Page 1
1.57 = 10
14 = 15
21 = 20
28 = 25
352.
193
= 386
= 579
= 7612
= 9515
3.2
13 = 4
26 = 6
39 = 8
52 = 10
654.
1713
= 3426
= 5139
= 6852
= 8565
1.100200
= 100 2200 2
÷÷ = 100 4
200 4÷÷ = 100 5
200 5÷÷ = 100 10
200 10÷÷
= 50100
= 2550
= 2040
= 1020
2.400240
= 400 2240 2
÷÷ = 400 4
240 4÷÷ = 400 8
240 8÷÷ = 400 10
240 10÷÷
= 200120
= 10060
= 5030
= 4024
3.36252
= 36 2252 2
÷÷ = 36 3
252 3÷÷ = 36 4
252 4÷÷ = 36 6
252 6÷÷
= 18126
= 1284
= 963
= 642
1.1836
= 18 1836 18
÷÷ = 1
22.
2149
= 21 749 7
÷÷ = 3
7
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 1
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 2
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 3
Àπ૬°“√‡√’¬π√Ÿâ∑’Ë 1
‡»… à«π·≈–∑»π‘¬¡
‡©≈¬
Page 2
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 22
3.88
121 = 88 11
121 11÷÷ = 8
114.
9135
= 91 735 7
÷÷ = 13
5
5.153187
=
153 17187 17
�
� = 9
11
1. (1)14
= 1 24 2
×× = 1 3
4 3×× (2)
35
= 3 25 2
×× = 3 3
5 3××
= 28
= 312
= 610
= 915
(3)47
= 4 27 2
×× = 4 3
7 3×× (4)
59
= 5 29 2
×× = 5 3
9 3××
= 814
= 1221
= 1018
= 1527
(5)116
= 11 26 2
×× = 11 3
6 3×× (6)
139
= 13 29 2
×× = 13 3
9 3××
= 2212
= 3318
= 2618
= 3927
2. (1)1218
=
12 218 2
�
�=
12 318 3
�
�(2)
8496
= 84 296 2
÷÷ = 84 4
96 4÷÷
= 69
= 46
= 4248
= 2124
(3)7030
=
70 530 5
�
�=
70 1030 10
�
�(4)
3248
=
32 248 2
�
�=
32 448 4
�
�
= 146
= 73
= 1624
= 812
(5)426768
=
426 2768 2
�
�=
426 3768 3
�
�(6)
12643216
= 1264 23216 2
÷÷ = 1264 4
3216 4÷÷
= 213384
= 124256
= 6321608
= 316804
3. (1) 27 (2) 65 (3) 16 (4) 8 (5) 72
(6) 9 (7) 60 (8) 20 (9) 84 (10) 77
4. (1)98
(2)45
(3)117
(4)67
(5)34
(6)1120
(7)1315
(8)1323
(9)1923
(10)53
·∫∫Ωñ°À—¥ 1
Page 3
3·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1.9
11 ·≈– 5
82. �
611
·≈– � 813
«‘∏’∑”9
11= 9 8
11 8×× = 72
88«‘∏’∑” �
611
= �6 1311 13
×× = �
78143
58
= 5 118 11
×× = 55
88�
813
= �8 11
13 11×× = �
88143
‡π◊ËÕß®“° 7288
� 5588
‡π◊ËÕß®“° �78
143� �
88143
¥—ßπ—Èπ 911
�58 μÕ∫ ¥—ßπ—Èπ 6
11�
813
μÕ∫
3.415
·≈– 364
4. � 81116
·≈– � 837
«‘∏’∑”415
= 41 45 4
×× = 164
20«‘∏’∑” �
1116
= �11 716 7
×× = �
77112
364
= 36 54 5
×× = 180
20�
37
= �3 167 16
×× = �
48112
‡π◊ËÕß®“° 16420
�18020
‡π◊ËÕß®“° �77
112� �
48112
¥—ßπ—Èπ 415
�364
μÕ∫ ¥—ßπ—Èπ �1116
� �37
π—Ëπ§◊Õ � 81116
� � 837
μÕ∫
·∫∫Ωñ°À—¥ 2
1. (1)4
11, 6
17
«‘∏’∑” ‡π◊ËÕß®“° 4 × 17 = 68
·≈– 6 × 11 = 66
¥—ßπ—Èπ 4 × 17 � 6 × 11
®–‰¥â«à“ 411
�6
17μÕ∫
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 4
Page 4
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 24
(2)9
13, 25
29
«‘∏’∑” ‡π◊ËÕß®“° 9 × 29 = 261
·≈– 25 × 13 = 325
¥—ßπ—Èπ 9 × 29 � 25 × 13
®–‰¥â«à“ 913
�2529
μÕ∫
(3)1517
, 285323
«‘∏’∑” ‡π◊ËÕß®“° 15 × 323 = 4,845
·≈– 17 × 285 = 4,845
¥—ßπ—Èπ 15 × 323 = 17 × 285
®–‰¥â«à“ 1517
= 285323
μÕ∫
(4) �79
, � 913
«‘∏’∑” ‡π◊ËÕß®“° (�7) × 13 = �91
·≈– (�9) × 9 = �81
