♦
482
xii)
01
124587 > 5421672
224384 > 654587602 xiv)
9542135 < 1245
354
Come to see the following type
Example 7: Complete the missing: i)
-million, 12-thousand and fifty-
= 401
0
2
Solution
-million , 12-thousand and fifty-
=4
0
2
Now, 4-million, 12-thousand and fifty-two = 4 012 0 5 2 Exercise 7: i) 2-millions
-thousand, 5-hundred and fifty=
ii)
-thousand and forty-
vii) viii)
=
-million, 12-thousand and thirty = 215
v) 105-million and vi)
2
-million and 12-hundred = 40100
iii) 650-million, iv)
011
=
= 21
0 003
-thousand and fifty-
-million, 2-hundred and fifty-
ix) 85-million,
6
000030
-million, 214-thousand and 5-million,
4010
-thousand and fifty-
=
0150
2
= 555000 =
4010
5 2
x)
-million, 12-thousand and fifty-
xi)
-million, 210-thousand and
xii)
-million and fifty = 2200000
♦
Dear pupil,
= 401
= 401
0
2
002
come to understand the following type
Example 8 : Complete as the pattern 24538765, 24548765,
,
,
,
Solution You note the ten-thousand digit is changed ascending by only 1 , so that we can complete as:
24538765, 24548765, ..24558765.., ..24568765.., ..24578765.. , ..24588765.. Exercise 8: Complete as the pattern: i)
20345876, 20345776,
,
,
,
ii)
85421367, 85431367,
,
,
,
iii)
201351468, 221351468,
iv)
22153000, 22154000,
,
,
,
v)
32000000, 35000000,
,
,
,
vi)
502494876, 502484876,
,
,
,
vii)
965432100, 965432600,
,
,
,
viii)
5468000321, 5466000321,
ix)
854600300, 884600300,
x)
548000325, 648000325,
,
,
, ,
, ,
,
, ,
, ,
,
,
,
,
♦ Dear pupil type
also come to understand the following
Example 9: Complete the missing: i) 2 3
6
5=
+ 5000000 +
+ 10000 +
+ 70 +
Solution Remember, the number of zeroes that after a digit equals the number of digits that come after this digit
2
3
6
5=
+ 500000 +
+ 1000 +
+ 70 +
Now, 2 5 3 1 6 7 5 = ..2 000 000.. + 500 000 + ..30 000.. + 1 000 + ..600.. + 70 + ..5.. Exercise 9: Complete the missing: i)
2 3 +1
5
= 5000 000 +
ii)
5 5 400 +
iii)
5 2 1 3 = 10 000 000 + 5000 + + 20 +
iv)
1
v)
4
vi)
6 1 +8
4 2 + 6000 + 2
4
2 1 5 + 40 000 +
=
+ 50 000 +
+ 1000 000 +
3 1= + 80 + 4=
+ 10 000 +
+
+ 800 000 +
+
+ 50 000 000 +
+ 200 000 +
+ 300 +
+ 200 000 +
+ 8 000 +
5 = 1 00 000 000 + + 600 + +5
+ 50 +
+ 5000 000 +
♦ Come to see the following type: Example 10: Complete the missing: i) The greatest number formed from 5, 2, 7, 9, 4, 1 is
ii)
The smallest 3-digit number formed from 5, 2, 7, 9, 4, 1 is
Solution Remember, the greatest number means arrangement of the digits 5, 2, 7, 9, 4, 1 descending.
Remember, the smallest 3-digit number means arrangement of the smallest 3 digits of 5, 2, 7, 9, 4, 1 ascending. i.e. 1, 2, 4
Now, Now, The greatest number formed The smallest 3-digit number from 5, 2, 7, 9, 4, 1 is 975421 formed from 5, 2, 7, 9, 4, 1 is 124 Exercise 10: Complete the missing: i)
The greatest number formed from 5, 3, 7, 8, 4, 0 is
ii)
The smallest number formed from 8, 2, 6, 9, 0, 1 is
iii)
The greatest number formed from 5, 2, 3, 0, 9, 4, 1 is
iv)
The smallest number formed from 6, 2, 8, 3, 4, 7 is
v)
The greatest number formed from 2, 1, 8, 7, 9, 4, 3 is
vi)
The smallest 3-digit number formed from 1, 2, 3, 9, 4, 6 is
vii)
The greatest 2-digit number formed from 5, 3, 7, 6, 4, 1 is
viii) The smallest 5-digit number formed from 3, 8, 7, 9, 0, 1 is ix)
The greatest 3-digit number formed from 6, 2, 7, 9, 4, 1 is
x)
The smallest 1-digit number formed from 5, 2, 4, 8, 3, 1 is
xi)
The greatest 1-digit number formed from 3, 2, 0, 8, 4, 5 is
xii)
The smallest 8-digit number formed from 9, 2, 5, 1, 3, 7, 4, 6 is
xiii) The smallest 2-different digit number is xiv)
The greatest 4-same digit number is
xv)
The smallest 3-different digit number is
xvi)
The greatest 9-same digit number is
xvii) The smallest 3-same odd digit number is xviii) The greatest 4-different even digit number is xix)
The greatest 5-different odd digit number is
xx)
The smallest 6-same even digit number is
♦
Also
come to understand the following type:
Example 11: Arrange the following group of numbers in descending order: 2468736, 387060, 27364, 54287060, 294308364 Remember, If there are two numbers are equal , compare between the first two digits or the second 7-digits
5-digits
Remember, firstly compare between the number of digits in each number
Solution
5-digits
8-digits
9-digits
2468736, 87060, 27364, 54287060, 294308364 Now, 294308364,
54287060,
2468736,
87060,
27364
Exercise 11: A- Arrange each of the following groups in ascending: i)
2315487, 25403548, 124587, 65498721, 12458732
ii)
98754213, 2154685, 54987254, 325401240, 654210
iii)
54621358, 85462215, 65421387, 54987215, 85420147
iv)
21247573, 21243573, 21245573, 21248573, 21240573
v)
65402347, 65402547, 65402747, 65402947, 65402147 B- Arrange each of the following groups in descending:
vi)
2135487, 54687, 5468721, 546872301, 5428
vii)
4587902, 12458763, 250120148, 45218, 9875
viii) 45876213, 34521387, 54216872, 36421801, 90021547 ix)
56482137, 56082137, 56582137, 56882137, 56282137
x)
542013876, 542017876, 542010876, 542012876
♦
Dear pupil
come to see the following type
Example 12: Complete the missing: 12-million =
-thousand =
-hundred =
-ten =
-unit
Solutions 2 000 000
2 000 000
2-million =
-thousand =
2 000 000
2 000 000
-hundred =
2 000 000
-ten =
-unit
Now, 2-million = ..2 000-thousand = ..2 000 0-hundred = ..2 000 00ten = ..2 000 000-unit
Exercise 12: Complete the missing: i) ii)
12-million = unit
-thousand =
-hundred =
-million = 65000-thousand = -unit
-ten =
-hundred =
-
-ten =
iii)
-million = = -unit
-thousand = 124 000 0-hundred =
-ten
iv)
-million = = -unit
-thousand =
v)
-million = 000 000-unit
-thousand =
-hundred =
-ten = 32
vi)
605-million = unit
-thousand =
-hundred =
-ten =
vii)
-million = -unit
-thousand = 3 000 0-hundred =
viii)
-million = -unit
-thousand =
-hundred = 201 000 00-ten
-
-ten =
-hundred = 500 000 -ten =
ix)
-million = 000 000 -unit
-thousand =
-hundred =
-ten = 321
x)
524-million = unit
-thousand =
-hundred =
-ten =
-
Operations on Large Numbers Addition As you ve known from the previous year how to add two or more numbers, but here the numbers are greater than the previous numbers Well, come to understand how to carry out addition operation. ♦
Vertically
Example 13: Add: a)
65024035 + 3254879
Remember, We add from the left 5 + 9 = 14. We write 4 and carry up 1 and so on
Solution 1 1
65024035 + 3254879 68278914 Exercise 13: Find the result for each of the following: i)
65420017 + 3215407
ii)
231000548 + 215487005
iii)
54002149 + 365004987
iv)
93215400 + 21549876
v)
120365487 + 1245783
vi)
12456 + 124500698
vii)
321546987 viii) 365 ix) + 9654879 + 987546321 +
x)
65400987 + 654987021
♦
xi)
654009875 + 987600546
xii)
698540021 654
9 + 987654219
Horizontally
Example 14: Remember, you can solve it in the draft or direct: 4 + 9 = 13
Find the result: i)
32154914 + 125870219 Solution 1 1
1
Draft
1
3 2 1 5 4 9 1 4 + 1 2 5 8 7 0 2 1 9 = 158025133 Exercise 14:
11
1
Find the result: i) 5421 + 215490036
ii) 321005498 + 58
iii) 10 + 65429871
iv) 9870 + 235498798
v) 36540098 + 55
vi) 3698754 + 2564987
vii)36542 + 65987421
viii) 98 + 65421309
ix) 21548736 + 96543214
x) 23154987 + 985
xi) 635214987 + 254873691
xii)54620098 + 21005648
xiii) ( 5249 + 3654879 ) + 254 xiv)
65987 + ( 3654 + 98756402 )
xv)( 654381 + 3659874) + 654287 xvi)
1
32154914 +1 2 5 8 7 0 2 1 9 158025133
( 3215870 + 12987 ) + ( 21698754 + 6959 )
xvii) ( 65987 + 96005870 ) + ( 6598721 + 60 000 000 )
Subtraction As you ve known from the previous year how to subtract two, but here the numbers are greater than the previous numbers. Well, come to understand how to carry out subtraction operation. ♦
Vertically
Remember that we subtract from the left. 1 7 is impossible, so that we ll borrow 1 from 8 with 10, then 1 becomes 11 and 8 becomes 7. Now, 11 7 = 4 and so on.
Example 15: Subtract i)
32651481 - 2654987 Solution
11 15 14 10 13 17
2 1 5 4 0 3 7 11
32651481 - 2654987 29996494 Exercise 15: Find the result i)
65420017 - 3215407
ii)
531000548 - 215487005
iii)
iv)
93215400 - 21549876
v)
120365487 1245783
vi)
vii) x)
-
-
548742456 - 124500698
321546987 viii) 653211365 ix) 9654879 87546321 -
900000987 - 654987021
xi)
654009875 - 187600546
xii)
54002149 5004987
698540021 654
624500089 - 187654219
♦
Horizontally
Example 16:
Remember, you can solve it in the draft or direct.
Find the result: i)
93210005 - 3265487 Solution
7 12 1110 8 2 1 0 9 9 9 15
7 12 11 10 8 2 1 0 9 9 9 15
93210005
3 2 6 5 4 8 7 = 79944518
-
9 3 2 1 0 0 0 5 3 2 6 5 4 8 7 7 9 9 4 4 5 1 8
Exercise 16: Find the result: i) 542100210 - 490036
ii) 321005400 - 58
iii) 1000 000 - 629871
iv) 98700000 - 498798
v) 36540000 - 55
vi) 369875401 - 2564987
vii)365429810 - 65987421
viii) 98000000 - 421309
ix) 21548736
x) 23154987 - 985
9643214
xi) 635214987 - 254873691 xiii) ( 5249 + 3654879 ) xiv)
1005648
254
65987654 - ( 3654987 - 56402 )
xv)( 654381223 xvi)
xii)54620098
3659874) + 654287
( 3215870 - 12987 ) + ( 21698754 - 6959 )
xvii) ( 65987 + 96005870 ) - ( 659871 + 60 000 ) xviii) ( 1245 + 32000150 ) xix)
698712000
( 6542154
( 1245780 - 987009 )
xx)698732100 + ( 12457800
564821 )
65421 )
Multiplication Multiplying by 2-digit Number Well, dear pupil comes to understand how to multiply a number by 2-digit number. Example 17: Multiply a) 235 × 23 Solution
1
1
235 × 2 470
1
1
235 × 23 4700 + 705 5405
11
×
We ve added 0 because 2 is a tens and tens has a 0
235 3 705
Add 0 then multiply 2 by 235, also as in the previous Multiply 3 by 235 . Add 705 and 4700
1
Exercise 17:
+
A- Find the result: i)
652
ii)
× 25
iv)
654 × 65
4700 705 5405
531
iii)
× 85
v)
4571 × 12
98 × 64
vi)
6540 13 ×
vii)
654 × 59
viii)
5024 × 11
ix)
9600 57 ×
x)
5241 × 20
xi)
9870 22 ×
xii)
6004 55 ×
B- Find the product: xiii) 12 × 124
xiv)
Solve all the following type in the draft with putting the smallest under the greatest
542 × 66
xvi)
125 × 35
xvii) 21 × 540
xix)
33 × 9870
xx)
xv)
40 × 351
xviii) 1204 × 54
1205 × 55
xxi)
54 × 6508
Multiplying by 3-digit Number Well, dear pupil also comes to understand how to multiply a number by a 3-digit number Example 18: Multiply i) ×
654 123
We ve added because 4 hundreds hundreds has
00 is a and 00
We ve added 0 because 2 is a tens and tens has a 0
Solution 1 1 1 1 1
654 × 423 261600 + 13080 + 1962 276642
Add 00 , then multiply 4 by 654. Add 0 , then multiply 2 by 654. Multiply 3 by 654. Add 261600 , 3080 and 1962
Exercise 18: A- Find the result: i)
652
ii)
× 205
iv)
654
531
iii)
× 815
v)
× 605
4571
98 × 684
vi)
× 512
6540 × 213
vii)
654 × 519
viii)
5024 × 111
ix)
9600 × 527
x)
5241 × 200
xi)
9870 × 222
xii)
6004 × 515
B- Find the product: xiii) 102 × 124
xiv)
Solve all the following type in the draft with putting the smallest under the greatest
542 × 616
xvi)
125 × 325
xvii) 201 × 540
xix)
333 × 9870
xx)
xxii) 255 × 215
1205 × 555
xxiii) 5420 × 325
xv)
400 × 351
xviii) 1204 × 154 xxi)
540 × 6508
xxiv) 5402 × 548
C- Find the product: xxv)
562 × 200 =
xxvi)
xxvii)
255 × 3001 =
xxviii) 958 × 804 =
xxix)
9100 × 56 =
xxx)
1111 × 548 =
xxxi)
8301 × 9 =
xxxii)
5003 × 96 =
xxxiv)
54 × 5048 =
xxxiii) 87 × 84005 =
5000 × 500 =
Long Division Parts of Division What s your name ? It s Divided by or over.
46
÷
What s your name ? It s the Dividend.
What s your name ? It s the Quotient.
5
=
9
What s your name ? It s the Divisor.
What s your name ? It s the remainder.
+ r1 What s your name? It s equals or is equal to.
Can be read as: 46 is divided by or over 5, equals 9 and remainder 1 Exercise 19 : i)
If 355 ÷ 4 = 88 + r3. Then the dividend is , the divisor is , the quotient is and the remainder is
ii)
If 23550 ÷ 25 = 942. Then the divisor is , the quotient is , the dividend is and the remainder is
iii)
If 51870 ÷ 42 = 1235. Then the quotient is , the dividend is , the divisor is and the remainder is
iv)
3258 is divided by 6, equals 542 and remainder 1. Thus can you write the problem?
v)
6981 is divided by 13, equals 535 and remainder 2. Thus can you write the problem?
vi)
If the divisor is 52, the quotient is 125 and the dividend is 36500. Then can you write the problem?
vii)
If the remainder is 3, the dividend is 3745, the quotient is 532 and the divisor is 7. Then can you write the problem?
Dividing by 1-digit Number Dear pupil comes to learn how to carry out the long division by 1-digit number. Example 19: Divide i)
Remember, all these steps on your mind
693 ÷ 3 Solution 6 ÷ 3
3
=2
9 ÷ 3
=
3 ÷ 3
=
693 6 09
3
This line 3 × 2
9 brought dawn from the dividend.
9 03 1
This line 3 × 3 3 brought dawn from the dividend. This line 3 × 1
3
0
The quotient = 2 3 1 693 ÷ 3 = 23.1
Thus; Exercise 20:
Find the result for each of the following: i) iii)
482 ÷ 2
ii)
5555 ÷ 5
÷ 3
iv)
844
633
÷ 4
v)
505 ÷ 5
vi)
936
÷ 3
vii)
6248 ÷ 2
viii) 824
÷ 2
ix)
8642 ÷ 2
x)
÷ 3
903
Example 20: Find the result for each of the following: 232 ÷ 4 Solution 4 23 ÷ 4
≈5
2 32 2 0 0 32
32 ÷ 4 = 8 The quotient = 5 8 Thus;
Remember, all these steps on your mind
This line 4 × 5
2 brought dawn from the dividend.
3 2 0 0
This line 4 × 8
232 ÷ 4 = 58
Exercise 21: Find the result for each of the following: i)
125 ÷ 5
ii)
725
÷5
iii)
135 ÷ 3
iv)
532
÷ 4
v)
315 ÷ 7
vi)
3486 ÷ 2
vii)
483 ÷ 3
viii) 564
ix)
750 ÷ 6
x)
÷ 3
2952 ÷ 8
Device of the remainder In the following, we will show if the long division is infinite. Example 21: Find the result for each of the following 3217 ÷ 6
Remember, all these steps on your mind
6
32 ÷ 6 ≅ 5 21 ÷ 6
≅
37 ÷ 6 ≅
Remember, all these steps on your mind
Solution 3217 30 021
3
18
037
6
36
This line 6 × 5
1 brought dawn from the dividend. This line 6 × 3 7 brought dawn from the dividend. This line 6 × 6
01
The quotient = 5 3 6 and r 1 Thus; 3217 ÷ 6 = 5 36 and r1 Exercise 22: Find the result of each of the following i)
695 ÷ 3
ii)
1291 ÷ 5
iii)
2373 ÷ 4
iv)
876
v)
1952 ÷ 6
vi)
2569 ÷ 8
vii)
2585 ÷ 7
viii) 2155 ÷ 6
ix)
2356 ÷ 4
x)
÷ 7
2062 ÷ 3
Dividing by 2- digit Number To carry out the division by 2-digit, follow the following: Example 22: Find the result for each of the following: 1512 ÷ 12
Solution
Remember, all these steps on your mind
12
1 ÷ 1 = 3÷1
1
≅
2
1 5 12 12 031
Remember, all these steps on your mind This line 12 × 1
1 brought dawn from the dividend.