¥—ßπ—Èπ (�7) × 13 � (�9) × 9
®–‰¥â«à“ �79
� �9
13μÕ∫
(5) �4251
, � 5371
«‘∏’∑” ‡π◊ËÕß®“° (�42) × 71 = 2,982
·≈– (�53) × 51 = �2,703
¥—ßπ—Èπ (�42) × 71 � (�53) ×51
®–‰¥â«à“ �4251
� �5371
μÕ∫
(6) � 358
, � 3112133
«‘∏’∑” æ‘®“√≥“ � 58, � 112
133
‡π◊ËÕß®“° (�5) × 133 = �665
·≈– (�112) × 8 = �896
¥—ßπ—Èπ (�5) × 133 � (�112) × 8
Page 5
5·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
®–‰¥â«à“ �58
� �112133
®–‰¥â«à“ � 358
� � 3112133
μÕ∫
2. (1) = (2) � (3) � (4) � (5) � (6) =
(7) = (8) = (9) � (10) � (11) = (12) �
3. (1)35
, 56
, 47
, 78
, 69
«‘∏’∑” ‡ª≈’Ë¬π‡»… à«π¢â“ßμâπ„Àâ¡’μ—« à«π‡∑à“°—π ‚¥¬À“ §.√.π. ¢Õßμ—« à«π ·μà‡π◊ËÕß®“°69 = 2
3 ®÷ßÀ“ §.√.π. ¢Õß 5, 6, 7, 8 ·≈– 3
3 ) 3, 5, 6, 7, 8
2 ) 1, 5, 2, 7, 8
1, 5, 1, 7, 4
§.√.π. ¢Õß 3, 5, 6, 7 ·≈– 8 §◊Õ 3 × 2 × 5 × 7 × 4 = 840
35
= 3 3 2 7 45 3 2 7 4
× × × ×× × × × = 504
840
56
= 5 5 7 46 5 7 4
× × ×× × × = 700
840
47
= 4 3 2 5 47 3 2 5 4
× × × ×× × × × = 480
840
78
= 7 3 5 78 3 5 7
× × ×× × × = 735
840
69
= 23 = 2 2 5 7 4
3 2 5 7 4× × × ×× × × × = 560
840
¥—ßπ—Èπ ‡√’¬ß≈”¥—∫®”π«π®“°πâÕ¬‰ª¡“°‰¥â¥—ßπ’È 47
, 35
, 69
, 56
, 78
μÕ∫
(2) �79
, � 68
, � 1112
, � 2528
, � 23
«‘∏’∑” ‡ª≈’Ë¬π‡»… à«π¢â“ßμâπ„Àâ¡’μ—« à«π‡∑à“°—π ‚¥¬À“ §.√.π. ¢Õßμ—« à«π ·μà‡π◊ËÕß®“°
�68 = � 3
4 ®÷ßÀ“ §.√.π. ¢Õß 9, 4, 12, 28 ·≈– 3
3 ) 9, 4, 12, 28, 3
4 ) 3, 4, 14, 28, 1
3, 1, 01, 07, 1
§.√.π. ¢Õß 9, 4, 12, 28 ·≈– 3 §◊Õ 3 × 4 × 3 × 7 = 252
�79
= �7 4 79 4 7
× ×× × = �
196252
�68 = � 3
4= �
3 9 74 9 7
× ×× × = �
189252
Page 6
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 26
�1112
= �11 3 712 3 7
× ×× × = �
231252
�2528
= �25 928 9
×× = �
225252
�23
= �2 3 4 73 3 4 7
× × ×× × × = �
168252
¥—ßπ—Èπ ‡√’¬ß®”π«π®“°πâÕ¬‰ª¡“°‰¥â¥—ßπ’È � 1112
, � 2528
, � 79
, � 68
, � 23
μÕ∫
(3)32
, 76
, 109
, 87
, 1211
«‘∏’∑” ‡ª≈’Ë¬π‡»… à«π¢â“ßμâπ„Àâ¡’μ—« à«π‡∑à“°—π ‚¥¬À“ §.√.π. ¢Õßμ—« à«π
2 ) 2, 6, 9, 7, 11
3 ) 1, 3, 9, 7, 11
1, 1, 3, 7, 11
§.√.π. ¢Õß 2, 6, 9, 7 ·≈– 11 §◊Õ 2 × 3 × 3 × 7 × 11 = 1,386
32
=
3 3 3 7 112 3 3 7 11 � � � �
� � � �= 2079
1386
76
=
7 3 7 116 3 7 11 � � �
� � �= 1617
1386
109
=
10 2 7 119 2 7 11
� � �
� � �= 1540
1386
87
=
8 2 3 3 117 2 3 3 11
� � � �
� � � �= 1584
1386
1211
=
12 2 3 3 711 2 3 3 7
� � � �
� � � �= 1512
1386
¥—ßπ—Èπ ‡√’¬ß®”π«π®“°πâÕ¬‰ª¡“°‰¥â¥—ßπ’È 1211
, 109
, 87
, 76
, 32
μÕ∫
(4) �138
, � 1511
, � 1915
, � 2317
, � 187
«‘∏’∑” ‡ª≈’Ë¬π‡»… à«π¢â“ßμâπ„Àâ¡’μ—« à«π‡∑à“°—π ‚¥¬À“ §.√.π. ¢Õßμ—« à«π
§.√.π. ¢Õß 8, 11, 15, 17 ·≈– 7 §◊Õ 157,080
�138
= �
13 11 15 17 718 11 15 17 7
� � � �
� � � �= �
255255157080
�1511
= �
15 8 15 17 711 8 15 17 7
� � � �
� � � �= �
214200157080
�1915
= �
19 8 11 17 715 8 11 17 7
� � � �
� � � �= �
198968157080
�2317
= �
23 8 11 15 717 8 11 15 7
� � � �
� � � �= �
212520157080
�187
= �
18 8 11 15 177 8 11 15 17
� � � �
� � � �= �
403920157080
¥—ßπ—Èπ ‡√’¬ß®”π«π®“°πâÕ¬‰ª¡“°‰¥â¥—ßπ’È � 187
, � 138
, � 1511
, � 2317
, � 1915
μÕ∫
Page 7
7·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
·∫∫Ωñ°À—¥ 3
1. (1)45
56
78
+ +
«‘∏’∑” À“ §.√.π. ¢Õß 5, 6 ·≈– 8 ‰¥â 120
45
56
78
+ + = 4 245 24
5 206 20
7 158 15
×× + ×
× + ××
= 96120
100120
105120
+ +
= 301120
= 261
120μÕ∫
(2)1315
49
34
−⎛⎝
⎞⎠ +
«‘∏’∑” À“ §.√.π. ¢Õß 15, 9 ·≈– 4 ‰¥â 180
1315
49
34
−⎛⎝
⎞⎠ + = 13 12
15 124 209 20
3 454 45
×× − ×
×⎛⎝
⎞⎠ + ×
×
= 156180
80180
135180
−⎛⎝
⎞⎠ +
= 211180
= 131
180μÕ∫
(3)23
56
89
+⎛⎝
⎞⎠ −
«‘∏’∑” À“ §.√.π. ¢Õß 3, 6, 9 ‰¥â 18
23
56
89
+⎛⎝
⎞⎠ − = 2 6
3 65 36 3
8 29 2
×× + ×
×⎛⎝
⎞⎠ − ×
×
= 1218
1518
1618
+⎛⎝
⎞⎠ −
= 1118
μÕ∫
Page 8
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 28
(4) «‘∏’∑” 1035
656
+ = ( )10 635
56
+ + +⎛⎝
⎞⎠
= 163 65 6
5 56 5
+ ×× + ×
×⎛⎝
⎞⎠
= 161830
2530
+ +⎛⎝
⎞⎠
= 164330
+
= 171330
μÕ∫
(5) «‘∏’∑” 10 357
− = 10267
−
= 707
267
−
= 447
= 627
μÕ∫
(6) «‘∏’∑” 1135
67
15− = 58
59715
−
= 58 35 3
9715
×× −
= 17415
9715
−
= 7715
= 52
15μÕ∫
(7) «‘∏’∑” 1034
678
− = 434
558
−
= 43 24 2
558
×× −
= 868
558
−
= 318
= 378
μÕ∫
(8) «‘∏’∑” 1535
34
− = 785
34
−
= 78 45 4
3 54 5
×× − ×
×
= 31220
1520
−
= 29720
= 141720
μÕ∫
Page 9
9·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(9) «‘∏’∑” 121516
1034
538
−⎛⎝
⎞⎠ + = 207
16434
438
−⎛⎝
⎞⎠ +
= 20716
17216
8616
−⎛⎝
⎞⎠ +
= 12116
= 79
16μÕ∫
(10) «‘∏’∑” 212
314
5+⎛⎝
⎞⎠ − = 5
2134
5+⎛⎝
⎞⎠ −
= 104
134
204
+⎛⎝
⎞⎠ −
= 34
μÕ∫
(11) «‘∏’∑” 1234
138
512
− +⎛⎝
⎞⎠ = 51
4118
512
− +⎛⎝
⎞⎠
= 51 64 6
11 38 3
5 212 2
×× − ×
× + ××
⎛⎝
⎞⎠
= 30624
3324
1024
− +⎛⎝
⎞⎠
= 30624
4324
−
= 26324
= 102324
μÕ∫
(12) «‘∏’∑”5
1278
14
312
+ − +⎛⎝
⎞⎠ = 5 2
12 27 38 3
14
14
×× + ×
×⎛⎝
⎞⎠ − +⎛
⎝⎞⎠
= 1024
2124
12
+⎛⎝
⎞⎠ −
= 3124
1224
−
= 1924
μÕ∫
Page 10
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 210
(13) «‘∏’∑”
� � �58
712
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
= �58
�7
12
= �
5 38 3 �
��
7 212 2
�
�
= �1524
�1424
= �2924
= �15
24μÕ∫
(14) «‘∏’∑”
� �45
318
⎛⎝⎜
⎞⎠⎟ = �
4 185 18
3 518 5
�
��
�
�
= �
7290
1590
�
= �5790
= �1930
μÕ∫
(15) «‘∏’∑”
1221
1512
73
� � �⎛⎝⎜
⎞⎠⎟
=
12 421 4
15 712 7
7 283 28
�
�� �
�
��
�
�
⎛⎝⎜
⎞⎠⎟
=
4884
10584
19684
� � �⎛⎝⎜
⎞⎠⎟
= �25384
= � 31
84μÕ∫
(16) «‘∏’∑”
� � �821
15
38
⎛⎝⎜
⎞⎠⎟ =
�
�
��
�
��
�
�
8 4021 40
1 1685 168
3 1058 105
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
= �320840
�168840
�315840
= �173840
μÕ∫
(17) «‘∏’∑”
13
115
238
� �⎛⎝⎜
⎞⎠⎟
=
13
65
198
� �⎛⎝⎜
⎞⎠⎟
=
1 403 40
6 245 24
19 158 15
�
��
�
��
�
�
⎛⎝⎜
⎞⎠⎟
=
40120
144120
285120
� �⎛⎝⎜
⎞⎠⎟
=
40120
429120
�
= �389120
= � 329
120μÕ∫
Page 11
11·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(18) «‘∏’∑”
� � �35
89
713
⎛⎝⎜
⎞⎠⎟
=
��
��
�
��
�
�
3 1175 117
8 659 65
7 4513 45
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
= �351585
�520585
�315585
= �1186585
= � 216585
μÕ∫
2. μÕπ‡™â“𓬥”‡π‘π‡¥‘π∑“߉¥â 534