24
This line 12 × 2, note that we don t take 3 × 1 = 3 because 12 × 3 = 36 > 31
072
2 brought dawn from the dividend. This line 12 × 6, but don t take 7 ÷ 1 = 7, where 12 × 7 = 84 > 72
7 ÷ 1 ≅ 6 72 The quotient = 1 2 6 00 So that: 1512 ÷ 12 = 126 Exercise 23:
Find the result for each of the following: i)
3682
÷ 14
ii)
3564 ÷
11
iii)
3525
÷ 15
iv)
4048 ÷
16
v)
3250
÷ 10
vi)
2125 ÷
17
vii)
2250
÷ 18
viii) 4995 ÷
16
ix)
2028
÷ 13
x)
13574 ÷
11
Example 23: Find the result for each of the following: 20274 ÷ 31 Solution
Remember, all these steps on your mind
31 20 ÷ 3 ≅ 16 ÷ 3 ≅
6 5
Remember, all these steps on your mind
20274 186 0167
This line 31 × 6
155
This line 31 × 5
7 brought dawn from the dividend.
0124 12 ÷ 3 = 4 124 The quotient = 6 5 4 000 So that: 20274 ÷ 31 = 654
4 brought dawn from the dividend. This line 31 × 4
Exercise 24: Find the result for each of the following: i)
11844
÷
21
ii)
70335 ÷
42
iii)
18054
÷
51
iv)
7625
÷
61
v)
1168
vi)
25276 ÷
71
vii)
2295
viii) 19184 ÷
44
ix)
25602 ÷
÷
55
xi)
14 3556
xiv)
54
÷
32 ÷
54 34
3672
x)
2354
xii)
23 8142
xiii)
45 2025
xv)
85
xvi)
52 340132
19890
Applications on Operations Different Types of Problems ♦
Come to understand the following type
Example 24: Rewrite each of the following in digital form, then find the result: i) 5-ten + 6-hundred 5-ten + 6-hundred
Change it into digital form
1
Solution 50 2 + 600
050 + 600 650
Make all digits equal in the above and the under by adding zeroes
Exercise 25: Rewrite each of the following in digital from, then find the result: i) 3-thousand + 4-tent
ii) 951-ten - 4-thousand
iii)
6-million + 321-ten
iv) 7-thousand - 50-hundred
v) 61-thousand + 7-million
vi) 712-ten - 5-thousand
vii) 3-ten + 51-hundred ix) 401-hundred ♦
viii) 9321-hundred
4-thousand
5-thousand
x) 532-ten + 6-thousand
Come to understand the following type
Example 25: A- Write the number that B- Write the number that comes after each of the comes before each of the following: following: i)
323157 ,
ii) Solution
Look at the last digit, then ask your self: what s the number after 7 ? It s 8
323157 ,
323158
, 4200216 Look at the last digit, then ask your self: what s the number before 6 ? It s 5
4200215 , 4200216
+1
1-
Exercise 26 : A- Write the number that comes before each of the following: i)
, 125401
ii)
, 100136
iii)
, 532201
iv)
, 214206
v)
, 3001
vi)
, 121312
vii)
, 45007
viii)
, 145008
ix)
, 2003
B- Write the number that comes after each of the following. i)
10014 ,
ii)
267005 ,
iii)
100172 ,
iv)
5017 ,
v)
30045 ,
vi)
2005 ,
vii)
130015 ,
viii)
22512 ,
ix)
10006 ,
♦
Come to see the following type
Example 26: A-Write the number that B-Write the number that comes between each of comes between each of the the following two numbers following two numbers i)
354136 ,
, 354138
Look at the last two digits in the given two number then; Ask your self, what s the number between 6, 8 ? It s 7
Solution
354136 , 354137 , 354138 +1
Or
ii) 4015
> 4029
v)
436
< 438 > 665039
< 10033
> 258719
B- Write the number that comes between each of the following two numbers: ix) xi) xiii) xv)
4021, , 4019 611051 , , 611049 755001, , 755003 636518, , 63652
x) xii) xiv) xvi)
585431, , 585429 62532, , 62534 50091, , 50089 165403, , 1654028
♦
Come to see the empty squares (addition)
Example 27: Complete the missing digits: i)
5
4
+ 3 3 1 9 6 1 2 9 Solution 5
4
+ 3 3 1 96 1 29 5
4
I have 1 and I want to reach 9 It s 8
8
+3 3 1 961 2 9 5
1
4
8
+ 3 38 1 9612 9 1 1 5 7
4
8
+ 3 3 1 96 1 2 9 8
1 1 7
5
4
Remember, in case of addition, ask your self I want to reach the number down
8
+ 30 3 8 1 96 1 2 9
I have 4 and I want to reach 2 . impossible because 4 >2. Then consider 2 as 12 and carry up above the following and again I have and I want to reach 12 . It s 8
It s we 1 4
1 + 3 = 4, I have 4 and I want to reach 1 . It s impossible then 1 becomes 11 , again I have 4 and I want to reach 11 . It s 7
1 + 5 = 6, I have 6 and I want to reach 6 . It s 0 .
I have 3 and I want to reach 9 . It s 6 .
+
1 1 7 4 8 5 6 30 38 1
9 6 1 2 9
Exercise 28: Complete the missing digits: i)
5
+
3 5 3 1 9 11 2 9
ii)
iv) 3. . 5 +3 1 2 9 2. 0 9 1
v)
vii)
3 . 5 + 2 2 88 2 91
iii) 3 3 1 + 2 2 . 8 6 21 5 vi)
3
3
+5 2 4 92011 ix) + 6 5 8 1 8 0 1 9
+ 3562228 = 9213531
xi)
53654386 +
xii)
= 713254103
+ 531354246 = 931843401
xiii)
3215436
xiv)
+
= 9365103
+ 4352006 = 7654863
xv)
53652481 +
xvi)
= 90352013
+ 59654406 = 79654105
xvii) 15254306 + ♦
4
viii) 4 3 5 3 6 + . 8 9 6 8 1
+ 39138 68081 x)
3 3 +4 2 4 9 16 16
= 192541101
Come to see the empty squares (subtraction)
Example 28 Complete the missing digits: i)
3
2
-3 2 1 5 4 81 3
Don t forget, in case of subtraction, look the square, if it s above, we add and if it s under, we subtract
Solution 3
2
4
-3 2 1 5 4 8 13
is above, then we add: 1+ 3 = 4
3 2 4 -3 2 1 1 5 4 8 1 3
is under, subtract 2-1 = 1
3 0 24 -3 2 1 1 5 41 8 1 3
is above, add 2+8=10, we can t write 10 in but, we write 0 and carry down 1 under the following.
3 0 24 -3 8 2 11 1 5 41 8 1 3
1 + 4 =5, is under, subtract; 3 - 5 is impossible, then 3 becomes 13 and carry down 1 under the following, again is under, subtract 13 5 = 8
9 3 0 24 -3 8 2 1 1 1 5 41 8 1 3
1 + 5 = 6,
is above, add 6 + 3 = 9
Exercise 29: Complete the missing digits: i) 3 2 1 - 2 3 1 312 1
ii)
iv) 3 8 -5 3 2 184 3 1
v) -
vii)
viii) 6 8 2 1 3 1 8 7 5 1
-68 893 16 293
3
2
- 2 1 2 25 2 31 56 . 6 3 1 5 1 2
iii) 2 3 1 2 2 . 1 5 323 vi)
3 -
ix)
2 1 2 32 68 3 21 3 6 7 3 6 7 . 1 62 1 7 3
ix) x)
- 35654361 = 11654186 563654108 -
- 4325386 = 36321381
xi) xii)
14368386 - 432541
xiii) xiv)
3653659 -
5553333 - 456007
xvii) ♦
= 4510419 = 3613243 = 13131486
- 53006 = 111212111
xv) xvi)
= 138654619
= 2222222 = 15654001
Come to see multiplication in the following form
Example 29: Complete × 13 = 1599
i)
We can rewrite the problem as 1599 ÷ 13 by the long division, we can get 133
..133.. × 13 = 1599
ii) Solution
5×
= 725
We can rewrite the problem as 725 ÷ 5 by the long division, we can get 145
5 × ..145 .. = 725 Don t forget: We always multiplication into division
change
Exercise 30: × 4 = 260
i) iii)
53 ×
= 5194
ii) iv)
56 ×
= 1176 × 5 = 315
the
× 13 = 169
v)
vi)
vii)
43 ×
= 1849
viii)
ix)
10 ×
= 4350
x)
× 10 = 36400
xi) xiii)
1000 ×
♦
xiv)
= 975 × 6 = 888
100 ×
= 163400 × 100 = 14000
xii)
= 436000
× 7= 3738
xv)
15 ×
10 ×
= 36100 × 54= 53028
xvi)
Come to see division in the following form
Example 30: Complete 540 ÷
i)
= 12
Here, if the space in the second the sign ÷ still ÷ . I.e. 540 ÷ 12 by long division, we can get 45
÷ 32 = 325
ii) Solution
Here, if the space in the ÷ first the sign change into × . I.e. 325 × 32 by multiplication, we can get 10400
10400 ÷ 32 = 325
540 ÷ 45 = 12 Exercise 31: Complete the missing: i)
225 ÷
iii)
92
÷
= 45
ii)
1845 ÷
=4
iv)
252
÷
=5 = 36
v)
4092 ÷
= 33
vi)
1368 ÷
=3
vii)
738 ÷
=6
viii)
1849 ÷
= 43
ix)
÷ 123 = 982
x)
÷ 54
= 361
xi)
÷8
xii)
÷ 123
= 36
xiii)
÷ 546 = 548
xiv)
÷ 650
= 361
xv)
÷ 12
xvi)
÷8
= 3541
♦
= 1549
= 54
Come to see how to complete as the pattern
Example 31: Complete as the pattern: i) 3622, 3626, 3630,
,
,
,
Solution Firstly: note is the problem in ascending order or descending order? It s in ascending
Secondly: Ask yourself, how did he add in each one time? By subtracting, it s 4
323622, 323626, 323630 , 323634 , 323638 , 323642 , 323646 In the draft calculate each one. For example 323630 + 4 323634
Note that: if the biggest came after the smallest. Then it is in ascending order and vice versa.
Exercise 32: Complete as the pattern: i)
654436, 654441,
ii)
36431, 36437,
iii)
32324, 323251,
iv)
3616855, 3616862,
v)
235131, 2351321,
,
,
,
,
, ,
, ,
,
, ,
, ,
, ,
+
323634 4 323638
vi)
6346, 6338,
vii)
7135, 71342,
viii)
549813, 546598,
ix)
85411361, 5481135,
x)
552638, 552631,
♦
,
, ,
, ,
,
,
, ,
,
, ,
,
, ,
Using or =
Example 32: i)
Put the suitable sign < , > or = : 3636 + 25127 352 × 636 Solution 3636 + 25127 352 × 636 Calculate it in the draft
91361772 8-digits
127776 >
6-digits
Thus; 3636 + 25127 > 352 × 636 Exercise 33: Put the suitable sign < , > or = : i)
36130 ÷
ii) iii)
32525 21542 +
iv) v) vi)
10
1253
252 361238 +
49
÷
13
5473
×
10
32525 ×
5
3613
2315 52482 -
2123
783543 -
606 126
vii)
223
viii)
4638
×
65
ix)
437328 -
1306
x)
52506 ÷
6
1683
÷
9
7363
+
2383
16416 ÷
24
×
321
98
xi)
368
×
324
44874 ÷
54
xii)
4313
+
7336
9636541 -
13486
xiii)
435
+ 73115
1131
×
25
xiv)
5638
×
8
56314 +
481
xv)
2704
÷
52
6553
-
461
Word Problems ♦
Addition
Example 33: Mohamed has bought two boxes of oranges for L.E 1375 and three boxes of apples for L.E 12175. Calculate the total cost of oranges and apples. Solution The total cost of oranges and apples = 1375 + 12175 You can calculate = L.E 13550 it in the draft
1375 + 12175 13550
Exercise 34: i)
Nadia has got 3dozens of stocks for L.E 135 and 5dozens of shoes for L.E 1452. Calculate the total cost for what Nadia has.
ii)
Ramy bought 5boxes of pencils for L.E 325 and 2boxes of pens for L.E105. Find the total cost for what Ramy buy.
iii)
If Bassem has got 364265 metres of material for making trousers and 336545 metres for making jackets. Then what are the total metres for making the suits?
iv)
If Abd El Rahman bought 50bags of rice for L.E 1575 and 6cans of butters for L.E 615. Then calculate the total cost that he paid.
v)
Amr paid 1000 pounds for 31325 kilos of banana and 2000 pounds for 165415 kilos of watermelon. Calculate the total kilos that he bought.
vi)
Marwa bought 5dozens of notebooks for L.E 550, 4dozens of pencils for L.E 275 and 3boxes of pens for L.E 635. Find the total cost of what she buy.
vii)
Mohamed has got three coloured cars, the green one L.E654321, the yellow one L.E 562135 and the blue one L.E 5632143. Find the total cost of Mohamed s cars.
♦ Subtraction Example 34: Mohamed has got two containers, the first weighs 254175kg and the second weighs 123457kg calculate the difference between the weight of the two containers.
Solution The difference = 254175 - 123457 = 130718 kg
Remember the previous words that denote to the subtraction problem
254175 - 123457 130718
Exercise 35: i)
Nabila has got 3535 metres of cloth for making dresses and 26545 metres for making blouses. Calculate the difference between the cloth of the dresses and that of the blouses.
ii)
Ramy s tall is 1752 mm and Amr s tall is 2012 mm. How much is Amr s tall greater than Ramy s tall?
iii)
If Bassem s house height is 431465 mm and Ahmed s house height is 503654 mm. Then how much is Bassem s house height less than Ahmed s house height?
iv)
If Madiha has got 134215 pounds and her brother has got 933575 pounds. How much did they have together?
v)
Marwa has got a tree whose height is 50356 mm and Mona has another one whose height is 131215 mm. How much does Mona tree increase Marwa s tree?
vi)
If Abd El Rahman has got P.T 1525, he bought three kilos of apples for P.T 126. How much money was left?
vii)
If Mohamed has got P.T 565413 and his sister has got P.T 35565. How much did Mohamed has more than his sister?
viii)
Donia has bought 32125 grams of banana and 13325 grams of potatoes. How much did the banana decrease the potatoes?
♦
Addition and Subtraction togather
Example 35: i) Doaa has got P.T 2542, she bought three kilos of banana for P.T 575 and two kilos of apples for P.T 123. How much money was left? Solution The cost of banana and apples = 575 + 123 = P.T 698 The left money = 2042 698 = P.T 1344
You can carry out it in the draft
575 + 123 698 20.20 -18.05 2.15
Exercise 36: i)
Bassma has got P.T 565, she bought a pencil for P.T 125 and a pen for P.T 27. How much money was left?
ii)
If Omr has got 54735 grams of banana, he gave Hany 2451 grams and Mona 5421 grams. How many bananas were left?
iii)
Abd El Rahman has took 32154 mm of cloth from his father, he made a trousers by 3214 mm and a jacket by 4615 mm. How much millimetre was left?
iv)
Marwa s mother has got 654871 grams of a cake, she gave Marwa 1225 and her brother 54215. What s the remainder from the cake?
v)
Mahmoud has took P.T 325 from his father and P.T2127 from his mother, he bought a pencil for P.T 545. How much money was left?
vi)
Nagy has took 87542 grams of sweets from his father and 2154 kilos from his mother, he gave his brother 1715 grams. How much sweet was left with him?
vii)
Magda has took 311 cm of cloth from her brother and 135cm from her sister, she made a dress by 275 cm.. How much cloth was left?
viii)
Mohamed has took P.T 535 from his father and P.T 352 from his mother, he bought a pencil for P.T 253 and a pen P.T 375. How much money was left?
♦
Multiplication
Example 36: i)
If Mohamed has bought 100 kilos of rice for P.T 175 each. What s the total cost? Solution The total cost = 100 × 175 Put the two zeroes of = 17500 100 and multiply 1 by = P.T 17500 175
Exercise 37: i)
Maged has got 15 metres of material for L.E 74 each. What s the total cost of the material?
ii)
What s the total cost of 654 kilos of banana, if you knew the price of one kilo P.T 65?
iii)
Mohamed has bought a pencil for P.T 275. What s the price of 85 pencils?
iv)
If you knew the price of a metre of cloth P.T 351. Then can you calculate the total price of 5421metres?
v)
Mona has got 53 kilos of sweets, the price of a kilo L.E14. What s the total cost?
♦
Division
Example 37: i)
Mohamed has got 45 kilos of watermelon for P.T 5535. What the price of one kilo?
Solution The price of one kilo = 5535 ÷ 45 = P.T 123 Exercise 38: i)
How many 23s are there in 529?
ii)
How many 8s are there in 36696?
iii)
How many nines are there in 58878?
iv)
How many twenty-threes are there in 12558?
v)
If Ahmed has got 71 metres of cloth for P.T 46434. Then what s the price of each metre?
vi)
Mona bought 7 kilograms of apples for P.T 4578. What is the price of each one?
vii)
Hoda bought 35 metres of a certain material for making a balloon, she paid P.T 2275, what s the price of each metre?
viii)
Ramy has got 6 toys of cars for P.T 32526. What is the price of each one?
Divisibility Definition Well, dear pupil divisibility.
comes to understand the meaning of
Look, 45 ÷ 5 = and remainder i.e. the quotient without remainder. So that we say: 45 is divisible by 5 But 25 ÷ 4 = and remainder i.e. the quotient with remainder. So that we say: 25 isn t divisible by 4
Rule The number is divisible by another if the quotient without remainder Example 38: Complete using divisible or not divisible i)
56 is
by 8
ii)
72 is
by 7
Solution Since 56 ÷ 8 = 7 and r0 So that: 56 is divisible by 8
Since 72 ÷ 7 = 10 and r2 So that:72 is not divisible by 7
Exercise 39: A-Complete using divisible or not divisible i)
50 is
iii)
150 is
v)
240 is
vii)
3446 is
by 5
ii)
45 is
by 4
iv)
245 is
by 15
by 35
vi)
288 is
by 6
viii) 861 is
by 7
by 53
by 9
ix)
1963 is
by 3
x)
1092 is
by 13
Divisibility by 2 Well, dear pupil comes to find out if the number is divisible by 2 or not without carrying out division.