°‘‚≈‡¡μ√
μÕπ∫à“¬‡¥‘π∑“߉¥â 456
°‘‚≈‡¡μ√
∂÷ßμ”∫≈‡¢“™—¬ π
(1) «‘∏’∑” μ”∫≈‚§°¡à«ß·≈–μ”∫≈‡¢“™—¬ πÕ¬ŸàÀà“ß°—π
534
456
+ = 91912
+ °‘‚≈‡¡μ√
= 107
12°‘‚≈‡¡μ√
¥—ßπ—Èπ μ”∫≈‚§°¡à«ß·≈–μ”∫≈‡¢“™—¬ πÕ¬ŸàÀà“ß°—π 107
12 °‘‚≈‡¡μ√ μÕ∫
(2) «‘∏’∑” ¢“°≈—∫‡¥‘π∑“߉¥â√–¬–∑“ß 325 °‘‚≈‡¡μ√ ®÷ßπ—Ëßæ—°
®ÿ¥∑’Ëπ—Ëßæ—°Àà“ß®“°μ”∫≈‚§°¡à«ß 107
123
25
− = 71160
°‘‚≈‡¡μ√
¥—ßπ—Èπ ®ÿ¥∑’Ëπ—Ëßæ—°Àà“ß®“°μ”∫≈‚§°¡à«ß 71160
°‘‚≈‡¡μ√ μÕ∫
3. «‘∏’∑” 𓬷¢Áß·√ßμ—°πÈ”„πμÕπ‡™â“‰¥â 313
ªïö∫
μÕπ∫à“¬μ—°πÈ”‰¥âÕ’° 434
ªïö∫
√«¡μ—°πÈ”‰¥â 313
434
+ = 71312
+ = 81
12 ªïö∫
μÕπ§Ë”„™âπÈ”∑’Ëμ—°‰«â‰ª 235
ªïö∫
¬—߇À≈◊ÕπȔլŸà 81
122
35
− = 52960
ªïö∫
¥—ßπ—Èπ ¬—ß‡À≈◊ÕπȔլŸà 52960
ªïö∫ μÕ∫
Page 12
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 212
4. «‘∏’∑” ¡’πÈ”¡—πÕ¬Ÿà 1634
∂—ß
„™âπÈ”¡—π‰ª 525
∂—ß
¬—߇À≈◊ÕπÈ”¡—π 1634
525
− = 117
20∂—ß
π”πÈ”¡—π¡“‡æ‘Ë¡Õ’° 956
∂—ß
¥—ßπ—Èπ ®–¡’πÈ”¡—π‡ªìπ 117
209
56
+ = 211160
∂—ß μÕ∫
5. «‘∏’∑” ©“ߢ⓫¡’¢â“«Õ¬Ÿà 65 ∂—ß
«—π·√°μ«ß¢â“«ÕÕ°‰ª 1423
∂—ß
«—π∑’Ë Õßμ«ß¢â“«ÕÕ°‰ªÕ’° 1612
∂—ß
¥—ßπ—Èπ ®–‡À≈◊բ⓫ 65 1423
1612
−⎛⎝⎜
⎞⎠⎟− = 33
56
∂—ß
«—π∑’Ë “¡π”¢â“«¡“„ à≈ß©“ß 2134
∂—ß
¥—ßπ—Èπ ®–¡’¢â“«„π©“ß 3356
2134
+ = 557
12∂—ß μÕ∫
6. ‡π◊ËÕß®“°πÈ”Àπ—°°√–‡ªÜ“√«¡°—∫ ‘ËߢÕß A, B, C, D, E ·≈– F ¡’πÈ”Àπ—°√«¡
312
47
106
34
835
345
525
315
+ + + + + + = 351920
°‘‚≈°√—¡
μâÕß°“√πÈ”Àπ—°√«¡°√–‡ªÜ“Àπ—°¡“°∑’Ë ÿ¥ ·μà‰¡à‡°‘ππÈ”Àπ—°∑’Ë°”Àπ¥
(1) 30 °‘‚≈°√—¡
«‘∏’∑” ®–μâÕ߇≈◊Õ° ‘ËߢÕß A, C, D, E ·≈– F ´÷Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ
351920
634
− = 2915 °‘‚≈°√—¡
¥—ßπ—Èπ ‡≈◊Õ° ‘ËߢÕß A, C, D, E ·≈– F μÕ∫
(2) 25 °‘‚≈°√—¡
«‘∏’∑” ®–μâÕ߇≈◊Õ° ‘ËߢÕß C, D, E ·≈– F ´÷Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ
351920
47
106
34
− +⎛⎝
⎞⎠ = 24
12 °‘‚≈°√—¡
¥—ßπ—Èπ ‡≈◊Õ° ‘ËߢÕß C, D, E ·≈– F μÕ∫
Page 13
13·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(3) 20 °‘‚≈°√—¡
«‘∏’∑” ®–μâÕ߇≈◊Õ° ‘ËߢÕß A, C ·≈– F ÷́Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ
312
47
208
35
315
+ + + = 191320
°‘‚≈°√—¡
¥—ßπ—Èπ ‡≈◊Õ° ‘ËߢÕß A, C ·≈– F μÕ∫
(4) 15 °‘‚≈°√—¡
«‘∏’∑” ®–μâÕ߇≈◊Õ° ‘ËߢÕß A ·≈– B ÷́Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ
312
47
106
34
+ + = 141920
°‘‚≈°√—¡
¥—ßπ—Èπ ‡≈◊Õ° ‘ËߢÕß A ·≈– B μÕ∫
1.25
12
× = 2 15 2
×× 2.
45
27
× = 4 25 7
××
= 15
= 835
3.
��
35
29
⎛⎝
⎞⎠ =
( )
� �
�
3 25 9
4.
23
47
� �⎛⎝⎜
⎞⎠⎟
=
2 43 7
( )
� �
�
=
� 215
=
� 821
5.89
711
× = 8 79 11
×× 6.
13
35
× = 1 33 5
××
= 5699
= 15
7.
35
79
��⎛
⎝⎞⎠ =
3 75 9
( )
� �
�8.
��
�
1112
34
⎛⎝
⎞⎠
⎛⎝
⎞⎠ =
� � �
�
( )
11 312 4
=
� 715
= 1116
9.
��
�
25
45
⎛⎝
⎞⎠
⎛⎝
⎞⎠ =
� � �
�
2 45 5
10.
��
56
58
⎛⎝
⎞⎠ =
( )
� �
�
5 56 8
= 825
=
� 2548
11.25
45
× = 2 45 5
×× 12.
35
49
× = 3 45 9
××
= 825
= 415
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 5
1
1
4
1
1
3
1
3
1
3
1
1
Page 14
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 214
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 6
1
1
6
1
1. 53
11× = 5
13
11× 2.