Rule The number is divisible by 2 if its unit-digit is one of 2, 4, 6, 8 .
0,
Example 39: Complete using divisible by 2 or not divisible by 2 i)
82647 is
ii)
Here, the unit-digit 7 isn t one of 0, 2, 4, 6, 8
82647 So that: 82647 divisible by 2
is
Solution
95134 is Here, the unit-digit 4 is one of 0, 2, 4, 6, 8
95134 not So that: 95134 is divisible by 2
Exercise 40: A-Complete using divisible by 2 or not divisible by 2 i)
2154 is
ii)
9842 is
iii)
86 is
iv)
452176 is
v)
8 is
vi)
86 is
vii)
53 is
viii) 9 is
ix)
192 is
x)
3214 is
xi)
12 is
xii)
32154841 is
B-Encircle the numbers that are divisible by 2: 321, 52, 8, 87543, 542, 6, 861, 73, 942, 54621, 85, 987216
Divisibility by 3 Well, dear pupil comes also to find out if the number is divisible by 3 or not without carrying out the division.
Rule The number is divisible by 3 if the sum of its digits is divisible by 3. Example 40: Complete using divisible by 3 or not divisible by 3 i)
647 is
Here, the sum = 6 + 4 + 7 = 17 and 17 isn t divisible by 3
ii) Solution
5154 is Here, the sum = 5 + 1 + 5 + 4 = 15 and 15 is divisible by 3
5154 647 So that: 647 is not divisible So that: 5154 is divisible by by 3 3 Exercise 41: A-Complete using divisible by 3 or not divisible by 3 i)
2154 is
ii)
9842 is
iii)
86 is
iv)
452176 is
v)
8 is
vi)
86 is
vii)
53 is
viii) 9 is
ix)
192 is
x)
3214 is
xi)
12 is
xii)
32154841 is
B-Encircle the numbers that are divisible by 3: 321, 52, 8, 87543, 542, 6, 861, 73, 942, 54621, 85, 987216
Divisibility by 5 Well, dear pupil comes also to find out if the number is divisible by 5 or not without carrying out the division.
Rule The number is divisible by 5 if its unit-digit is one of 0, 5 . Example 41: Complete using divisible by 5 or not divisible by 5 i)
45 is
Here, the unit-digit 5 is one of 0, 5
ii) Solution
45 So that: 45 is divisible by 5
5551 is Here, the unit-digit 1 is not one of 0, 5
5551 So that: 5551 is not divisible by 5
Exercise 42: A-Complete using divisible by 5 or not divisible by 5 i)
2154 is
ii)
9840 is
iii)
86 is
iv)
452175 is
v)
8 is
vi)
80 is
vii)
53 is
viii) 5 is
ix)
190 is
x)
3005 is
xi)
12 is
xii)
3000000 is
B-Encircle the numbers that are divisible by 5: 325, 52, 8, 87545, 540, 6, 865, 73, 940, 54625, 85, 987210
Divisibility by 6 Well, dear pupil comes also to find out if the number is divisible by 6 or not without carrying out the division.
Rule The number is divisible by 6 if it is divisible by 2and 3 together. Example 42: Complete using divisible by 6 or not divisible by 6 i)
642 is
Here, the unit-digit 2 is one of 0, 2, 4, 6, 8 and the sum = 6 + 4 + 2 = 12 and 12 is divisible by 3
ii) Solution
3524 is Here, the unit-digit 4 is one of 0, 2, 4, 6, 8 but the sum = 3 + 5 + 2 + 4 = 14 and 14 is not divisible by 3
3524 642 So that: 642 is not divisible So that: 5154 is divisible by by 6 6 Exercise 43: A-Complete using divisible by 6 or not divisible by 6 i)
2154 is
ii)
9843 is
iii)
86 is
iv)
452176 is
v)
8 is
vi)
87 is
vii)
53 is
viii) 9 is
ix)
195 is
x)
3214 is
B-Encircle the numbers that are divisible by 6: 321, 52, 8, 87543, 542, 6, 861, 73, 942, 54621, 85, 987216
Divisibility by 10 Well, dear pupil comes also to find out if the number is divisible by 10 or not without carrying out the division.
Rule The number is divisible by 10 if its unit-digit is one of 0 or it is divisible by 2 and 5 together. Example 43: Complete using divisible by 10 or not divisible by 10 i)
640 is
Here, the unit-digit 0 is 0
ii) Solution
5154 is Here, the unit-digit isn t 0
4
5154 640 So that: 647 is divisible by So that: 5154 is not divisible 10 by 10 Exercise 44: A-Complete using divisible by 10 or not divisible by 10 i)
2150 is
ii)
9842 is
iii)
80 is
iv)
452170 is
v)
8 is
vi)
10 is
vii)
50 is
viii) 9 is
ix)
192 is
x)
xi)
10 is
xii) 32154840 is
3210 is
B-Encircle the numbers that are divisible by 10: 321, 50, 8, 87543, 540, 2, 861, 70, 942, 54621, 80, 987210
Prime Numbers Defn: If a number is divisible by only itself and one. Then it is called a prime number.
Don t forget isn t P.N
Example 44:
1
Get out the prime numbers between 11 and 25. Solution 11
,
,
,
,
,
,
,
,
,
,
,
,
, 25
12 13 14 15 16 17 18 19 20 21 22 23 24 The P.N are
13
,
17
,
19
12, 14, 16, 18, 20, 22, 24 are divisible by 2, then they aren t P.N
,
23
15, 21are divisible by 3, then they aren t P.N
Exercise 45: A- Get out the prime numbers between the following each two numbers i)
1, 10
ii)
10, 21
iii)
20, 31
iv)
30, 42
v)
41, 55
vi)
54, 67
vii)
66, 77
viii)
75, 86
ix)
85, 90
x)
89, 100
B- Encircle the prime number in the following table: 1 20 21 40 41 60 61 80 81 100
2 19 22 39 42 59 62 79 82 99
3 18 23 38 43 58 63 78 83 98
4 17 24 37 44 57 64 77 84 97
5 16 25 36 45 56 65 76 85 96
6 15 26 35 46 55 66 75 86 95
7 14 27 34 47 54 67 74 87 94
8 13 28 33 48 53 68 73 88 93
9 12 29 32 49 52 69 72 89 92
10 11 30 31 50 51 70 71 90 91
C-Complete using prime number or not prime number i)
14 is
ii)
13 is
iii)
17 is
iv)
91 is
v)
25 is
vi)
99 is
vii)
33 is
viii)
46 is
ix)
85 is
x)
100 is
xi)
121 is
xii)
245 is
xiii)
4600 is
xiv)
5986 is
xv)
487 is
xvi)
4512 is
xvii)
4521 is
xviii)
4575 is
Applications on Divisibility Different Types of Problems ♦ Come to see the following type 2 : Example 45: i)
Write two 3-digit numbers are divisible by 2.
You note the required 3digit number, so that we write any 2digits.
Exercise 46:
ii)
Write two 3-digit numbers aren t divisible by 2.
Solution
150 532
The third digit is one of 0, 2, 4, 6, 8 .
You note the required 3-digit number, so that we write any 2digits.
151 535
The third digit isn t one of 0, 2, 4, 6, 8 .
i) ii)
Write three 3-digit numbers are divisible by 2. Write two 2-digit numbers aren t divisible by 2.
iii)
Write four 2-digit numbers are divisible by 2.
iv)
Write two 4-digit numbers aren t divisible by 2.
v)
Write four 3-digit numbers aren t divisible by 2.
vi)
Write two 2-digit numbers are divisible by 2.
vii)
Write five 2-digit numbers aren t divisible by 2.
viii) Write five 3-digit numbers aren t divisible by 2.
ix)
Write two 5-digit numbers are divisible by 2.
x)
Write three 5-digit numbers aren t divisible by 2.
♦
Come to see the following type 3 :
Example 46: i)
Write two 3-digit numbers are divisible by 3.
You note the required 3digit number, so that we write any 3-digits such that their sum is one of 3, 6, 9, 12, 15,
Exercise 47:
ii)
Write two 3-digit numbers aren t divisible by 3.
Solution 1+2+0= 3
120 231
2+3+1= 6
You note the required 3digit number, so that we write any 3-digits such that their sum isn t one of 3, 6, 9, ,
1+2+2= 5
122 230 2+3+0= 5
i)
Write three 3-digit numbers are divisible by 3.
ii)
Write two 2-digit numbers aren t divisible by 3.
iii)
Write four 2-digit numbers are divisible by 3.
iv)
Write two 4-digit numbers aren t divisible by 3.
v)
Write four 3-digit numbers are divisible by 3.
vi)
Write two 2-digit numbers are divisible by 3.
vii)
Write five 2-digit numbers aren t divisible by 3.
viii) Write five 3-digit numbers aren t divisible by 3. ix)
Write two 5-digit numbers are divisible by 3.
x)
Write three 5-digit numbers aren t divisible by 3.
♦
Come to see the following type 5 :
Example 47: i)
Write two 3-digit numbers are divisible by 5.
You note the required 3digit number, so that we write any 2-digits.
ii)
Write two 3-digit numbers aren t divisible by 5.
Solution 150 535
You note the required 3digit number, so that we write any 2-digits.
The third digit is one of 0, 5 .
Exercise 48:
152 533
The third digit isn t one of 0, 5 .
i)
Write three 3-digit numbers are divisible by 5.
ii)
Write two 2-digit numbers aren t divisible by 5.
iii)
Write four 2-digit numbers aren t divisible by 5.
iv)
Write two 4-digit numbers are divisible by 5.
v)
Write four 3-digit numbers are divisible by 5.
vi)
Write two 2-digit numbers aren t divisible by 5.
vii)
Write five 2-digit numbers are divisible by 5.
viii) Write five 3-digit numbers aren t divisible by 5. ix)
Write two 5-digit numbers are divisible by 5.
x)
Write three 5-digit numbers aren t divisible by 5.
♦
Come to see the following type 3 and 2 or 6 :
Example 48: i)
Write two 3-digit numbers are divisible by 3 and 2 together.
ii)
Write two 3-digit numbers aren t divisible by 3 and 2 together.
You note the required 3-digit number, so that we write any 3digits such that their sum is one of 3, 6, 9, 12, 15, and the third digit is one of 0, 2, 4, 6, 8
Exercise 49:
Solution 1+2+0 =3
120 132 1+3+2 =6
You note the required 3-digit number, so that we write any 3digits such that their sum is one of 3, 6, 9, 12, 15, and the third digit isn t one of 0, 2, 4, 6, 8 or vice versa
1+1+1 =3
111 123 1+2+3 =6
i)
Write three 3-digit numbers are divisible by 3 and 2 together.
ii)
Write two 2-digit numbers are divisible by 6.
iii)
Write four 2-digit numbers aren t divisible by 3 and 2 together.
iv)
Write two 4-digit numbers are divisible by 2 but aren t divisible by 3.
v)
Write four 3-digit numbers aren t divisible by 6.
vi)
Write two 2-digit numbers aren t divisible by 2 but are divisible by 3.
vii)
Write five 2-digit numbers are divisible by 6.
viii) Write five 3-digit numbers are divisible by 3 but aren t divisible by 2. ix)
Write two 5-digit numbers aren t divisible by 6.
x)
Write three 5-digit numbers aren t divisible by 2 and 3 together.
♦
Come to see the following type 5 and 2 or 10 :
Example 49: i)
Write two 3-digit numbers are divisible by 5 and 2 together.
You note the required 3digit number, so that we write any 2-digits.
Exercise 50:
ii)
Write two 3-digit numbers aren t divisible by 5 but are divisible by 2.
Solution 150 530
The third digit is 0 .
You note the required 3digit number, so that we write any 2-digits.
154 532
The third digit isn t 0 but one of 2, 4, 6, 8
i)
Write three 3-digit numbers are divisible by 2 and 5.
ii)
Write two 2-digit numbers are divisible by 10.
iii)
Write four 2-digit numbers aren t divisible by 5 but are divisible by 2.
iv)
Write two 4-digit numbers are divisible by 10.
v)
Write four 3-digit numbers are divisible by 5 and 2.
vi)
Write two 2-digit numbers aren t divisible by 10.
vii)
Write five 2-digit numbers are divisible by 2 but are divisible by 5.
viii) Write five 3-digit numbers are divisible by 10. ix)
Write two 5-digit numbers are divisible by 5 but are divisible by 2.
x)
Write three 5-digit numbers aren t divisible by 10.
♦
Come to see the following type 3 and 5 :
Example 50: i)
Write two 3-digit numbers are divisible by 3 and 5 together.
You note the required 3-digit number, so that we write any 3digits such that their sum is one of 3, 6, 9, 12, 15, and the third digit is one of 0, 5
Exercise 51:
ii)
Write two 3-digit numbers aren t divisible by 3 but are divisible by 5.
Solution 1+2+0 =3
120 105 1+0+5 =6
You note the required 3-digit number, so that we write any 3digits such that their sum is one of 3, 6, 9, 12, 15, and the third digit isn t one of 0, 5
1+0+2 =3
102 123 1+2+3 =6
i)
Write three 3-digit numbers are divisible by 3 and 5 together.
ii)
Write two 2-digit numbers aren t divisible by 3 but are divisible by 5.
iii)
Write four 2-digit numbers aren t divisible by 5 and 3 together.
iv)
Write two 4-digit numbers aren t divisible by 5 but are divisible by 3.
v)
Write four 3-digit numbers are divisible by 3 and 5 together.
vi)
Write two 2-digit numbers are divisible by 5 and 3 together.
vii)
Write five 2-digit numbers aren t divisible by 3 but are divisible by 5.
viii)
Write five 3-digit numbers aren t divisible by 3 and 5 together.
ix)
Write two 5-digit numbers are divisible by 3 and 5 together.
x)
Write three 5digit numbers are divisible by 5 but aren t divisible by 3.
♦
Come to see the following type 3, 2 and 5 :
Example 51: i)
Write two 3-digit numbers are divisible by 3, 2 and 5 together. Solution You note the required 3-digit number, so that we write any 3-digits such that their sum is one of 3, 6, 9, 12, 15, and the third digit is 0
1+2+0=3
120 150 1+0+0=6
Exercise 52: i)
Write three 3-digit numbers are divisible by 3, 2 and 5.
ii)
Write two 2-digit numbers are divisible by 3, 2 and 5.
iii)
Write four 2-digit numbers aren t divisible by 5, 2 and 3 together.
iv)
Write two 4-digit numbers aren t divisible by 5 and 2 together but are divisible by 3.
v)
Write four 3-digit numbers are divisible by 3, 2 and 5 together.
vi)
Write two 2-digit numbers are divisible by 5, 2 together but aren t divisible by 3.
vii)
Write five 2-digit numbers are divisible by 3, 2 and 5 together.
viii)
Write five 3-digit numbers aren t divisible by 3, 2 together but are divisible by 5.
ix)
Write two 5-digit numbers are divisible by 3, 2 and 5 together.
x)
Write three 5-digit numbers are divisible by 5, 2 together but aren t divisible by 3.
♦
Come to understand the following type:
Example 52: Complete the missing digits: i)
321
is divisible by 2
ii)
12
5 is divisible by 3 1+2+1+5=9
Solution 321 2
12 1 5 Divisibility by 3, it must the sum of the digits is one of 3, 6, 9, 12, .
Divisibility by 2, it must the unit digit is one of 0, 2, 4, 6, 8 .
Exercise 53: Complete the missing digits: i)
9
iii)
01is divisible by 2
ii)
58
5 is divisible by 3
305is divisible by 5
iv)
63
2 is divisible by 6
1 is divisible by 2
vi)
2
80 is divisible by 10
v)
70
vii)
42 0 is divisible by 2 viii) 205 and 5 and 2
ix)
153
xi)
321 is divisible by 2, 3 xii) 125 is divisible by 3 and 5 and 5
is divisible by 2
x)
1
is divisible by 3 05 is divisible by 3
xiii) 401
is divisible by 6
xiv)
1
20 is divisible by 10
xv)
4 is divisible by 2 xvi)
5
87 is divisible by 5
25
Example 53: Complete the missing digits: i)
1
is a prime number
ii)
7 is a prime number
Solution 13
17 Here, add any prime number such that its units 7
Here, add any prime number such that its tens 1
Exercise 54: Complete the missing digits: i)
6
is a prime number
ii)
7 is a prime number
iii)
5
is a prime number
iv)
3 is a prime number
v)
4
is a prime number
vi)
2 is a prime number
vii)
2
is a prime number
viii)
ix)
1
is a prime number
x)
♦
1 is a prime number 9 is a prime number
Come to understand the following type
Example 54: Complete the missing: i)
The smallest 3-digit number divisible by 2 is
ii) The greatest 4-digit number divisible by 5 is
Solution 100 The unit digit must be the smallest of 0, 2, 4, 6, 8 , the tens digit is 0 the smallest digit, and the hundreds digit is 1 because 0 will make all the number 0.
9995 The unit digit must be greatest of 0, 5 and tens, hundreds, thousands digits must be the greatest digit.
the the and 9
Exercise 55: Complete the missing: i)
The greatest 3-digit number divisible by 5 is
ii)
The smallest 2-digit number divisible by 3 is
iii)
The greatest 3-digit number divisible by 2 is
iv)
The smallest 4-digit number divisible by 10 is
v)
The greatest 2-digit number divisible by 6 is
vi)
The smallest 4-digit number divisible by 3 and 2 is
vii)
The greatest 3-digit number divisible by 3 and 5 is
viii)
The greatest 5-digit number divisible by 2 and 5 is
ix)
The smallest 4-digit number divisible by 6 is
x)
The smallest 6-digit number divisible by 2 is
Factorization of Numbers Factors or Numbers Well, dear pupil comes to understand how to know and to get out the factors of a number.