8
58
��⎛
⎝⎞⎠ =
81
58
��⎛
⎝⎞⎠
= 1511
= �5
= 14
11
3.23
18× = 23
181
× 4.12
9× = 12
91
×
= 12 = 92
= 412
5. ( )
� �
�10
35
⎛⎝
⎞⎠ =
��
�
101
35
⎛⎝
⎞⎠
⎛⎝
⎞⎠ 6.
9
56
��⎛
⎝⎞⎠ =
91
56
��⎛
⎝⎞⎠
= �6 = �
152
= � 7
12
7.
�� �
( )
1518
9⎛⎝
⎞⎠ =
��
�
1518
91
⎛⎝
⎞⎠
⎛⎝
⎞⎠ 8.
��
23
15⎛⎝
⎞⎠ =
��
23
151
⎛⎝
⎞⎠
= 152
= �10
= 712
9. ( ) � �27
79
=
��
271
79
⎛⎝
⎞⎠ 10.
58
13× = 58
131
×
= �21 = 658
= 818
11.
79
12 ( )� � =
79
121
��⎛
⎝⎞⎠ 12.
56
14× = 56
141
×
= �283
= 353
= � 913
= 1123
2
1
2
1
3
1
4
3
3
2
5
1
7
3
Page 15
15·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1.23
412
× = 23
92
× 2.
� � 15
623
⎛⎝
⎞⎠ =
� �
15
203
⎛⎝
⎞⎠
= 93
= �43
= 3
3.
910
1012
� �⎛⎝
⎞⎠ =
910
212
��⎛
⎝⎞⎠ 4.
� � � 2
14
23
⎛⎝
⎞⎠
⎛⎝
⎞⎠ =
� � �
94
23
⎛⎝
⎞⎠
⎛⎝
⎞⎠
=
� 18920
=
( ) ( )� � �3 24
= � 99
20= 3
2
5.34
213
× = 34
73
× 6.45
416
× = 45
256
×
= 74
= 4 56×
= �134
= 103
= 313
7.
� � 725
34
⎛⎝
⎞⎠ =
��
375
34
⎛⎝
⎞⎠ 8.
13
312
× = 13
72
×
=
� 11120
= 76
= � 51120
= 116
9.
� � 12
125
⎛⎝
⎞⎠ =
� �
12
72
⎛⎝
⎞⎠ 10.
34
213
� �⎛⎝
⎞⎠ =
34
73
� �⎛⎝
⎞⎠
= �74
= �74
= �134
= �134
11.45
425
× = 45
225
× 12.
�� �
67
213
⎛⎝
⎞⎠
⎛⎝
⎞⎠ =
� � �
67
73
⎛⎝
⎞⎠
⎛⎝
⎞⎠
= 8825
= 2
= 31325
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 7
1
13
1
1
1
4
1
3
1 1
2
5
1 2
3
1
1
2
11
1
Page 16
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 216
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 8
15
2
2
1
1
1
1
3
1
1
1. 314
427
× = 134
307
× 2. 3 234
× = 31
114
×
= 19514
= 334
= 131314
= 814
3. 5
310
114
� �⎛⎝⎜
⎞⎠⎟
=
5310
54
� � ⎛⎝⎜
⎞⎠⎟
4. 259
112
× = 239
32
×
= �538
= 236
= � 658
= 356
5. 213
612
× = 73
132
× 6.
�312
2⎛⎝⎜
⎞⎠⎟
× = �72
21
×
= 916
= �7
= 1516
7. 112
2× = 32
21
× 8.
� �114
114
⎛⎝⎜
⎞⎠⎟
× ⎛⎝⎜
⎞⎠⎟
=
� �54
54
⎛⎝⎜
⎞⎠⎟
× ⎛⎝⎜
⎞⎠⎟
= 3 = 2516
= 19
16
9. 214
125
× = 94
75
× 10. 30 ×
�456
⎛⎝⎜
⎞⎠⎟ = 30
1×
�
296
⎛⎝⎜
⎞⎠⎟
= 6320
= �145
= 33
20
11.
�918
⎛⎝⎜
⎞⎠⎟
×
�627
⎛⎝⎜
⎞⎠⎟
= �
738
×
� 447
⎛⎝⎜
⎞⎠⎟
12. 12 812
× = 121
172
×
= 80314
= 102
= 575
14
11
2
5
1
6
1
Page 17
17·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1.116
2. �4
133. 15 4. �
117
5. �179
6.523
7. �1649
8.200
3
1.56
29
÷ = 56
92
× 2.
12
34
÷ ⎛⎝⎜
⎞⎠⎟
� =
12
43
×⎛⎝⎜
⎞⎠⎟
�
= 154
= �23
= 334
3.38
516
÷ = 38
165
× 4.23
56
÷ = 23
65
×
= 65
= 45
= 115
5.
� 54
⎛⎝⎜
⎞⎠⎟
÷ 32
= �54
23
× 6.72
32
÷ = 72
23
×
= �56
= 73
= 213
7.
� � �35
310
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
=
� � �35
103
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
8.35
310
÷ = 35
103
×
= 21
= 105
= 2 = 2
9.29
16
÷ = 29
61
× 10.
14
78
×⎛⎝⎜
⎞⎠⎟
� =
14
87
×⎛⎝⎜
⎞⎠⎟
�
= 43
= �27
= 113
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 9
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 10
2
1
3
2
2
1
2
1
1
2
1
1
1
1
1
1
2
3
2
1
2
1
2
1
Page 18
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 218
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 11
1
2
2
1
2
1
1
1
11.23
58
÷ = 23
85
× 12.
� �56
113
⎛⎝⎜
⎞⎠⎟
÷ ⎛⎝⎜
⎞⎠⎟
=
� � �56
311
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
= 1615
= 522
= 11
15
1.9
102÷ = 9
1012
× 2. 10
109
÷ ⎛⎝⎜
⎞⎠⎟
� =
101
910
× ⎛⎝⎜
⎞⎠⎟
�
= 920
= �9
3.37
8÷ = 37
18
× 4. 1515
÷ = 151
51
×
= 356
= 75
5.