Defn: a is called a factor of c, if there is a number b such that a × b=c
For example 6 = 3 × 2 so that 3 and 2 are called two factors of 6 ♦
Knowing the factors of a number
Example 55: Choose the correct answer: a)
is a factor of 9
Remember, It is considered 1 is a factor of any number because 1 times any number gives the same number.
[ 2, 4, 3, 5] Solution
..3.. is a factor of 9
[ 2, 4, 3, 5]
There is no any number times 2, 4 or 5 gives 9, but 3 × 3 = 9 or 9 ÷ 3 =3 so that 3 is a factor of 9
Exercise 56: Choose the correct answer: i)
is a factor of 25
[ 2, 4, 3, 5]
ii)
is a factor of 27
[ 2, 4, 3, 5]
iii)
is a factor of 4
[ 7, 4, 3, 5]
iv)
is a factor of 63
[ 7, 4, 6, 5]
v)
is a factor of 81
[ 7, 4, 1, 5]
Remember that it is considered any number is a factor of itself.
vi)
5 is a factor of
[ 2, 4, 3, 5]
vii)
2 is a factor of
[ 42, 25, 3, 5]
viii)
6 is a factor of
[ 2, 4, 3, 54]
ix)
12 is a factor of
[ 2, 48, 30, 16]
x)
27 is a factor of
[ 2, 14, 81, 56]
xi)
5 is a factor of
[ 2, 4, 3, 5]
xii)
3,2 are factors of
[ 6, 2, 3, 5]
xiii)
6, 1 are factors of
[ 2, 6, 3, 54]
xiv)
2, 3, 5 are factors of
[ 2, 48, 30, 16]
xv)
1, 2, 3, 5 are factors of
[ 2, 30, 8, 56]
xvi)
,
are factors of 25
[ 2,5 - 4,1 - 1, 5]
xvii)
,
are factors of 27
[ 2, 9 - 4, 3 - 3, 9]
xviii)
,
are factors of 4
[ 7, 1
xix)
,
are factors of 63
[ 7, 4 - 9, 6 - 9, 7]
xx)
,
,
[2,9,7 - 4,27,1 - 3,27,9]
are factors of 81
1, 4
3, 5]
♦
Seeing how to get out all factors of a number
Example 56: i)
Find all factors of 18. Solution
Remember, all numbers are factors of 0 other than 0
Ask your self 18 = × , × and so on
Since 18 = 1 × 18 , = 3 × 6 , = 2 × 9
,
×
So that: All factors of 18 are 1, 18, 6, 3, 2 and 9 Exercise 57: Find all factors of each of the following: i)
4
ii)
6
iii)
5
iv)
v)
24
vi)
20
vii)
16
viii) 12
ix)
45
x)
56
xi)
36
xii)
10
xiii) 81
xiv)
121
xv)
23
xvi)
7
xvii) 21
xviii) 40
xix)
63
xx)
90
xxi)
xxii) 25
xxiii) 14
♦
32
15
xxiv) 100
Knowing how to get out the prime factors of a number
Example 57:
It must convert each number into its factors
Find the prime factors of 18 Solution Since 18 = 2 × 9 3 × 3 Thus 18 = 2 × 3 × 3
or
18 = 3 × 6 2 × 3 18 = 3 × 2 × 3
So that, the prime factors of 18 are 2 and 3
Note: All prime numbers have only one factor that is the same number. Exercise 58: Find the prime factors of each of the following: i)
4
ii)
6
iii)
5
iv)
v)
24
vi)
21
vii)
16
viii) 12
ix)
45
x)
56
xi)
36
xii)
10
xiii) 81
xiv)
13
xv)
23
xvi)
7
xvii) 20
xviii) 40
xix)
63
xx)
90
xxi)
xxii) 25
xxiii) 14
♦
32
15
xxiv) 100
Come to see how to factorize a number
Example 58: Factorize 36, then find the prime factors. 36 ÷ 2 no
18 and so
Solution
Another method to get out the prime factors of a number
÷
Since 36 18 9 3 1
2 2 3 3
or
24 = 2 × 18 2 × 9 3 × 3 24 = 2 × 2 × 3 × 3
Thus 24 = 2 × 2 × 3 × 3 The prime factors of 24 are 2 and 3. Exercise 59: A- Complete the missing: i)
32 2
ii)
28
Don t write the same factor more than one time.
iii) 12 2
iv) 24
2 2 2 2 v)
5 1
14 7 1
3 1
vi) 15
vii) 6 2
x)
1
42
xi)
16
xii) 21
7 1
7 1 xiii) 63 3
viii) 9 3
5 1
ix) 56 28 2
12 2 2 3 1
1 1
xiv) 44
xv) 81
xvi) 49 1
1
1 1
B- Factorize, then write the prime factors of each one. xvii) 7
xviii) 27
xix)
xxi) 84
xxii)
25
xxiii) 33
xxiv) 14
xxv) 46
xxvi) 64
xxvii) 72
xxviii) 26
xxix) 70
xxx)
xxxi) 50
xxxii) 100
60
30
xx)
20
Highest Common Factor (H.C.F) Well, dear pupil comes to understand how to get out the H.C.F of two or more numbers and also the preferable name is called greatest common factor (G.C.F), the word greatest is preferred with the numbers.
Defn: The greatest common factor between two or more numbers means the greatest number, which is a factor of the two or more numbers together ♦
Two numbers
Example 59: Get out or find the H.C.F of 18 and 24. Solution Firstly: We factorize the two numbers. 18 2 9 3 3 3 1
24 12 6 3 1
18 = 2 × 3 × 3
2 2 2 3
24 = 2 × 2 × 2 × 3
Now, we can get out the H.C.F as follows: 18 = 2 × 3 × 3 24 = 2 × 3 × - × 2 × 2 H.C.F = 2 × 3 = 6
Put the similar digits under to each other
You ve noticed: Each similar two digits we took one, where each two 2 we took one and each two 3 we took one
Exercise 60: A- Find the H.C.F of each of the following: i)
5, 10
ii)
4, 8
iii)
6, 8
iv)
12, 8
v)
10, 18
vi)
20, 6
vii)
21, 27
viii)
32, 42
ix)
16, 24
x)
30, 81
xi)
30, 26
xii)
24, 28
xiii)
25, 50
xiv)
15, 35
xv)
28, 42
xvi)
52, 40
xvii) 49, 28
xviii) 56, 81
B- Complete xix)
The H.C.F of 46 and 38 is
xx)
The H.C.F of 81 and 32 is
xxi)
The H.C.F of 94 and 52 is
xxii) The H.C.F of 63 and 28 is xxiii) The H.C.F of 46 and 64 is xxiv) The H.C.F of 32 and 64 is ♦
Three numbers or more
Example 60: Get out or find the H.C.F of 18, 12 and 24. Solution Firstly: We factorize the two numbers. 18 2 9 3 3 3 1 18 = 2 × 3 × 3
12 2 6 2 3 3 1 12 = 2 × 2 × 3
24 12 6 3 1
2 2 2 3
24 = 2 × 2 × 2 × 3
Now, we can get out the H.C.F as follows: 18 = 2 × 3 × 3 12 = 2 × 3 × - × 2 24 = 2 × 3 × - × 2 × 2 H.C.F = 2 × 3 = 6
Put the similar digits under to each other
You ve noticed: Each similar three digits we took one, where each three 2 we took one and each three 3 we took one
Exercise 61: A- Find the H.C.F of each of the following: i)
5, 10, 15
ii)
4, 10, 8
iii)
6, 14, 8
iv)
12, 22, 8
v)
10, 28, 18
vi)
20, 12, 6
vii)
21, 27, 14
viii)
33, 42, 81
ix)
16, 24, 9
x)
30, 81, 26
xi)
30, 25, 26
xii)
24, 28, 20
xiii)
25, 60, 50
xiv)
15, 36, 35
xv)
28, 30, 42
xvi)
52, 40, 90
xvii) 49, 28, 56
B- Complete xix)
The H.C.F of 46, 20 and 38 is
xx)
The H.C.F of 81, 56 and 32 is
xxi)
The H.C.F of 94, 50 and 52 is
xxii) The H.C.F of 63, 64 and 28 is xxiii) The H.C.F of 46, 45 and 64 is xxiv) The H.C.F of 32, 58 and 64 is xxv) The H.C.F of 25, 95 and 58 is xxvi) The H.C.F of 21, 56 and 58 is xxvii) The H.C.F of 25, 100 and 90 is
xviii) 56, 81, 32
Multiples Well, dear pupil comes to understand how to get multiples of a number.
Defn: If numbers are divisible by a number. Then they are called the multiples of it. For example 6 is a multiple of 3, where 6 ÷ 3 = 2. Also it is called the second multiple of 3 because 3 × 2 = 6 Example 61:
Remember, the zero multiple of 3 is 3 × 0=0
Complete the missing: i)
The first multiples of 4 are
three
ii) The fifth multiple of 3 is
Solution The first three multiples of 4 The fifth multiple of 3 is 15 are 0, 4, 8 The zero: 4 × 0=0 The first: 4 × 1=4 The second: 4 × 2 = 8
Exercise 62: Complete the missing: i)
The third multiple of 2 is
ii)
The sixth multiple of 5 is
iii)
The second multiple of 8 is
iv)
The fourth multiple of 4 is
v)
The first four multiples of 6 are
vi)
The third multiple of 9 is
vii)
The zero multiple of 10 is
The fifth: 3 × 5 = 15
viii)
The smallest multiple of 8 is
ix)
The multiples of 4 between 20 and 50 are
x)
8 is
multiple of
xi)
27 is
multiple of
xii)
The tenth multiple of 8 is
xiii)
120 is
xiv)
The first ten multiples of 4 are
xv)
45 is
multiple of multiple of
Lowest Common Multiple (L.C.M) Well, dear pupil comes to learn how to get out or take out the lowest common multiple of two or more numbers. It s preferred to call least common multiple, where the word least is preferred to use with the numbers.
Defn: The least common multiple of two or more numbers means the least number, which is a multiple of the two or more numbers together ♦
Two numbers
Example 62: Find the least common multiple of 8, 12. Solution First method: Can you complete? The first seven multiples of 8 are , , , , , The first six multiples of 12 are , , , , ,
,
What is the least common multiple of 8 and 12 together? Yes, it s 24 Second method: Firstly: we factorize the two number 8 and 12. 8
12
1 8 =
1 ×
12 =
×
×
×
Now, we can get out the L.C.M as follows: 8 = 2 × 2 × 2 12 = 2 × 2 × - × 3 L.C.M = 2 × 2 × 2 × 3 = 24 Exercise 63:
Put the similar digits under to each other You ve noticed: Each similar two digits we took one, where each two of 2, 2 we took one and also each one of 3, 2 we took one
A- Find the L.C.M of each of the following: i)
5, 10
ii)
4, 8
iii)
6, 8
iv)
12, 8
v)
10, 18
vi)
20, 6
vii)
21, 27
viii)
32, 42
ix)
16, 24
x)
30, 81
xi)
30, 26
xii)
24, 28
xiii)
25, 50
xiv)
15, 35
xv)
28, 42
xvi)
52, 40
xvii) 49, 28
B- Complete xix)
The L.C.M of 46 and 38 is
xx)
The L.C.M of 81 and 32 is
xxi)
The L.C.M of 94 and 52 is
xxii) The L.C.M of 63 and 28 is
xviii) 56, 81
xxiii) The L.C.M of 46 and 64 is xxiv) The L.C.M of 32 and 64 is ♦
Three numbers or more
Example 63: Get out or find the L.C.M of 8, 12 and 18. Solution Firstly: We factorize the two numbers. 18
12
8
1
1
1
18 = ... × ... ×
12 = ×
8 =
×
×
×
Now, we can get out the L.C.M as follows: 18 = 2 × 3 × 3 12 = 2 × 3 × - × 2 8 = 2 × - × - × 2 × 2 L.C.M = 2 × 3 × 3 × 2 × 2 = 72 Exercise 64:
Put the similar digits under to each other
You ve noticed: Each similar three of 2 we took one, each similar two of 3, 2 we took one and each one of 3, 2 we took one.
A- Find the L.C.M of each of the following: i)
5, 10, 15
ii)
4, 10, 8
iii)
6, 14, 8
iv)
12, 22, 8
v)
10, 28, 18
vi)
20, 12, 6
vii)
21, 27, 14
viii)
33, 42, 81
ix)
16, 24, 9
x)
30, 81, 26
xi)
30, 25, 26
xii)
24, 28, 20
xiii)
25, 60, 50
xiv)
15, 36, 35
xv)
28, 30, 42
xvi)
52, 40, 90
xvii) 49, 28, 56
xviii) 56, 81, 32
B- Complete xix)
The L.C.M of 46, 20 and 38 is
xx)
The L.C.M of 81, 56 and 32 is
xxi)
The L.C.M of 94, 50 and 52 is
xxii) The L.C.M of 63, 64 and 28 is xxiii) The L.C.M of 46, 45 and 64 is xxiv) The L.C.M of 32, 58 and 64 is xxv) The L.C.M of 25, 95 and 58 is xxvi) The L.C.M of 21, 56 and 58 is xxvii) The L.C.M of 25, 100 and 90 is
Pencil Set-square
Protractor Ruler
The universe has geometrical instruction. Batlimous
Angles Measuring* Angles Well, dear pupil angle.
comes to learn how to measure an
It s known that the unit of measuring angle is arc degree or angular degree and its sign is o which writes above on the right of the number of degrees. For example if we said: m ( ∠ ABC) = 40o can be read as measure of ABC angle equals 40 degrees Example 64: Measure the following angle:
A
B
C Solution
Now, to measure this angle, follow the following steps: A 1)
Put the centre of the protractor at the vertex of the angle B C
*
Types of angles are acute angle , right angle and obtuse angle
B
A 2) Make 0 of the protractor coincide on the BC B
C A 3) Look to which BA denotes on the protractor, you ll find it denoting to 50o. So that we can write m( ABC) = 50o C
B
Exercise 65: Measure each of he following angles: A i) ii)
B iii) A
m( ∠ ABC) =
A m( ∠ BAC) =
C C
B
C
iv)
Z
m( ∠ XYZ) = B m( ∠ CAB) =
X
Y
L
v)
vi) Y m( ∠ YXZ) =
N m( ∠ LMN) = vii) A
M
X viii)
Z
A
C B
C
m( ∠ ABC) =
m( ∠ CAB) =
B
B
ix)
m( ∠ ABC) = m( ∠ BCD) = A
C
D
x) C
A m( ∠ ABC) = m( ∠ BCD) = m( ∠ CDE) = m( ∠ DEB) = m( ∠ EBA) = m( ∠ CBE) =
D
B E
Types of Angles Well, dear pupil comes to how to recognize the type of an angle if we ve known the measure of it.
Defn: We ll define the types of angles as the following: 1)
The acute angle is an angle whose measure is less than 90o and greater than 0o.
2)
The right angle is an angle whose measure equals 90o.
3) The obtuse angle is an angle whose measure is greater than 90o and less than 180o. 4)
The straight angle is an angle whose measure equals 180o.
Example 65: Complete a)
m( ∠ ABC) = 80o, its b) type is
m( ∠ XYZ) = 100o, its type is
Solution m( ∠ ABC) = 80o, its type is m( ∠ XYZ) = 100o, its type is an acute angle an obtuse angle 80o is less than 90o, so that it is an acute angle.
Exercise 66: A-Complete i)
m( ∠ LMN) = 50o, its type is
100o is greater than 90o and less than 180o, so that it is an acute angle.
ii)
m( ∠ YXZ) = 120o, its type is
iii)
m( ∠ BAC) = 90o, its type is
iv)
m( ∠ CAB) = 180o, its type is
v)
m( ∠ XYZ) = 89o, its type is
vi)
m( ∠ LYN) = 91o, its type is
vii)
m( ∠ EFG) = 170o, its type is
viii)
m( ∠ ABC) = 88o, its type is
B-Show the type of the angle for each of the following: 10o, 120o, 56o, 83o, 90o, 180o, 150o, 80o, 100o, 105o, 64o
Drawing Angles Well, now dear pupil comes to learn how to draw an angle whose measure is known. Example 66: Draw ∠ ABC whose measure is 120o Solution Follow the following steps: 1)
Draw the ray BA horizontally.
2)
Put the centre of your protractor at B and its 0 on BA.
B
A
A
B
C o
3)
Put a sign at 120 on the measure tape of the protractor, say C A
4)
Carry up the protractor, then join C with B to make a ray BC.
Exercise 67:
B C
120o
A
B
A- Draw each of the following angles: i)
m( ∠ ABC) = 80o
ii)
m( ∠ ACB) = 50o
iii)
m( ∠ WVZ) = 180o
iv)
m( ∠ CAB) = 65o
v)
m( ∠ NBV) = 90o
vi)
m( ∠ XYZ) = 150o
vii)
m( ∠ FED) = 95o
viii)
m( ∠ LMN) = 100o
ix)
m( ∠ OPQ) = 106o
x)
m( ∠ ZYX) = 90o
xi)
m( ∠ FHG) = 135o
xii)
m( ∠ BCA) = 90o
xiii)
m( ∠ FGH) = 10o
xiv)
m( ∠ NLM) = 75o
xv)
m( ∠ DEF) = 180o
xvi)
m( ∠ YZX) = 35o
B- Draw angles with the following measures, stating their types: 30o, 102o, 45o, 90o, 87o, 125o, 180o, 60o, 78o, 93o, 155o
Triangles Sum of Measures of Angles of Triangles Well, dear pupil comes to know the sum of measures of angles of triangles.
Q: Form the opposite figures: Can you complete the following by using your protractor? 1)
m( ∠ ABC) = , m( ∠ ACB) = , m( ∠ BAC) = Sum =
C
A
B
Z
2)
m( ∠ XYZ) = , m( ∠ YXZ) = , m( ∠ XZY) = Sum =
X
L 3)
m( ∠ LMN) = , m( ∠ LNM) = , m( ∠ MLN) = Sum =
Y
N
M
What did you notice? Yes, ♦
Practical experiment:
1)
Take a board paper
2)
Draw any triangle on it and number the angles by 1, 2, 3
1
3
3)
Separate the triangle about the board paper
2
1
3
2
4)
1
Cut the angles as you see 3
5) 6)
Put them beside to each other as you see You ll note them make a straight angle.