� �16
2⎛⎝⎜
⎞⎠⎟÷ =
� �
16
12
⎛⎝⎜
⎞⎠⎟
÷ ⎛⎝⎜
⎞⎠⎟
6. 634
÷ = 61
43
×
= 112
= 8
7. 234
÷ = 21
43
× 8.
�43
2⎛⎝⎜
⎞⎠⎟
÷ =
� �43
12
⎛⎝⎜
⎞⎠⎟
= 83
= �23
= 223
9. 725
÷ = 71
52
× 10.
� �38
4⎛⎝⎜
⎞⎠⎟÷( ) =
� � �
38
14
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
= 352
= 332
= 1712
Page 19
19·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1. 1217
427
÷ = 857
307
÷ 2. 435
23
÷⎛⎝⎜
⎞⎠⎟
� =
235
32
×⎛⎝⎜
⎞⎠⎟
�
= 857
730
× = �6910
= 8530
= � 69
10
= 176
= 256
3. 156
45
÷ = 116
54
× 4. 514
310
÷ = 214
103
×
= 5524
= 7 104
×
= 27
24= 35
2
= 1712
5.
�325
315
⎛⎝⎜
⎞⎠⎟÷ = �
175
165
÷ 6.
� �1012
134
⎛⎝⎜
⎞⎠⎟÷⎛
⎝⎜⎞⎠⎟
=
� �212
74
⎛⎝⎜
⎞⎠⎟÷⎛
⎝⎜⎞⎠⎟
= �175
516
× = �
212
47
×⎛⎝⎜
⎞⎠⎟
�
= �1716
= 3 42×
= �11
16= 6
·∫∫Ωñ°À—¥ 4
1. (1)4
112233
× = 4 2211 33
×× (2)
518
0× = 0
= 833
(3)
4213
39168
×⎛⎝⎜
⎞⎠⎟
� = �42 3913 168
×× (4)
4517
68180
× = 45 6817 180
××
= �42 3168
× = 6868
= �34
= 1
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 12
1
1 17
6
7
1
5
2
3
2
1
11
1
2
1
3
11
4
1
1
1
4
Page 20
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 220
(5) (�15) ×13
45
⎛⎝⎜
⎞⎠⎟ =
� �
151
1345
⎛⎝⎜
⎞⎠⎟
(6)1213
91× = 1213
911
× 5
= �133
= 84
= � 413
(7)
� �7
1524
⎛⎝⎜
⎞⎠⎟×( ) =
� �
715
241
⎛⎝⎜
⎞⎠⎟×⎛
⎝⎜⎞⎠⎟
(8)3
148
25
× = 314
425
×
= 565
= 95
= 1115
= 145
(9) 1423
711
× =
443
711
� (10)49
81
10× = 4
98110
×
= 283
= 4 910×
= 913
= 185
= 335
(11)
� �413
515
⎛⎝⎜
⎞⎠⎟×⎛
⎝⎜⎞⎠⎟
=
� �133
265
⎛⎝⎜
⎞⎠⎟×⎛
⎝⎜⎞⎠⎟
(12) 827
25
29× = 58
76329
×
= 33815
= 2 637
×
= 228
15= 18
2. (1)175
(2) �1615
(3)14
(4)16
(5) �5
44(6)
329
(7)211
(8) �629
3. (1)56
1517
÷ = 56
1715
× (2)
89
481
÷ ⎛⎝⎜
⎞⎠⎟
� =
89
814
× ⎛⎝⎜
⎞⎠⎟
�
= 1718
= �2 81
9×
= �18
(3)76
5636
÷ = 76
3656
× (4) 401013
÷ = 401
1310
×
= 7 656× = 52
= 34
(5) ( ) � �48
1217
÷ ⎛⎝⎜
⎞⎠⎟
=
� �481
1712
⎛⎝⎜
⎞⎠⎟
× ⎛⎝⎜
⎞⎠⎟
(6) 71213
÷ = 71
1312
×
= 68 = 9112
= 77
12
1
3
7
1
8
5
3
1
4
1
2
1
4
1
4
1
2
13
1
9
1
9
1
6
1
8
1
5
2
9
1
Page 21
21·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(7)
�73
149
⎛⎝⎜
⎞⎠⎟÷ =
�
73
914
⎛⎝⎜
⎞⎠⎟× (8)
34
81÷ = 34
181
×
= �96
= 1108
= �32
= �112
(9)1213
336
÷ = 1213
363
× (10) 523
34
÷ = 173
43
×
= 14413
= 689
= 111
13= 7
59
(11)
� �435
613
⎛⎝⎜
⎞⎠⎟÷⎛
⎝⎜⎞⎠⎟
=
� �235
193
⎛⎝⎜
⎞⎠⎟÷⎛
⎝⎜⎞⎠⎟
(12) 18 534
÷ = 18234
÷
=
� � �235
319
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
= 181
423
×
= 6995
= 7223
= 3323
(13) 4
14
214
÷⎛⎝⎜
⎞⎠⎟
� =
174
94
÷⎛⎝⎜
⎞⎠⎟
� (14) 456
8÷ = 296
18