2 3
1
2
Rule The sum of measures of angles of triangles is 180o. Example 67: Complete a)
ABC is a triangle, m( ∠ A) = 50o and m( ∠ B) = 70o. Thus m( ∠ C) = Solution
ABC is a triangle, m( ∠ A) = 50o and m( ∠ B) = 70o. Thus m( ∠ C) = 180o ( 50o + 70o) = 180o 120o = 60o You ve noticed: we subtract the sum of the two given angles from 180o because the sum of measures of angles of triangles is 180o.
Exercise 68: Complete i)
ABC is a triangle, m( ∠ C) = 45o and m( ∠ B) = 60o. Thus m( ∠ A) =
ii)
ABC is a triangle, m( ∠ A) = 30o and m( ∠ C) = 100o. Thus m( ∠ B) =
iii)
XYZ is a triangle, m( ∠ X) = 20o and m( ∠ Y) = 70o. Thus m( ∠ Z) =
iv)
LMN is a triangle, m( ∠ M) = 80o and m( ∠ N) = 20o. Thus m( ∠ L) =
v)
DEF is a triangle, m( ∠ D) = 50o and m( ∠ E) = 60o. Thus m( ∠ F) =
vi)
GHK is a triangle, m( ∠ K) = 40o and m( ∠ H) = 50o. Thus m( ∠ G) =
vii)
OPQ is a triangle, m( ∠ O) = 50o and m( ∠ P) = 10o. Thus m( ∠ Q) =
Types of Triangles Well, dear pupil comes to understand how to recognize the type of a triangle according to the measures of angles or according to the lengths of sides. ♦ According to the measures of angles: Ok, can you compete by using your protractor? 1)
The first type
C
Z
A
B
X
Y
M( ∠ A)=.. ,m( ∠ B)=.. ,m( ∠ C)=.. m( ∠ X)=.. ,m( ∠ Y)=.. ,m( ∠ Z)=..
The angle > 90o is
The angle > 90o is
Thus the triangle that has an obtuse angle is called obtuse angled-triangle. 2)
The second type Z
C
B
A
X
Y m( ∠ A)=.. ,m( ∠ B)=.. ,m( ∠ C)=.. m( ∠ X)=.. ,m( ∠ Y)=.. ,m( ∠ Z)=..
The angle = 90o is
The angle = 90o is
Thus the triangle that has a right angle is called right angled-triangle 3)
The third type C
Z
B
A
X Y M( ∠ A)=.. ,m( ∠ B)=.. ,m( ∠ C)=.. m( ∠ X)=.. ,m( ∠ Y)=.. ,m( ∠ Z)=..
The angle = 90o is And the angle > 90o is
The angle > 90o is And the angle = 90o is
Thus the triangle that hasn t a right angle or an obtuse angle is called acute angled-triangle Note: Any triangle has at least two acute angles
Example 68: Complete The measures of angles of a triangle are 40o, 50o, 90o. Thus its type is Solution
i)
The measures of angles of a triangle are 40o, 50o, 90o. Thus its type is a right angled-triangle You ve noticed: there is an angle whose measure is 90o, so that the triangle is a right angled-triangle.
Exercise 69: Complete i)
The measures of angles of a triangle are 30o, 70o, 80o. Thus its type is
ii)
The measures of angles of a triangle are 25o, 55o, 100o. Thus its type is
iii)
The measures of angles of a triangle are 40o, 30o, 110o. Thus its type is
iv)
The measures of angles of a triangle are 45o, 60o, 75o. Thus its type is
v)
The measures of angles of a triangle are 20o, 70o, 90o. Thus its type is
vi)
The measures of angles of a triangle are 10o, 60o, 100o. Thus its type is
vii)
The measures of angles of a triangle are 30o, 100o, 50o. Thus its type is
viii)
The measures of angles of a triangle are 92o, 53o, 35o. Thus its type is
ix)
ABC is a triangle in which m( ∠ A) = 45o and m( ∠ B) = 50o. Thus m( ∠ C) = and the type of ABC is
x)
XYZ is a triangle in which m( ∠ Y) = 100o, m( ∠ X) = 45o. Thus m( ∠ Z) = and the type of XYZ is
xi)
LMN is a triangle in which m( ∠ L) = 50o, m( ∠ N) = 65o. Thus m( ∠ M) = and the type of LMN is
♦ According to the lengths of sides Well, can you compete by using your ruler? 1)
The first type
Z
C
A AB =
B , BC =
, AC = ...
What do you notice? Yes
X YX =
Y , ZY =
, XZ =
What do you notice? Yes
Thus the triangle that has equal sides is called equilateral triangle. 2)
The second type C
Z
A AB =
B , BC =
, AC =
What do you notice? Yes
Y
X ZX =
, XY =
, YZ =
What do you notice? Yes
Thus the triangle that has two equal sides is called isosceles triangle
3)
The third type C
Z
B
A AB =
, BC =
, AC =
What do you notice? Yes
X ZX =
Y , XY =
, YZ =
What do you notice? Yes
Thus the triangle that has different sides is called scalene triangle Example 69: Complete i)
The lengths of sides of a triangle are 4cm , 5cm, 4cm. Thus its type is Solution
The lengths of sides of a triangle are 4cm, 5cm, 4cm. Thus its type is an isosceles triangle You ve noticed: there are two sides of 4cm, so that the triangle is an isosceles triangle.
Exercise 70: A- Complete i)
The lengths of sides of a triangle are 5cm, 7cm, 8cm. Thus its type is
ii)
The lengths of sides of a triangle are 10cm, 5cm, 10cm. Thus its type is
iii)
The lengths of sides of a triangle are 4cm, 3cm, 5cm. Thus its type is
iv)
The lengths of sides of a triangle are 5cm, 5cm, 5cm.
Thus its type is v)
The lengths of sides of a triangle are 5cm, 7cm, 9cm. Thus its type is
vi)
The lengths of sides of a triangle are 10cm, 6cm, 6cm. Thus its type is
vii)
The lengths of sides of a triangle are 3cm, 3cm, 3cm. Thus its type is
viii)
The lengths of sides of a triangle are 9cm, 5cm, 5cm. Thus its type is
B- Show the type of each of the following triangles according to the lengths of sides Z i) ii) C
B
A
Y
X iii)
C
A
iv)
B
N
M
L
Drawing Triangles Well, dear pupil comes to learn how to draw a triangle by two given side lengths and the included angle measure between them or by two given angle measures and their side length.
By two given side lengths and the included angle measure between them
♦
Well, come to see the following example Example 70: a)
Draw the triangle ABC in which AB = 5cm, BC = 6cm and m( ∠ ABC) = 50o. Solution
Follow the following steps: 1)
2)
Draw one of the two given segments, say AB = 5cm A
Put the centre of your protractor at A B and its 0 on AB
B
B We ve put the centre of protractor at B because the given angle ∠ ABC.
3)
4)
5)
Put a sign at 50o on the protractor because A m( ∠ ABC) = 50o. Carry up the protractor.
B
B
A
Join B with the sign
B
A
6)
Put 0 of your ruler at B and put a sign on the ray at 6cm which denote to C A
B
C
7)
Carry up the ruler. B
A C 8)
Join C with A by your ruler to get the required triangle. B
A C
9)
6cm
You ll get the required triangle as in the opposite figure.
50o
A
5cm
B
Exercise 71: 1)
Draw a triangle ABC in which:
2) 3)
4)
i)
AB = 3cm, BC = 2cm and m( ∠ ABC) = 55o.
ii)
CB = 4cm, AC = 7cm and m( ∠ BCA) = 100o.
iii)
AC = 5cm, AB = 2cm and m( ∠ BAC) = 120o.
iv)
BC = 6cm, CA = 6cm and m( ∠ BCA) = 90o.
v)
AB = 8cm, CB = 4cm and m( ∠ ABC) = 85o.
vi)
CB = 5cm, AC = 3cm and m( ∠ BCA) = 60o.
vii)
BA = 2cm, AC = 5cm and m( ∠ BAC) = 75o. Draw a triangle ABC in which BC = BA = 4cm and m( ∠ ABC) = 70o. Measure ∠ BAC Draw a triangle XYZ in which XY = 5cm, YZ = 7cm and m( ∠ XYZ) = 80o. Measure XZ and find the perimetre of ∆ XYZ. Draw a triangle LMN in which LM = 4cm, MN = 5cm and m( ∠ LMN) = 65o. Find m( ∠ MLN) and m( ∠ LNM).
By two given angle measures and its side length.
♦
Example 71: Draw a triangle ABC in which m( ∠ ABC) = 60o, m( ∠ BAC) = 70o and AB = 4cm. Solution Follow the following steps: 1) Draw the given segment AB = 4cm
A
B
2) Draw one of the two given angles, say m( ∠ ABC) = 60o, put the centre of the protractor at B and 0 on AB, then put a sign at 60o on the protractor.
A
B
3) Join B with sign by a ray
60o
A
4) Draw the other angle m( ∠ BAC) = 70o, put the centre of the protractor at A and 0 on AB, then put a sign at 70o on the protractor.
B
60o
A
B
5) Join A with the sing by a ray 70o
A
60o
B
C 6) Expand the two rays until meet to each other at a point which is C. 60o
70o
A
B
C
7) The required triangle as in the opposite one 60o
70o
Exercise 72: 1)
2)
Draw a triangle ABC in which:
A
4cm
B
i)
m( ∠ ABC) = 55o, m( ∠ ACB) = 45o and BC = 3cm.
ii)
m( ∠ ACB) = 78o, m( ∠ ABC) = 65o and CB = 4cm.
iii)
m( ∠ ABC) = 25o, m( ∠ CAB) = 100o and BA = 5cm.
iv)
m( ∠ BAC) = 70o, m( ∠ ABC) = 80o and AB = 3cm.
v)
m( ∠ BCA) = 82o, m( ∠ CBA) = 65o and CB = 6cm.
vi)
m( ∠ CBA) = 63o, m( ∠ ACB) = 40o and BC = 4cm.
vii)
m( ∠ CAB) = 90o, m( ∠ ACB) = 35o and AC = 10cm.
Draw a triangle LMN in which LM = 6cm, m( ∠ LMN) = 120o and m( ∠ MLN) = 30o. Calculate m( ∠ LNM) and the length of NM.
Squares and Rectangles Perimetres Well, dear pupil comes to understand how to calculate the perimetres of squares and rectangles
Defn: As you ve known from the previous year that the perimetre of a shape is the sum of lengths of the outer sides. ♦
Perimetres of Squares
Can you complete using your ruler and the following squares? L Z D C
A
B
X Y AB = = = = cm XY = = = = cm Sum = AB + BC + CD + DA Sum = XY + YZ + ZL + LX = + + + = + + + = cm = the perimetre = cm = the perimetre But, But, 4 × AB = 4 × = cm 4 × XY = 4 × = cm , 4 × BC = 4 × = cm , 4 × YZ = 4 × = cm
, 4 × CD = 4 × , 4 × DA = 4 ×
= =
, 4 × ZL = 4 × , 4 × LX = 4 ×
cm cm
= =
cm cm
What did you notice? Yes,
Rule The perimetre of a square = 4 × side length Example 72: Calculate the perimetre of ABCD square in which AB = 5cm. Solution The perimetre of ABCD square = 4 × side length = 4 × AB = 4 × 5 = 20 cm Exercise 73: 1)
Calculate the perimetre of ABCD square in which: i)
2)
AB = 2cm
ii) AB = 8dm
iii) BC = 4cm
iv) DA = 10cm
v) BC = 9m
vi) BC = 12m
vii)CB = 7m
viii)AC = 3cm
ix) BC = 11dm
Find the perimetre of a square whose side length is : x)
1m
xi)
8dm
xii)
4m
xiii)
13cm
xiv)
9m
xv)
20cm
xvi) 7dm 3)
Pay attention, how to solve and the unit is the same of the given length
xvii) 30cm
xviii) 21cm
Complete : xix)ABCD is a square in which AB = 6m. Thus CD = cm xx) LMNZ is a square in which LM = 30 cm. Thus NZ = dm
♦
Perimetres of Rectangles
Can you complete using your ruler and the following squares? Z L D C
A
B
AB = = cm , BC = = cm Sum = AB + BC + CD + DA = + + + = cm = the perimetre But, (AB + BC) × 2 = ( + )× = ×2 = cm ,(BC + CD) × 2 = ( + )× = ×2 = cm ,(CD + DA) × 2 = ( + )× = ×2 = cm ,(DA + AB) × 2 = ( + )× = ×2 = cm What did you notice? Yes,
X Y XY = = cm , YZ = = cm Sum = XY + YZ + ZL + LX = + + + = cm = the perimetre But, 2 (XY + YZ) × 2 = ( + )× = ×2 = cm 2,(YZ + ZL) × 2 = ( + )× = ×2 = cm 2,(ZL + LX) × 2 = ( + )× = ×2 = cm 2,(LX + XY) × 2 = ( + )× = ×2 = cm
2 2 2 2
Well, dear pupil, if each of the longest two sides AB, CD or YZ, LX is called length that is abbreviated by L . Also each of the shortest two sides BC, DA or XY, ZL is called width or breadth that is abbreviated by W , so that we can write the following rule.
Rule The perimetre of a rectangle = (L + W) × 2 Example 73: Calculate the perimetre of ABCD rectangle in which AB = 2cm and AD = 5cm. Solution The perimetre of ABCD rectangle = (L + W) ×2 = (AD + AB) × 2 = (5 + 2) ×2 = 7 ×2 = 14 cm Exercise 74: 1)
Calculate the perimetre of ABCD rectangle in which: i)
2)
AB = 4cm, BC = 5cm
ii)
AD = 6cm, CD = 1cm
iii) AD = 2m, AB = 10m
iv)
DC = 5m, CB = 9m
v) CB = 1dm, CD = 7dm
vi)
AB = 6dm, CB = 2dm
vii)AD = 10m, DC = 5m
viii)
CD = 11m,CB = 9m
Find the perimetre of a rectangle whose dimensions * are: ix) 4cm, 15cm
x)
60cm, 10cm
xi) 20m, 10m
xii)
50m, 9m
xiii)
xiv)
6dm, 20dm
1dm, 70dm
xv)10m, 12m *
Dimensions means the length and the width
xvi) 11m, 90m
•
Well, dear pupil, if the required is the length or the width. We ll deduce the following rules from the previous one. perimetre -W 2 perimetre And the width W = -L 2
The length L =
Example 74: i)
The perimetre of a rectangle is 20 cm and its width is 3cm. Find its length.
ii)
The perimetre of a rectangle is 16 cm and its length is 5cm. Find its width.
Solution perimetre -W 2 20 - 3 = 2
The length L =
= 10 = 7 cm
-3
perimetre -L 2 16 - 5 = 2
The width W =
= 8 -5 = 3 cm
Exercise 75: A- Calculate the length of a rectangle in each of the following: i)
Its perimetre = 30cm and its width = 5cm.
ii)
Its perimetre = 20m and its width = 3m.
iii)
Its perimetre = 42dm and its width = 4dm.
iv)
Its perimetre = 12cm and its width = 2cm.
v)
Its perimetre = 40m and its width = 8m.
vi)
Its perimetre = 80cm and its width = 15cm.
vii)
Its perimetre = 34cm and its width = 6dm.
B- Calculate the width of a rectangle in each of the following: viii)
Its perimetre = 30cm and its length = 10cm.
ix)
Its perimetre = 50m and its length = 20m.
x)
Its perimetre = 44dm and its length = 15dm.
xi)
Its perimetre = 30cm and its length = 12cm.
xii)
Its perimetre = 16m and its length = 6m.
xiii)
Its perimetre = 24cm and its length = 9cm.
xiv)
Its perimetre = 80dm and its length = 30dm.
Areas Area is a measurement of surface, such as the floor, the wall or a tabletop . The area has nothing to do with the thickness of a surface. Well, dear pupil comes to understand how to calculate the areas of squares and rectangles.
Defn: As you ve known in the previous year that the area of a shape is the number of small squares of unit side length * in this shape.
♦ Units Of Areas 1cm 1cm
1cm
Look, the small square has sides each 1cm. We say that it measures 1cm by 1cm. It has an area of 1 square centimetres that is abbreviated by 1cm2 *
Unit side length means side length of 1cm, 1dm, 1m,
The shape has 7 squares each 1 square centimetres. It has an area of 7 square centimetres that is abbreviated by 7cm2 Right, we can deduce that the units of area are square units, which means 1cm2, 1dm2, 1m2, and can be read as 1 square centimetre, 1 square decimetre, 1 square metre, respectively. Can you complete?
1cm
1dm
Area =
Area =
1cm 1m
Area =
Area =
1m 1dm
Area =
Area =
1cm
1cm
Area =
Area =
Note: We can convert from square units to other by the following table: cm2
dm2 100
m2
km2
100
1000000
Can you complete? i)
1m2 =
cm2
ii)
1dm2 =
cm2
iii)
1km2 =
m2
iv)
1km2 =
dm2
v)
1m2 =
dm2
vi)
1km2 =
cm2
vii)
5m2 =
cm2
viii)
3dm2 =
cm2
ix)
10dm2 =
x)
10m2 =
m2
xi)
125km2 =
xii)
124m2 =
cm2
xiii)
6km2 =
xiv)
7km2 =
xv)
42405cm2 =
xvi)
2154dm2 =
cm2 m2 m2
xvii) 5400354m2 = xviii) 4563cm2 =
m2, m2,
cm2 dm2
km2, dm2,
xix)
4500246m2 =
km2,
xx)
254601cm2 =
m2,
m2 cm2 m2 dm2,
cm2
dm2
♦ Areas of Squares Can you complete using your ruler and the following squares? L Z D C
A
B
AB = = = = cm Number of small squares = cm2 = the area But, BC × AB = = cm2 × , CD × BC = = cm2 × , DA × CD = = cm2 × , AB × DA = = cm2 ×
X Y XY = = = = cm Number of small squares = cm2 = the area But, YZ × XY = = cm2 × , ZL × YZ = = cm2 × , LX × ZL = = cm2 × ,XY × LX = = cm2 ×
What did you deduce? Yes,
Rule Area of a square = side length × side length Example 75: Calculate the area of a square whose side length 6cm. Solution Area of a square = side length × side length =6 × 6 = 36 cm2
Exercise 76: A- Calculate the area of a square whose side length: i)
2m
ii)
20dm
iii)
100cm
iv)
5km
v)
50cm
vi)
12m
vii)
45dm
viii)
60m
ix)
46dm
x)
80cm
xi)
2km
xii)
3m
B- Calculate the area and the perimetre of the square ABCD in which: xiii)
AB = 30m
xiv)
xvi)
CD = 50km
xvii) AD = 70cm
xviii) AD = 10m
xix)
BC = 55dm
xx)
xxi)
xxii) BA = 90cm
BC = 2dm CD = 600m
xxiii) AD = 2km
xv)
DC = 100cm CD = 40dm
xxiv) AB = 300m
♦ Areas Of Rectangles Can you complete using your ruler and the following squares? L Z D
C
A
B X
AB = = Length = cm ,BC = = Width = cm Number of small squares = cm2 = the area
XY = = Width = cm ,YZ = = Length = cm Number of small squares = cm2 = the area
Y
But, BC × AB = , CD × BC = , DA × CD = , AB × DA =
× × × ×
= = = =
2
cm cm2 cm2 cm2
But, YZ × XY = , ZL × YZ = , LX × ZL = ,XY × LX =
× × × ×
= = = =
cm2 cm2 cm2 cm2
What did you deduce? Yes,
Rule Area of a rectangle = L × W Example 76: Calculate the area of a rectangle whose dimensions 6cm and 3cm. Solution Area of a rectangle = L × W =6 × 3 = 18 cm2 Exercise 77: A- Calculate the area of a rectangle whose dimensions are : i)
2m, 5m
ii)
10dm, 20dm iii)
20cm ,100cm
iv)
10km, 5km
v)
100cm, 50cm vi)
30m, 12m
vii)
10dm, 45dm viii)
10m, 60m
1dm, 46dm
x)
80cm, 20cm
100km, 2km xii)
xi)
ix)
3m, 3m
B- Calculate the area and the perimetre of the rectangle ABCD in which: xiii) AB = 30m, BC = 2dm xv)
xiv) DC = 100cm, AD = 10cm
CD = 5km, AD = 70cm xvi)
AD = 10m, AB = 20m
xvii) BC = 5dm, CD = 600m xviii) CD = 40dm, BC = 30dm xix)
BA = 90cm, AD = 2km
xx)
AB = 300m, BC = 30m
•
Well, dear pupil, if the required is the length or the width. We ll deduce the following rules from the previous one. Area W Area And the width W = L
The length L =
Example 77: i)
The area of a rectangle is 20cm2 and its width is 4cm. Find its length.
ii)
The area of a rectangle is 16cm2 and its length is 8cm. Find its width.
Solution Area W 20 = 4
The length L =
Area L 16 = 8
The width W =
= 5 cm
= 2 cm
Exercise 78: A- Calculate the length and the perimetre of a rectangle in each of the following: i)
Its area = 30cm2 and its width = 5cm.
ii)
Its area = 20m2 and its width = 2m.
iii)
Its area = 42dm2 and its width = 6dm.
iv)
Its area = 12cm2 and its width = 2cm.
v)
Its area = 40m2 and its width = 5m.
vi)
Its area = 80cm2 and its width = 8cm.
vii)
Its area = 34cm2 and its width = 2dm.
B- Calculate the width and the perimetre of a rectangle in each of the following: viii)
Its area = 30cm2 and its length = 10cm.
ix)
Its area = 50m2 and its length = 10m.
x)
Its area = 44dm2 and its length = 11dm.
xi)
Its area = 30cm2 and its length = 10cm.
xii)
Its area = 16m2 and its length = 8m.
xiii)
Its area = 24cm2 and its length = 6cm.
xiv)
Its area = 80dm2 and its length = 10dm.
Please Dear pupil Review all syllabuses before solving each self-test and each exam, determine for your-self the time of exam and solve each exam more than once. Hassan A. Shoukr
Self-Tests Self-Test I Complete the missing: 1)
-millions and
-thousands = 1005000
2)
-millions and
-hundreds = 6000100
3)
-millions, 53601305
thousands,
-hundreds
4)
5-millions, 31-thousands and 8 =
5)
61-millions, 36-hundreds and 5 =
6)
95-millions, 35-thousands and 3-tens =
7)
356186 + 36193 =
8)
53893 + 3893196 =
9)
560038
93638 =
10)
480009
138899 =
11)
563863 +
12) 13) 14)
= 938696
+ 46389 = 86183 530061 -
= 163815
- 93685 = 41363
15)
500 ÷ 4 =
16)
25284 ÷ 7 =
17)
8073 ÷ 13 =
18)
8832 ÷ 23 =
19)
42175 ÷ 35 =
20)
436 × 5 =
and
=
21)
7 × 6301 =
22)
538 × 12 =
23)
37 × 409 =
24)
-millions,
25)
5-millions and
26)
931 × 463 =
27)
938 × 100 =
28)
1000 × 500 =
29)
368 × 100 =
30)
12541945 < 1254
31)
98721
32)
12459876 = 1245
33)
1000 × 530 =
34)
From the numbers 423, 1505, 632, 912, 96 :
35)
The numbers divisible by 3 are
36)
The numbers divisible by 2 are
37)
The numbers divisible by 5 are
38)
The numbers divisible by 6 are
39)
The numbers divisible by 10 are
40) 1 600 +
42)
=
5
945
123 > 987215123
2 3 +1
41) 5 50 000 +
-thousands and thirty-one =123000
5
876
=
+ 4000 000 +
5 5 5 = 500 000 000 + + 500 +
The prime numbers from 1 to 21 are
43) The prime numbers from 11 to 37 are
+ 20 000 +
+
+ 5000 000 +
+
44)
The factors of 28 are
45)
The factors of 56 are
46)
The prime factors of 18 are
47)
The prime factors of 36 are
48)
The H.C.F of 24 and 15 is
49)
The H.C.F of 18, 36 and 63 is
50)
The L.C.M of 10 and 30 is
51)
The L.C.M of 16, 36 and 64 is
52)
The greatest number formed from 1, 8, 9, 2, 0, 4 is
53)
The smallest number formed from 5, 4, 8, 9, 3, 4, 1 is ...
54)
The measure of an obtuse angle is
55)
The measure of an acute angle is
56)
The measure of a right angle is
57) The greatest 3-digit number formed from 5, 6, 4, 2, 1, 8, 0, 7 is 58) The smallest 5-digit number formed from 3, 2, 8, 9, 7, 5, 0, 1, 4 is 59)
The sum of measures of angles of a triangle is
60)
The equilateral triangle has
61)
The isosceles triangle has
62)
The scalene triangle has
63)
The greatest 5-digit number is
64)
The smallest 4-digit number is
65)
Any triangle has at least
sides sides sides
acute angles
66) The type of the triangle whose sides-lengths 5cm, 5cm, 3cm is
67)The type of the triangle whose sides-lengths 6m, 6m, 6m is 68)
The greatest 3-digit number divisible by 5 is
69)
The smallest 3-same digit number is
70)
The greatest 6-same digit number is
71)
The greatest 2-digit number divisible by 2 is
72) The type of the triangle whose sides-lengths 4cm, 3cm, 5cm is 73) In the triangle ABC, the measure of ∠ A = 60o and the measure of ∠ B = 50o, then the measure of ∠ C = 74)
In the opposite figure: A a) The measure of ∠ A = And its type is b) The measure of ∠ B = And its type is c) The measure of ∠ C = C And its type is d) The type of ∆ ABC according to its angles is C 75) In the opposite figure: a) The length of AB = cm b) The length of BC = cm c) The length of AC = cm d) The type of ∆ ABC according to A the lengths of its sides is 76) 77)
The perimetre of a rectangle = 2 × (
B
B
)
The area of a rectangle =
78)
The smallest 4-digit number divisible by 3 is
79)
The two 3-digit numbers that are divisible by 3 are
80)
The perimetre of a square =
81)
The area of a square =
,
82)
The three 4-digit numbers that are divisible by 5 are
,
, 83) The area of a rectangle whose dimensions 4cm 5cm is
and
84) The perimetre of a rectangle whose width 6cm its length 10cm is
and
85)
The area of a square whose side-length 7cm is
86) ,
The three 2-digit numbers that are divisible by 2 and 3 are ,
87)
The perimetre of a square whose side-length 10cm is
88)
15 kilograms and 25grams =
89)
135 kilometres and 5 metres =
90)
The three 3-digit numbers that are divisible by 2 are
grams metres ,
, 91)
13 metres and 4 decimetres =
decimetres
92)
18 metres, 5 decimetres and 6 centimetres =
93)
15 m2 =
94)
136 m2 =
cm2 dm2
95)
10 dm2 =
cm2
96)
13 km2 =
m2
97) The numbers that are divisible by 5 from 321, 50, 87543, 540, 861, 70, 94, 54820 are , , , 98)
14 km2 =
dm2
99)
19 km2 =
cm2
100)
The number between 4455700216 and 4455700218 is ..
101)
21346546m2 =
km2,
102)
12453687cm2 =
m2,
m2 dm2,
cm2
103)
The number is divisible by 2, if
104)
560000479
122)
524-millions + 8-thousands + 5-tens =
123)
125m2 + 5642cm2 =
124)
The number after 54216899 is
125)
21546802 +
> 5432847 cm2
= 98003241
126)
+ 65400089 = 987140021
127)
- 5468700 = 6542138
128)
654879213 -
= 6542187
129)
The number before 54621380 is × 4 = 260
130)
131) If a number is divisible only by itself and one. Then it s called 132)
56 ×
133)
1368 ÷
= 1176 =3
134) The numbers that are divisible by 2and 5 from 320, 50, 87543, 540, 861, 70, 94, 520 are , , , , 135)
The numbers that are prime numbers from 20 to 30 are ÷ 123 = 982
136)
Complete the missing digits: 137)
2 +
139)
3
3 5 4 5 6 8 . 72 25 598 5
1
0 + 1 9 2 . 72 25 598
141)
138)
2 -
140)
9 -
1 1 6 1 . 12 25 198
-
2 3 5 4 5 7 . 1 2 2 5 1 9
+
2 3 5 4 5 7 . 9 2 2 5 1 9
142) + 5 6 8 2 5 3 7 2 2 5 5 9
143)
144) - 5 6 8 2 5 3. 3 2 2 5 5 9
3 5 4 5 6 8 . 12 251 98 3
0
,
145) +
351 20 6 4 1 9 2 . 8 2 5 9
146)
3 5 4 ×6 2 9
148)
147) + +
-
8 7 5 5 9 × +
.
.
149)
7 4 5 7 8 . -
150)
. .
-
0 0 151)
401
152)
4
153)
is divisible by 5 01 is divisible by 3 50 is divisible by 5 and 2
154)
6
155)
3217
156)
is a prime number is divisible by 2
7is a prime number 98
23 1 2 4 6 6 . -
-
157)
9 3 01 5 6 1 1 6 1 5 2 2 5
72 is divisible by 3 and 2
Complete using or = : 158)
3254187
3252187
159)
3254164
10546687
160)
32154 + 12456
161)
62135487
162)
1215 × 5
21546872
12458793 1254834
211548
. . 0 0
125 × 21
163)
1254879
164)
125 × 213
165)
738 ÷ 6
166)
215
167)
18404 ÷ 43
168)
H.C.F of 10, 15
12548981 124
5472 ÷ 12 7831 L.C.M of 10, 15
169) The sum of measures of angles of ∆ ABC measures of angles of ∆ XYZ. 170)
Obtuse angle
acute angle.
171)
Acute angle
right angle.
172)
2-millions
3254-thousands
173)
3215-units
54-hundreds
174)
Right angle
obtuse angle.
175)
56-millions
452001-hundreds
the sum of
176) Area of a rectangle area of a square whose side length is the width of that rectangle. 177) Area a rectangle area of a square whose side length is the length of that rectangle. Complete as the pattern: 178)
21548732, 21547732,
179)
54218791, 54228791,
, ,
, ,
, ,
180)
,
,
,
, 22456871, 12456871
181)
,
,
,
, 20 000 000, 10 000 000
Self-Test II Put the suitable sign ( √ ) or ( X ) 1)
2000001 are 20-millions and one
( )
2)
5-millions and 5-hundreds are 5000500
( )
3)
12cm, 12cm and 10cm are the lengths of an isosceles triangle. ( )
4)
65000251 is 65-millions, 2-hundreds and fifty-one
( )
5)
120-millions are 12000000
( )
6)
125-tens are 1250
( )
7)
100o, 50o and 30o are the measures of obtuse angledtriangle. ( )
8)
The area of ABCD rectangle in which AB = 2cm and DA = 3 is 10cm2 ( )
9)
4521-units are 4521.
( )
10) 5468-hundreds are 4568.
( )
11) The obtuse angled triangle has at most one obtuse angle. ( ) 12) 5400-thousands are 54000.
( )
13) The area of a square in which a side = 5m is 25m.
( )
14) 1-million, 5-thousands, 3-tens and five is 15035.
( )
15) The equilateral triangle has all equal sides
( )
16) The prime factors of 20 are 4 and 5
( )
17) 45-millions, 2-thousands, 2-hundreds and three is 452203. ( ) 18) The right angled-triangle has at least one right angle.
( )
19) 5 in 4215367 is unit-thousand digit.
( )
20) The measures of angles of a triangle are 20o, 90o, 70o. Thus its type is obtuse angled-triangle. ( ) 21) The ten-million digit in 452187340 is 5.
( )
22) 8 in 5400084 is ten digit.
( )
23) The unit digit in 65400084 is 6.
( )
24) The length of a rectangle whose perimetre 30cm and its width 5cm is 10cm. ( ) 25) The place value of 7 in 7000000 is hundred-million digit. ( ) 26) All factors of 18 are 1, 18, 6 and 2
( )
27) The unit million digit has 6-zeros.
( )
28) ABC is a triangle in which AB = BC = AC. Thus it s scalene triangle. ( ) 29) The triangle has at least two acute angles.
( )
30) The thousands have 3-zeros.
( )
31)
( )
The hundred-million digit has 7-zeros.
32) 124587 > 124527.
( )
33) 124500054 < 12450054.
( )
34) If the sum of measures of two angle of a triangle is 120o. Then the third one is acute angle. ( ) 35)
The H.C.F of 18 and 24 is 6
( )
36) 451233 + 100000 = 551233.
( )
37) 3000000
( )
1245671 = 1754319.
38) Any triangle has at least two right angles.
( )
39)
( )
The greatest 3-digit number is 999.
40) The H.C.F of 18, 12 and 24 is 24
( )
41) The smallest 5-digit number 10 000.
( )
42) The scalene triangle has different sides in lengths.
( )
43) 0 is a multiple of all numbers
( )
44) The greatest 2-same digit number is 88.
( )
45) The sum of measures of angles of a triangle is 180o
( )
46) The smallest 3-different digit number is 102.
( )
47) The type of ∠ ABC whose measure 45o is acute angle. ( ) 48) The smallest 2-different odd digit number is 13.
( )
49) The third multiple of 5 is 15
( )
50)
The obtuse angled-triangle has only two acute angles ( )
51) The greatest 4-different even digit number is 2046.
( )
52) The units of area are squared units.
( )
53) The smallest 5-same odd digit number is 11111.
( )
54) 45-millions = 45000-thousands = 450000-hundreds
( )
55) 521-millions = 521000000-units
( )
56) The width of a rectangle whose perimetre 50m and its length 20 is 500cm. ( ) 57) 6300000-tens = 63000-thousands = 63-millions
( )
58) 125483 + 1240000 = 24925483
( )
59) The first three multiple of 2 are 2, 4, 6
( )
60) 90o is a measure of an acute angle.
( )
61) 5321100045
( )
62) 63)
65400012 = 5255700033
6, 1 are factors of 6 (524524 + 100000)
( ) 421300 = 203224
( )
64) The acute angled- triangle has three acute angle
( )
65) 4 is the first multiple of itself
( )
66) 9 is the multiple of 9
( )
67) 36-millions + 52-tens = 36000520
( )
68) The perimetre of a square is twice of length plus length. ( ) 69) 231-tens + 5421-units + 3-millions = 3007731
( )
70) The number before 1245380 is 1245379
( )
71) The perimetre of a rectangle is twice of length plus width. ( ) 72) The number after 652136989 is 652136988
( )
73) The number between 124587604 and 124587606 is 124587605. ( ) 74) The right angle whose measure is greater than 90o
( )
75) 12548650 > 12548651 > 12548652
( )
76) The area of a rectangle, the lengths of two adjacent sides 4 and 5 is 18cm. ( ) 77) 30002219 < 30002220 < 30002221
( )
78) The right angled-triangle has only one acute angle
( )
79) 50 is divisible by 5
( )
80) 3446 is divisible by 53
( )
81) The obtuse angle whose measure is 180o
( )
82) 86 is divisible by 2
( )
83) The L.C.M of 18, 12 and 8 is 72
( )
84) 3216 is divisible by 2 and 3
( )
85) The perimetre of ABCD square in which AB = 3cm is 12cm2. ( )
86) 5212 is divisible by 3
( )
87) The perimetre of a square whose side length 6cm is 24cm. ( ) 88) The straight angle is 180o
( )
89) 17 is a prime number
( )
90) the obtuse angled-triangle has two obtuse angles
( )
91) 5640 is divisible by 10
( )
92) 180o is a measure of a straight angle.
( )
93) 98721 is divisible by 6
( )
94) 89o is a measure of an obtuse angle
( )
95) 45 is the ninth multiple of 5
( )
96) 5cm, 6cm and 4cm are formed a triangle.