×
=
174
49
� �⎛⎝⎜
⎞⎠⎟
= 2948
= �189
(15) 1223
59
÷ = 383
95
×
= 1145
= 2245
·∫∫Ωñ°À—¥ 5
1. «‘∏’∑” 324
123
235
+ +⎛⎝
⎞⎠ = 3
24
123
235
+ + + +⎛⎝
⎞⎠ 2. «‘∏’∑” 3
24
456
212
+⎛⎝
⎞⎠ × = 7
12
56
52
+ +⎛⎝
⎞⎠ ×
= 324
31915
+ +⎛⎝
⎞⎠ = 8
13
52
+⎛⎝
⎞⎠ ×
= 312
44
15+⎛
⎝⎞⎠ + +⎛
⎝⎞⎠ = 25
352
×
2
1
27
1
4
1
1
1
3
1
3
2
Page 22
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 222
= 712
415
+ +⎛⎝
⎞⎠ = 125
6
= 72330
μÕ∫ = 2056
μÕ∫
3. «‘∏’∑” 856
712
74
−⎛⎝
⎞⎠ ÷ 4. «‘∏’∑” 45
34
1812
54−⎛⎝
⎞⎠ ÷
= 856
712
74
+⎛⎝
⎞⎠ − +⎛
⎝⎞⎠
⎡⎣⎢
⎤⎦⎥
÷ = 4534
1812
54+⎛⎝
⎞⎠ − +⎛
⎝⎞⎠
⎡⎣⎢
⎤⎦⎥
÷
= ( )8 756
12
74
− + −⎛⎝
⎞⎠
⎡⎣⎢
⎤⎦⎥
÷ = ( )45 1834
12
54− + −⎛⎝
⎞⎠
⎡⎣⎢
⎤⎦⎥
÷
= 43
47
× = 2714
54÷
= 1621
μÕ∫ = 1094
154
×
= 109216
μÕ∫
5. «‘∏’∑” 1025
356
213
− −⎛⎝
⎞⎠ 6. «‘∏’∑” 4
16
112
335
−⎛⎝
⎞⎠ ×
= 1025
356
213
− +⎛⎝
⎞⎠ − +⎛
⎝⎞⎠
⎡⎣⎢
⎤⎦⎥
= 376
112
185
+⎛⎝
⎞⎠ − +⎛
⎝⎞⎠
⎡⎣⎢
⎤⎦⎥
×
= 525
3 256
13
( ) − − + −⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
= ( )3 176
12
185
− + −⎛⎝
⎞⎠
⎡⎣⎢
⎤⎦⎥
×
= 525
32
− = 83
185
×
= 8910
= 485
= 89
10μÕ∫ = 9
35
μÕ∫
7. «‘∏’∑” 1012
3 312
212
+⎛⎝
⎞⎠ − ×⎛
⎝⎞⎠ 8. «‘∏’∑” 1
45
235
5 712
+⎛⎝
⎞⎠ ×⎧
⎨⎩
⎫⎬⎭
−
= 212
372
52
+⎛⎝
⎞⎠ − ×⎛
⎝⎞⎠ = 9
5135
5152
+⎛⎝
⎞⎠ ×⎧
⎨⎩
⎫⎬⎭
−
= 194
= 225
5152
×⎛⎝
⎞⎠ −
= 434
μÕ∫ = 442
152
−
= 292
= 1412
μÕ∫
6
1
Page 23
23·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
9. «‘∏’∑” 316
512
257
11
14+⎛
⎝⎞⎠ ÷ −⎛
⎝⎞⎠ 10. «‘∏’∑” 4
15
33
1012
12
54
−⎛⎝
⎞⎠ + ÷⎛
⎝⎞⎠
= 816
12
157
114
+ +⎛⎝
⎞⎠
⎡⎣⎢
⎤⎦⎥
÷ + −⎛⎝
⎞⎠
⎡⎣⎢
⎤⎦⎥
= 215
3310
252
54
−⎛⎝
⎞⎠ + ÷⎛
⎝⎞⎠
= 823
19
14÷ = 9
10252
45
+ ×⎛⎝
⎞⎠
= 263
1423
× = 910
10010
+
= 36469
= 10910
= 51969
μÕ∫ = 109
10μÕ∫
11. «‘∏’∑” 81
102
45
515
11
12÷ + ×⎛
⎝⎞⎠
⎧⎨⎩
⎫⎬⎭
12. «‘∏’∑” 823
216
1412
÷⎛⎝
⎞⎠ ÷
= 8110
145
13
1312
÷ + ×⎛⎝
⎞⎠
⎧⎨⎩
⎫⎬⎭
= 263
136
292
÷⎛⎝
⎞⎠ ÷
= 8110
14 365 36
13 536 5
÷ ×× + ×
×⎛⎝
⎞⎠ = 26
36
13229
×⎛⎝
⎞⎠ ×
= 8110
504180
65180
÷ +⎛⎝
⎞⎠ = 4
1229
×
= 8110
569180
÷ = 829
μÕ∫
= 8110
180569
×
= 1458569
= 2320569
μÕ∫
·∫∫Ωñ°À—¥ 6
1. «‘∏’∑” ·∫à߉¡â¬“« 8 ‡¡μ√ ÕÕ°‡ªìπ∑àÕπ —ÈπÊ ∑àÕπ≈– 34 ‡¡μ√
®–·∫à߉¥â 834
÷ = 81
43
× ∑àÕπ
= 1023
∑àÕπ
π—Ëπ§◊Õ ·∫à߉¥â 10 ∑àÕπ ‡À≈◊Õ‡»…¬“« 23
34
× = 12 ‡¡μ√ μÕ∫
2 2
11
Page 24
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 224
4000
1
39
5
1
1
2. «‘∏’∑” “¬‰ø¢¥Àπ÷Ë߬“« 14345 ‡¡μ√ ·∫à߇ªìπ¢¥ —ÈπÊ ¢¥≈– 8
12 ‡¡μ√
®–·∫à߉¥â 14345
812
÷ = 7195
172
÷
= 7195
217
×
= 143885
= 167885
¥—ßπ—Èπ ®–·∫àß “¬‰ø‰¥â 16 ¢¥ ·≈–‡À≈◊Õ‡»… 7885
172