( )
97)
3m2 = 100dm
( )
98) The area of a rectangle is length by width.
( )
99) 98725 is divisible by 5
( )
100) The L.C.M of 8 and 12 is 4
( )
101) 5km2 = 1000 000m2
( )
102) The acute angle is less than 90o and it can be 0o.
( )
103) 46 is a prime number
( )
104) 125o is a measure of an acute angle
( )
105) 2m2 = 200cm2
( )
106) The area of a square is length by length.
( )
107) The length of a rectangle whose area 80cm2 and its width 8cm is 10cm. ( )
Self-Test III Choose the correct answer(s): 1) 3-millions and 1 = a) 3000001 b) 30001
c) 300001
2) 42-millions, 4-hundreds and 5 = a) 4245 b) 4200405
c) 42000405
3) 200-millions, 4-thousands and 2 = a) 200040002 b) 200004002
c) 2000004002
4) 20000000 is a) 2
c) 200
5) 30000000 is a) 30-millions
-millions. b) 20 b) 30-thousands
c) 300-hundreds
6) The ten-million digit in 45200123 is a) 2 b) 5 c) 4 7) The hundred-million digit in 213564897 is a) 2 b) 1 c) 3 8) The unit-million digit in 32100548 a) 3 b) 2
c) 1
9) The place value of 3 in 56423178 is the unit- digit a) million b) thousand c) hundred 10) The place value of 7 in 789654123 is digit a) unit b) ten-million c) hundred-million 11) 54213687 a) < 12) 546987002 a)
54698702 b) >
13) 45213687 > 45213 87 a) 7 b) 6
c) = c) = c) 5
14) 452 a) 0
30078 < 452130078 b) 1
15) 46-millions, a) 5
c) 2
-thousands and 5 = 46003005 b) 3 c) 6
16) -millions, 100-thousands and 3-hundreds = 4100300 a) 4 b) 1 c) 3 17) 5000000 + 3000 + 20 = a) 500320 b) 5003020 18) 65325411 = a) 650000
c) 503020
+ 300000 + 20000 + 5000 + 400 + 10 + 1 b) 6500000 c) 65000000
19) The greatest number formed from 5, 3, 7, 8, 4, 0 is a) 875430 b) 034578 c) 304578 20) The smallest number formed from 5, 4, 6, 0, 2, 9, 3 is a) 9654320 b) 0234569 c) 2034569 21) The smallest 3-digit number formed from 5, 6, 4, 9, 0, 2, 8 is a) 024 b) 204 c) 420 22) The greatest 4-digit number formed from 6, 5, 4, 8, 2, 0, 9, 3 is a) 023 b) 203 c) 320 23) The smallest 3-same number is a) 000 b) 111
c) 333
24) The greatest 2-different digit number is a) 98 b) 89 c) 109 25) The smallest 4-different digit number is a) 0123 b) 1023 c) 1234 26) The greatest 5-same digit number is a) 1010101010 b) 99999 c) 88888
27) The smallest 2-same even digit number is a) 11 b) 22 c) 88 28) The greatest 3-different odd digit number is a) 135 b) 351 c) 531 29) 23-millions = thousands a) 230 b) 2300
c) 23000
30) 4510-thousands = -hundreds a) 451000 b) 4510000
c) 45100000
31) 451 + 453000 = a) 453351 b) 443451
c) 453451
32) 12000000 + 12 = a) 24000000 b) 12000012
c) 1200012
33) 12459700 a) 12456485
3215 = b) 12455485
c) 12456484
34) 451200032 a) 449689032
1521000 = b) 449679032
c) 448679032
35) 124 × 12 = a) 1588
b) 1478
c) 1488
36) 54 × 543 = a) 29322
b) 28322
c) 29222
37) 450 × 587 = a) 264250
b) 264150
c) 254150
38) 2260 ÷ 5 = a) 462
b) 453
c) 452
39) 11186 ÷ 47 = a) 238 b) 237
c) 228
40) 4-millions - 5-thousands = a) 3994000 b) 3995000
c) 4995000
41) 12-million + 54-ten = a) 12000530 b) 12000540
c) 12010540
42) The number before 1245871 is a) 1245870 b) 1245871
c) 1245872
43) The number after 4568721 is a) 4568720 b) 4568721
c) 4568722
44) The number between 12458792 and 12458790 is a) 12458792 b) 12458791 c) 12458790 45) 21548901 < a) 21548900
< 21548902 b) 21548901
c) 21548902
46) + 1245780 = 45120687 a) 43874807 b) 43874907
c) 43884907
47) 452198 + a) 89580249
= 90032547 b) 88580349
c) 89580349
= 213547 b) 12245155
c) 12245255
48) 12458702 a) 12244155
49) - 54681 = 1245671 a) 1300352 b) 1300351
c) 1301352
× 124 = 1240 50) a) 1 b) 10
c) 100
51) 12 × a) 458
c) 448
= 5496 b) 457
÷ 23 = 542 52) a) 12476 b) 12466
c) 22466
53) 11300 ÷ a) 26
c) 35
= 452 b) 25
54) How many 7s are there in 875? There are a) 135 b) 125 c) 126
55) How many 56s are there in 34048? There are a) 607 b) 68 c) 608 56) 150 is by 5 a) divisible b) not divisible
c) otherwise
57) 288 is by 6 a) divisible b) not divisible
c) other wise
58) 86 is divisible by a) 2 b) 3
c) 5
59) 129 is divisible by a) 2 b) 3
c) 5
60) 235 is divisible by a) 2 b) 3
c) 5
61) 2232 is divisible by a) 2, 3 b) 2,5
c) 3,5
62) 520 is divisible by a) 2, 3 b) 2, 5
c) 3, 5
63) 2205 is divisible by a) 2, 3 b) 2, 5
c) 3, 5
64) 5210 is divisible by a) 6 b) 10
c) 15
65) 405 is divisible by a) 6 b) 10
c) 15
66) 3012 is divisible by a) 6 b) 10
c) 15
67) 550 is by 10 a) divisible b) not divisible
c) otherwise
68) 1215 is a) divisible
c) Otherwise
by 15 b) not divisible
69) 17 is a) prime
number. b) not prime
c) otherwise
70) 1 is number. a) prime b) not prime
c) otherwise
71) The 3-digit number that divisible by 2 is a) 123 b) 231 c) 312 72) The 4-digit number that divisible by 3 is a) 2304 b) 2324 c) 2314 73) The 2digit number that divisible by 5 is a) 25 b) 52 c) 53 74) 12 a) 1
5 is divisible by 3 b) 2
c) 3
75) 1452 a) 0
is divisible by 5 b) 1
c) 2
76) 2153 a) 0
is divisible by 2 b) 1
c) 2
77) 3 is a prime number. a) 2 b) 3
c) 6
78) 2 is a prime number. a) 1 b) 2
c) 3
79) The greatest 3-same digit number divisible by 2 is a) 444 b) 666 c) 888 80) The smallest 3-different digit number divisible by 3 is a) 231 b) 321 c) 123 81) The greatest 2-same odd digit number divisible by 5 is .. a) 55 b) 77 c) 99 82)
The smallest 3-different even digit number divisible by 2 is a) 246 b) 426 c) 624
83) a) 2
is a factor of 9 b) 4
c) 3
84) 5 is a factor of a) 3 b) 4
c) 5
85) , a) 2, 9
are factors of 27 b) 4, 3
c) 3, 9
86) , a) 2, 9, 7
,
are factors of 81 b) 4, 27, 1
c) 3, 27, 9
87) The prime factors of 12 are a) 3, 4 b) 3, 2
c) 4, 12, 3, 2, 1
88) All factors of 18 are a) 2, 3, 9, 18, 1 b) 3, 6, 18, 1
c) 3, 2, 6, 18, 9, 1
89) The H.C.F of 18 and 24 is a) 6 b) 3
c) 2
90) The L.C.M of 8, 12 is a) 12 b) 24
c) 8
91) The H.C.F of 18, 12 and 24 is a) 6 b) 12
c) 18
92) The L.C.M of 18, 12 and 8 a) 24 b) 72
c) 18
93) The third multiple of 2 is a) 2 b) 4
c) 6
94) The zero multiple of 10 is a) 0 b) 10
c) 20
95) The acute angle is an angle whose measure is less than and greater than o a) 0 , 90o b) 90o, 0o c) 180o, 90o
96) The obtuse angle is an angle whose measure is greater than and less than o o a) 180 , 90 b) 0o, 90o c) 90o, 180o 97) The right angle whose measure is a) 0o b) 90o
c) 180o
98) The straight angle whose measure is a) 0o b) 90o c) 180o 99) An angle whose measure 80o, its type is angle a) acute b) obtuse c) right 100) An angle whose measure 90o, its type is angle a) acute b) obtuse c) right 101) An angle whose measure 125o, its type is angle a) acute b) obtuse c) right 102) An angle whose measure 180o, its type is angle a) acute b) straight c) right 103) The sum of measures of angles of a triangle is a) 90o b) 180o c) 170o 104) ABC is a triangle in which m( ∠ A) = 45o and m( ∠ B) = 90o. Thus m( ∠ C) = a) 90o b) 45o c) 40o 105) XYZ is a triangle in which m( ∠ X) = 50o and m( ∠ Z) = 60o. Thus ∠ Y is angle a) acute b) obtuse c) right 106) The measures of angles of a triangle are 50o, 60o, 70o. Thus its type is -angled triangle. a) acute b) obtuse c) right 107) ABC is a triangle in which m( ∠ A) = 120o and m( ∠ B) = 20o. Thus the type of ∆ ABC is angled triangle a) acute b) obtuse c) right
108) The lengths of sides of a triangle are 5cm, 7cm and 4cm. Thus its type is triangle a) equilateral b) isosceles c) scalene 109) The lengths of sides of a triangle are 9cm, 9cm and 6cm a) equilateral b) isosceles c) scalene 110) The perimetre of a square = a) width × length b) (width + length) × 2 c) 4 × side length 111) The perimetre of rectangle = a) width × length b) 4 × side length c) (width + length) × 2 112) The area of a square = a) 4 × side length b) width × length c) side length × side length 113) The area of a rectangle is = a) 4 × side length b) (width + length) × 2 c) width × length 114) The perimetre of a square whose side length 4cm is a) 20 b) 16 c) 24 115) The area of a square whose side length 0.06m is a) 36cm b) 3.6m c) 0.36m 116) The perimetre of a rectangle whose dimensions 3m and 2m is a) 6cm b) 6m c) 10m 117) The length of a rectangle whose perimetre 30cm and its width 5cm is a) 15cm b) 10cm c) 5cm 118) The area of a rectangle whose dimensions 2m, 5m is a) 10cm b) 10m c) 14m
Self-Test IV Different types of Problems: 1)
Arrange the following in ascending order: a) 215487, 1245873, 98745213, 124578, 231548 b) 21050046, 30002154, 21055498, 12458963, 21045457 c) 52314374, 52314174, 52314774, 52314574, 52314874
2)
Arrange the following in descending order: a) 52146330, 4521873, 21548701, 1245387, 542187 b) 21453687, 45876231, 4521876, 3254187, 9645421 c) 53209487, 53201487, 53204487, 53207487, 53200487
3)
4)
Amr paid 1000 pounds for 31325 kilos of bananas and 2000 pounds for 165415 kilos of watermelons. Calculate the total kilos that he bought. Draw ∠ ABC whose measure is 135o. State its type.
5)
Mona has took 87542 grams of sweets from his father and 2154 kilos from his mother, he bought a pencil for P.T 545. How much money was left?
6)
What is the total cost of 654 kilos of bananas, if you ve known the price of one kilo is P.T 65?
7)
Ramy s tall is 1752mm and Bassem s tall is 2012mm. How much is Bassem s tall greater than Ramy s tall?
8)
Mohammed has got three coloured cars, the green one L.E 321544, the yellow one L.E 3021548 and the blue one L.E 213546. Find the total cost of Mohammed s cars.
9)
Magda has got 53 kilos of sweets, the price of a kilo is L.E 14. What is the total cost?
10) Abd El Rahman has bought 3232154mm of cloth form the market, he made trousers by 332214mm and jackets by 4615mm. How much millimetres of cloth were left? 11) How many 53s are there in 8109? 12) Aiya has bought 3200125 grams of bananas and 1300325grams of potatoes. How much did the bananas decrease the potatoes? 13) Bassem bought 35metres of a certain material for making a balloon, he paid P.T 2275, what s the price of each metre? 14) Write in letters a) 12458003
b) 4512870
c) 45120084
d) 987546321
15) Write in digits a) 5-millions, 45-thousands, 8-hundreds and fifty-one b) 54-millions and two. c) 201-millions, 52-thousands and sixty-four. 16) Determine the ten-million digit in each of the following: a) 987654321
b) 123546987
c) 456321789
17) Determine the place value of 7 in each of the following: a) 456987123
b) 789654123
c) 321789654
18) Find the result in each of the following: a) 12546 + 32154698
b) 789654122
c) 452 × 231
d) 2568 ÷ 8
e) 548 × 54
f) 47523 ÷ 73
321547
19) Amr paid L.E 10000 for 64543221 kilos of bananas and L.E 20000 for 56421307 of watermelons. Calculate the total
kilos that he bought. 20) State all the prime numbers between 12 and 30. 21) Write three 3-digit numbers aren t divisible by 3 22) Write four 2-digit numbers are divisible by 2 23) Write five 3-digit numbers are divisible by 5 24) Write four 2-digit numbers are divisible by 3 25) Find the H.C.F for each of the following: a) 10 and 40
b) 36, 45 and 72
26) Find the L.C.M for each of the following: a) 18 and 24
b) 12, 32 and 56.
27) Measure each of the following angles: a)
C
A
b)
B m( ∠ ABC) =
Z
m( ∠ XYZ) =
Y
X
28) Draw ∠ ABC whose measure: a) 150o
b) 90o
29) Draw ∆ ABC in which: a) AB = BC = 7cm and m( ∠ ABC) = 80o b) m( ∠ ABC) = m( ∠ BAC) = 50o and AB = 6cm 30) Draw ∆ ABC in which AB = 3cm, BC = 5cm and m( ∠ ABC) = 60o. Measure ∠ BAC, what s the type of ∆ BAC according to the measures of the angles? 31) Draw ∆ XYZ in which m( ∠ XYZ) = 70o, m( ∠ ZXY) = 55o and XY = 6cm. Measure the length of ZY, then what s the
type of ∆ XYZ according to the lengths of the sides? 32) Calculate the perimetre of a square when its side length: a) 6cm
b) 10cm
33) Calculate the area and the perimetre of a rectangle when its dimensions : a) 9cm and 7cm
b) 6cm and 8cm
34) Calculate the length of a rectangle when: a) Its perimetre = 24cm and its width = 3cm b) Its perimetre = 20cm and its width = 3cm c) Its area = 63cm2 and its width = 7cm. d) Its area = 54cm2 and its width = 6cm. 35) Calculate the width of a rectangle when: a) Its perimetre = 36cm and its length = 10cm. b) Its perimetre = 54cm and its length = 20 c) Its area = 36cm2 and its length = 12cm. d) Its area = 120cm2 and its length = 12cm 36) Calculate the length of a rectangle, its perimetre when its area = 45cm2 and its width = 5cm. 37) Calculate the length of a rectangle, then its area when its perimetre = 50cm and its width = 10cm. 38) Calculate the width of a rectangle, then its perimetre when its area = 72cm2 and its width = 8cm. 39) Calculate the width of a rectangle, then its area when its perimetre = 60cm and its length = 20cm.
Self-Test V Selected problems: 1) Write a 2-digit number not divisible by 2. 2) Write a 5-digit number divisible by 2. 3) Write a 4-digit number divisible by 3. 4) Write a 2-digit number divisible by 5. 5) Write a 3-digit number divisible by 6. 6) Write a 4-digit number divisible by 10. 7) Write odd or even in the space: a)
Odd + odd =
b)
Odd
c)
Even + odd =
d)
Even + even =
e)
Odd × odd =
f)
Odd × even =
g)
Even × even =
h)
Even × odd =
even =
8) State the greatest odd 3-digit number formed from 6, 1, 3 9) State the greatest even 4-digit number formed from 4, 0, 5, 2. 10) Write the greatest and the smallest 3-digit number divisible by 3. 11) A chocolate factory packs 32 chocolate bars in each box. How many boxes are needed for packing 1728 chocolate bars?
12) In a chicken farm, 1710 eggs are packed in cartons. Each carton holds 30 egg. How many cartons are needed for packing the whole quantity of eggs? 13) A box containing 48 bars of soap costs P.T 3120. Find the cost of each bar of soap. 14) A grocer sold 200 eggs. 74 eggs were sold for 18 piasters each, the rest were sold for 20 piasters each. Find the total cost of all eggs. 15) Use your protractor to draw an angle of measure 60o, vertex B and side BA. 16) Draw a triangle ABC in which AB = 6cm, AC = 5cm and angle A has measure 80o. 17) Draw a triangle XYZ in which measure of ∠ X = 70o, measure of ∠ Y = 30o and XY = 6cm. 18) Draw a triangle ABC in which AB = AC = 5cm and the measure of ∠ A = 60o. Find the length of BC and the measure of ∠ C. What is the type of ∆ ABC according to the measures of its angles and according to the lengths of its sides? 19) Find the area and the perimetre of a rectangle whose length is 10cm and its width is 7cm. 20) Find the length of a rectangle whose area is 42cm2 and its width is 6cm. 21) Find the width of a rectangle whose perimetre is 30cm and its length is 10cm. 22) Find the area and the perimetre of a square whose side length is 6cm.
Examinations Exam Style Paper I 1 a) Find the results: i)
356186 +171567
ii) 710250 - 326743
iii) 356 × 174
iv) 25 35125
b) Complete i)
The greatest number formed from the digits 5, 1, 6, 3, 7 is
ii) Right angle, its measure is 2 a) Find L.C.M and H.C.F of each 36, 24, 72 b) Compete using or = : i)
351 × 100
ii)
Obtuse angle
351000 ÷ 10 acute angle
iii) 31-millions and 30-thousands
313000
3 a) Find three 3-digit number are divisible by 2 and 3. b) Find the prime factors of each of 28, 32. c) Write the number 3001506 in letters. d) Find the factors of 18, 56. e) Which is the greatest angle: right angle, obtuse angle, and acute angle? f) Find the price of one kilogram, if the price of 31 kilos of apples is P.T 17515, then find the price of 45 kilos.
C 4 Complete the table from the figure: Angle
m( ∠ A) m( ∠ B) m( ∠ C)
Type Type of triangle A
B
Exam Style Paper II Answer the following questions 1
2
a) Add
:38567 + 12496
b) Subtract
:587236
c) Multiply
:254 × 37
d) Divide
:8415 ÷ 45
51247
Complete a)
-millions,
-thousands = 7125000
b) 804 × 100 = 10 ×
=
c) From the numbers 326, 423, 984, 222, 657, 862. Find: i)The number divisible by 3 are ii)The number divisible by 2 are iii)The numbers divisible by 6 are 3
a) Write the prime numbers between 11 and 37. b) Find L.C.M and H.C.F for 12 and 18. c) In the triangle ABC, m( ∠ A) = 80o and m( ∠ B) = 70o. What s m( ∠ C)?
4
If the price of 26 metres of cloth is L.E 286. Then find the price of one metre, then the price of 20 metres of this cloth.
5
C
Measure the angles of the opposite triangle ABC, then complete a) m( ∠ A) = type is
and its
b) m( ∠ B) = type is
and its
c) m( ∠ C) = type is
and its B
A
d) The type of ∆ ABC according to the measures of the angles is
Exam Style Paper III Answer the following questions: 1
a) Complete: i) 3 7 1 8 7 ii) +
iii) 5 6 0 ×
. - 5 6 1 8 8 7 9 6 1
iv)
3 1569
1 5
1 5 5 3
b) Arrange the following numbers in ascending order: 351086, 361086, 46109, 515361, 138609 2
a) Find H.C.F and L.C.M for 18 and 24. b) Put the suitable sign or = : i) 563806
5631186
ii) 632 × 3
16070 ÷ 5
c) Draw the angles which whose measures are : i)
70o
ii)
90o
iii)
125
d) Complete as the pattern: 321545, 323545, 325545 3
,
,
,
a) Put the suitable sign ( √ ) or ( X ): i) 31068 31059 = 9
( ) ii)35618 ÷ 13 = 2501
( )
iii)3156 is divisible by 6 ( ) iv)6 is L.C.M for 18, 12 ( ) v)The sum of measures of angles of a triangle is 150o ( ) b) Ahmed had bought 13 kilos of oranges for P.T 135 each. Find the price of quantity. c) Find the prime numbers between 31 and 58. 4
a) Complete: i)
-millions,
-thousands and
-units = 35018613
ii)31-millions, 30-thousands and 315-hundreds = b) Measure, then complete from the figure: C Type of ABC triangle AB BC AC B
A
Exam Style Paper IV Answer the following questions: 1
2
Find the result: i)
563819 + 48631
ii)
560001 - 63891
iii)
538 × 531
iv)
80675 ÷ 35
Complete: i)
-millions,
-thousands and
-units = 5001007
3
ii)
35-millions, 25-hundreds and 1 =
iii)
35m2 =
iv)
The 3-digit number divisible by 2 is
.
cm2
a) Find L.C.M and H.C.F for 48 and 18 b) If the price of 23 books of mathematics is P.T 3588. Then find the price of one book, then the price of 45 books.
4
a) Draw the ∆ ABC in which AB = 5cm, m( ∠ A) = 70o and m( ∠ C) = 45o. Find the length of AC and BC and state the type of the triangle according to the lengths of the sides. b) Calculate the area and the perimetre of a square whose side length is 10cm.
5
Put the suitable sign ( √ ) or ( X ): i)
The isosceles triangle has two equal sides.
( )
ii)
Each triangle has at least two acute angles.
( )
iii) The area of a rectangle equals twice length plus width.
( )
iv)
6321 is divisible by 2 and 3.
( )
v)
33 is a prime number.
( )
vi)
3 and 7 are prime factors of 21.
( )
Exam Style Paper V Answer the following questions: 1
Complete: i) The place value of 245689 is
6
in 1634517 is
and in
ii) The factors of the number 18 are
,
,
,
,
,
iii)The smallest number formed by 6, 8, 7, 2, 5 is and the greatest number formed by these digits is and the difference between them is iv) The type of a triangle whose side lengths 4cm, 5cm and 4cm is and the type of a triangle whose angle o measures 50 , 100o and 30o is 2
3
Find the result of each of the following: i) 2896261 + 6149467
ii) 458 × 182
iii)216491
iv)1134 ÷ 27
85372
a) Complete: i)7-millions and 215-thousands = ii) -millions and
-thousands = 56047000.
b) Complete using or = : i)3420100
3381299
iii)Right angle 4
ii)25 × 61
6100 ÷ 4
obtuse angle.
a) Find the H.C.F and L.C.M between 16 and 24. b) A box containing 48 bars of soap costs P.T 3120. Find the cost of each bar. c) Calculate the area and the perimetre of a rectangle whose dimensions 5cm and 10cm.
5
a) Complete: i)The triangle whose sides 18cm, 18cm and 18cm is ii)Triangle ABC in which m( ∠ A) = 100o, m( ∠ C) = 30o, so m( ∠ B) = and the type of this triangle with respect to the measures of its angles is b) Draw the angle whose measure is 80o.
Exam Style Paper VI Answer the following questions: 1 Complete: i)
5-millions, 31-thousands and 8 =
ii)
53893 + 3893196 =
iii)
500 ÷ 4 =
iv)
The prime factors of 18 are
v)
The number before 54621380 is
2 Choose the correct answer: i) 200-millions, 4-thousands and 2 = a) 200040002 b) 200004002 c) 2000004002 ii) 546987002 a)
c) =
iii) The smallest 3-digit number formed from 5, 6, 4, 9, 0, 2, 8 is a) 024 b) 204 c) 420 iv) 451 + 453000 = a) 453351 b) 443451 v) a) 1
c) 453451
× 124 = 1240
b) 10
c) 100
3 Put the suitable sign ( √ ) or ( X ): i)
5-millions and 5-hundreds are 5000500
( )
ii) The area of ABCD rectangle in which AB = 2cm and DA = 3 is 10cm2
( )
iii) The place value of hundred-million digit.
7
in 7000000 is ( )
iv)
The H.C.F of 18 and 24 is 6
( )
v)
The third multiple of 5 is 15
( )
4 a) Arrange the following in ascending order: 215487, 1245873, 98745213, 124578, 231548 b) Mona has taken 87542 grams of sweets from his father and 2154 kilos from his mother, he bought a pencil for P.T 545. How much money was left? 5 a) Write odd or even in the space: i) Odd + odd =
ii) Odd
even =
iii) Even + odd =
iv) Even + even =
b) Use your protractor to draw an angle of measure 60o, its vertex is B and its side is BA.
Exam Style paper VII Answer the following questions: 1 Choose the correct answer: i) How many 56s are there in 34048? There are a) 607 b) 68 c) 608 ii) 405 is divisible by a) 6 b) 10
c) 15
iii) The greatest 2-same odd digit number divisible by 5 is a) 55 b) 77 c) 99 iv) The prime factors of 12 are a) 3, 4 b) 3, 2 v)
c) 4, 12, 3, 2, 1
The obtuse angle is an angle whose measure is greater than and less than a) 180o, 90o b) 0o, 90o c) 90o, 180o
2 Complete: i)
The greatest number formed from 1, 8, 9, 2, 0, 4 is
ii)
368 × 100 =
iii)
The sum of measures of angles of a triangle is
iv) The type of the triangle whose sides-lengths 6m, 6m, 6m is v) The area of a rectangle whose dimensions 4cm and 5cm is 3 Put the suitable sign ( √ ) or ( X ): i)
90o is a measure of an acute angle.
( )
ii) The perimetre of a rectangle is twice of length plus width.
( )
iii)
86 is divisible by 2
( )
iv)
3m2 = 100dm
( )
v)
The L.C.M of 18, 12 and 8 is 72
( )
4 a) Mohammed has got three coloured cars, the green one L.E321544, the yellow one L.E 3021548 and the blue one L.E 213546. Find the total cost of Mohammed s cars. b) Find the L.C.M and H.C.F for 18 and 24 5 a) Find the area and the perimetre of a square whose side length is 10cm. b) Draw a triangle XYZ in which measure of ∠ X = 70o, measure of ∠ Y = 30o and XY = 6cm.
Exam Style Paper VIII Answer the following questions: 1 Put the suitable sign ( √ ) or ( X ) : i) The area of a rectangle, the lengths of two adjacent sides 4 and 5 is 18cm.
( )
ii) 231-tens + 5421-units + 3-millions = 3007731
( )
iii) (524524 + 100000)
( )
421300 = 203224
iv) The greatest 4-different even digit number is 2046.
( )
v) The triangle has at least two acute angles.
( )
2 Choose the correct answer: The measures of angles of a triangle are 50o, 60o, 70o. Thus its type is -angled triangle. a) acute b) obtuse c) right
i)
ii) The area of a rectangle is = a) 4 × side length b) (width + length) × 2 c) width × length iii) The hundred-million digit in 213564897 is a) 2 b) 1 c) 3 iv) The place value of 3 in 56423178 is the unit- digit a) million b) thousand c) hundred v) The place value of 7 in 789654123 is digit a) unit b) ten-million c) hundred-million 3 Complete: i)
15 kilograms and 25grams =
ii)
19 km2 =
grams
cm2
iii) The numbers that are divisible by 10 from 321, 50, 87543, 540, 861, 70, 94, 54820 are , , ,
iv) v)
1368 ÷
=3
7 is a prime number
4 a) Draw ∆ ABC in which AB = 3cm, BC = 5cm and m( ∠ ABC) = 60o. Measure ∠ BAC, what s the type of ∆ BAC according to the measures of the angles? b) Calculate the area and the perimetre of a rectangle when its dimensions 9cm and 7cm. 5 a) A chocolate factory packs 32 chocolate bars in each box. How many boxes are needed for packing 1728 chocolate bars? b) State the greatest even 4-digit number formed from 4, 0, 5, 2, 9, 6, 7, 8.
Exam Style Paper IX Answer the following questions: 1 Complete : i)
480009
ii)
436 × 5 =
iii)
The numbers divisible by 2 are
iv)
5 5 + 50 000 +
138899 =
5 5 = 500 000 000 + + 500 +
+ 5000 000 +
v) In the triangle ABC, the measure of ∠ A = 60o and the measure of ∠ B = 50o, then the measure of ∠ C = 2 Put the suitable sign ( √ ) or ( X ): i) ABC is a triangle in which AB = BC = AC. Thus it s scalene triangle.
( )
ii)
( )
5 in 4215367 is unit-thousand digit.
iii)
The equilateral triangle has all equal sides
( )
iv)
5-millions and 5-hundreds are 5000500
( )
v)
All factors of 18 are 1, 18, 6 and 2
( )
3 Choose the correct answer: i) The place value of 7 in 789654123 is digit a) unit b) ten-million c) hundredmillion ii) 65325411 = 1 a) 650000
+ 300000 + 20000 + 5000 + 400 + 10 + b) 6500000
c) 65000000
iii) 11186 ÷ 47 = a) 238 b) 237
c) 228
iv) 225 is divisible by a) 2,5 b) 3, 2
c) 3, 5
v) , a) 2, 9, 7
c) 3, 27, 9
,
are factors of 81 b) 4, 27, 1
4 a) Calculate the length of a rectangle, then its perimetre when its area = 45cm2 and its width = 5cm. b) Draw ∆ XYZ in which m( ∠ XYZ) = 70o, m( ∠ ZXY) = 55o and XY = 6cm. Measure the length of ZY, then what s the type of ∆ XYZ according to the lengths of the sides? 5 a) A box containing 48 bars of soap costs P.T 3120. Find the cost of each bar of soap. b) Find the result in each of the following : i) 12546 + 32154698
ii) 452 × 231
Exam Style Paper X Answer the following questions: 1 Choose the correct answer: i) The L.C.M of 8 and 12 is a) 12 b) 24
c) 8
ii) The zero multiple of 10 is a) 0 b) 10
c) 20
iii)
The acute angle is an angle whose measure is less than and greater than a) 0o, 90o b) 90o, 0o c) 180o, 90o
iv) 546987002 a)
c) =
v)
The length of a rectangle whose perimetre 30cm and its width 5cm is a) 15cm b) 10cm c) 5cm
2 Put the suitable sign ( √ ) or ( X ): i)
The unit million digit has 6-zeros.
( )
ii)
Any triangle has at least two right angles.
( )
iii)
The units of area are square units.
( )
iv)
The first three multiple of 2 are 2, 4, 6
( )
v)
(524524 + 100000)
( )
421300 = 203224
3 Complete: i)
The smallest 4-digit number divisible by 3 is
ii)
The area of a square whose side-length 7cm is
iii)
136 m2 =
iv)
5432845 >
dm2 > 5432847
v) If a number is divisible only by itself and one. Then it s called 4 a) Draw ∠ ABC whose measure 150o b) Amr paid L.E 10000 for 64543221 kilos of bananas and L.E 20000 for 56421307 of watermelons. Calculate the total kilos that he bought. 5 a) Write five 3-digit numbers are divisible by 5 b) Calculate the length of a rectangle, then its area when its perimetre = 50cm and its width = 10cm.
Mathematical Terms A ﻋﻣﻠﯾﺔ اﻟﺟﻣﻊ ﻗﯾﺎس اﻟزاوﯾﺔ ﻣﺳﺎﺣﺔ ﺗﺻﺎﻋدﯾﺎ )ﻣن اﻟﺻﻐﯾر (اﻟﻰ اﻟﻛﺑﯾر ﺗرﺗﯾب
addition angle measure area ascending order arrangement B
ﻋرض اﻟﺷﻛل ﻗﺎع/أﺳﻔل اﻻﻋﻣدة اﻟﺑﯾﺎﻧﯾﺔ رﺳم ﺧط ﻣﻧﻛﺳر ﺑﺎﺳﺗﺧدام اﻟﻣﺳطرة
breadth bottom bar chart broken line graph C
ﻗﺎرن ﺣول اﻟﻰ اﺣﺳب
comparing convert calculate common denominator
اﻟﻣﻘﺎم اﻟﻣﺷﺗرك
E
quadrillion 10 division operation determine drawing divisible by divisibility divisor dividend quotient division operation dimensions data representation decimal
دﯾﺷﻠﯾون ﻋﻣﻠﯾﺔ اﻟﻘﺳﻣﺔ ﺣدد رﺳم ﯾﻘﺑل اﻟﻘﺳﻣﺔ ﻋﻠﻰ ﻗﺎﺑﻠﯾﺔ اﻟﻘﺳﻣﺔ ﻗﺎﺳم ﻣﻘﺳوم ﻧﺎﺗﺞ اﻟﻘﺳﻣﺔ ﻋﻣﻠﯾﺔ اﻟﻘﺳﻣﺔ اﺑﻌﺎد )اﻟﺷﻛل او (اﻟﻣﺟﺳم ﺗﻣﺛﯾل اﻟﺑﯾﺎﻧﺎت ﻛﺳر ﻋﺷري
ﺗﺳﺎوي
Equality F
ﻋﺎﻣل اﻟﻣﺿﺎﻋف اﻻول ﻛﺳر
factor first multiple fraction H
اﻟﻌﺎﻣل اﻟﻣﺷﺗرك اﻷﻋﻠﻰ
highest common factor (H.C.F) hundred-million digit horizontally
رﻗم ﻣﺋﺎت اﻟﻣﻼﯾﯾن اﻓﻘﯾﺎ I ﻏﯾر ﻣﺣدد اﻟزاوﯾﺔ اﻟﻣﺣﺻورة ﻛﺳر ﻏﯾر ﻓﻌﻠﻲ
infinite included angle improper fraction
D 15
ﻓﻲ ﺻورة اﻟﻛﺳر اﻟﻌﺷري
decimal form
L اﻟﻣﺿﺎﻋف اﻟﻣﺷﺗرك اﻷدﻧﻰ اﻟﻘﺳﻣﺔ اﻟﻣطوﻟﺔ رﺳم ﺧط ﺑﺎﺳﺗﺧدام اﻟﯾد طول
lowest common multiple long division line graph length M
ﻣﻠﯾون ﻣﻠﯾﺎر
million milliard multiplication operation multiple measure of
ﻋﻣﻠﯾﺔ اﻟﺿرب ﻣﺿﺎﻋف ﻗﯾﺎس ال ﻋدد وﻛﺳر )ﻋﺷري (او اﻋﺗﯾﺎدي
mixed number N
اﻷﻋداد اﻟﻌﺷرﯾﺔ
numeral decimals
Z اﻟﻘﯾﻣﺔ اﻟﻣﻛﺎﻧﯾﺔ ﻟﻠرﻗم اﻟﻌدد اﻻوﻟﻲ ﻣﺣﯾط ﻛﺳر ﻓﻌﻠﻲ
place value prime number perimeter proper fraction R
ﺑﺎﻗﻲ اﺧﺗزل او ﺑﺳط
remainder reduce S
طول اﻟﺿﻠﻊ ﺑﺳط اﺑﺳط ﺻورة ﻋﻣﻠﯾﺔ اﻟطرح
side length simplify simplest form subtraction T
ﺗرﯾﻠﯾون ﻋدد ﻣﻛون ﻣن ﺛﻼث ارﻗﺎم ﺛﻠﺛﯾن ﺛﻼث اﺧﻣﺎس اﻋﻠﻰ/ ﻗﻣﺔ ﺛﻼﺛﺔ ﻣن اﻷﻟف ﺗﻛﻠﻔﺔ ﻛﻠﯾﺔ
trillion 3-digit number 2- third 3-fifth top 3thousandth total cost U
اﺣﺎد اﻟﻣﻼﯾﯾن
unit-million digit V
راﺳﯾﺎ
vertically W
ﻋرش اﻟﺷﻛل
width
P zero multiple
اﻟﻣﺿﺎﻋف اﻟﺻﻔري
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