× = 745 ‡¡μ√ μÕ∫
3. «‘∏’∑” √«¡ “¡«—π𓬥‘π‰∂𓉥â 178
213
13
10+ + ‰√à
= 478
13
310
+ + +⎛⎝
⎞⎠ ‰√à
= 4181120
+ ‰√à
= 561
120‰√à
¥—ßπ—Èπ √«¡ “¡«—π𓬥‘π‰∂𓉥â 561
120 ‰√à μÕ∫
4. «‘∏’∑” ∂—ß„∫Àπ÷Ëß¡’πÈ”¡—πÕ¬Ÿà 15,000 ≈‘μ√ ‡¡◊ËÕ„™âπÈ”¡—π‰ª 35 ¢Õß∂—ß ‡À≈◊ÕπÈ”¡—πÕ¬Ÿà 3,000 ≈‘μ√
¥—ßπ—Èπ „™âπÈ”¡—π‰ª 15,000 − 3,000 = 12,000 ≈‘μ√
π—Ëπ§◊Õ πÈ”¡—π 35 ¢Õß∂—ß §‘¥‡ªìππÈ”¡—π 12,000 ≈‘μ√
πÈ”¡—π 1 ∂—ß §‘¥‡ªìππÈ”¡—π 12 00035
, ÷ ≈‘μ√
= 12000 53
× ≈‘μ√
= 20,000 ≈‘μ√¥—ßπ—Èπ ∂—ß„∫π’È®ÿπÈ”¡—π 20,000 ≈‘μ√ μÕ∫
5. «‘∏’∑” ∂—ß„∫Àπ÷Ëß®ÿπÈ”‰¥â 10,000 ≈‘μ√
·μà¡’πȔլŸà 35 ¢Õß∂—ß ‡∑à“°—∫ 3
510 000× , = 6,000 ≈‘μ√
„™âπÈ”‰ª 14 ¢Õß∂—ß ‡∑à“°—∫ 1
410 000× , = 2,500 ≈‘μ√
®–‡À≈◊ÕπȔլŸà 6,000 − 2,500 = 3,500 ≈‘μ√¥—ßπ—Èπ ‡À≈◊ÕπȔլŸà 3,500 ≈‘μ√ μÕ∫
6. «‘∏’∑” «—π‡ “√åπ“ß™«π™◊Ëπ„™â‡ß‘π‰ª 58 ¢Õ߇ߑπ∑’Ë¡’Õ¬Ÿà
®–‡À≈◊Շߑπ 88
58
− = 38 ¢Õ߇ߑπ∑’Ë¡’Õ¬Ÿà
«—πÕ“∑‘μ¬åπ“ß™«π™◊Ëπ„™â‡ß‘π 15 ¢Õ߇ߑπ∑’ˇÀ≈◊Õ
= 15
38
× ¢Õ߇ߑπ∑’Ë¡’Õ¬Ÿà
= 340
¢Õ߇ߑπ∑’Ë¡’Õ¬Ÿà
Page 25
25·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
600
1
31
1
¬—߇À≈◊Շߑπ 38
340
− = 1240
¢Õ߇ߑπ∑’Ë¡’Õ¬Ÿà
π“ß™«π™◊Ëπ¬—ߧ߇À≈◊Շߑπ 7,200 ∫“∑
‡ß‘π 1240
¢Õ߇ߑπ∑’Ë¡’Õ¬Ÿà §‘¥‡ªìπ‡ß‘π 7,200 ∫“∑
‡ß‘π 1 ¢Õ߇ߑπ∑’Ë¡’Õ¬Ÿà §‘¥‡ªìπ‡ß‘π 7 2001240
, ÷ = 7200 4012
× = 24,000 ∫“∑
¥—ßπ—È𠇥‘¡π“ß™«π™◊Ëπ¡’‡ß‘πÕ¬Ÿà 24,000 ∫“∑ μÕ∫
7. «‘∏’∑” π“¬πæ¥≈¢—∫√∂‰¥â√–¬–∑“ß 546 °‘‚≈‡¡μ√
„π‡«≈“ 523 ™—Ë«‚¡ß
π“¬πæ¥≈¢—∫√∂¥â«¬Õ—μ√“‡√Á«‡©≈’ˬ 546 523
÷ = 5463
17× °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
≈ 96.35 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
¥—ßπ—Èπ π“¬πæ¥≈¢—∫√∂¥â«¬Õ—μ√“‡√Á«‡©≈’ˬ™—Ë«‚¡ß≈– 96.35 °‘‚≈‡¡μ√ μÕ∫
8. «‘∏’∑” “¡„πÀâ“¢Õß 49 ¡“°°«à“Àπ÷Ëß„π ’Ë¢Õß 2
5 Õ¬Ÿà 3
549
14
25
×⎛⎝
⎞⎠ − ×⎛
⎝⎞⎠ = 12
452
20−
= 48 18180
−
= 30180
= 16
μÕ∫
9. «‘∏’∑” ‚√߇√’¬π·ÀàßÀπ÷Ëß¡’π—°‡√’¬π™“¬ 25 ¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥
¡’π—°‡√’¬πÀ≠‘ß 55
25
− = 35 ¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥
¡’π—°‡√’¬πÀ≠‘ß¡“°°«à“π—°‡√’¬π™“¬Õ¬Ÿà 480 §π
π—Ëπ§◊Õ 35
25
− = 15 ¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥
π—°‡√’¬π 15
¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥ ‡ªìππ—°‡√’¬π 480 §π
π—°‡√’¬π 1 ¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥ ‡ªìππ—°‡√’¬π 48015
÷ = 480 × 5 = 2,400 §π
¥—ßπ—Èπ ‚√߇√’¬ππ’È¡’π—°‡√’¬π∑—ÈßÀ¡¥ 2,400 §π μÕ∫
10. «‘∏’∑” π“¡√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“°«â“ß 1225
‡¡μ√
¡’¥â“𬓫¬“«°«à“ 112 ‡∑à“¢Õߥâ“π°«â“ßÕ¬Ÿà 4 ‡¡μ√
π—Ëπ§◊Õ ¡’¥â“𬓫‡∑à“°—∫ 32
625
4×⎡⎣⎢
⎤⎦⎥
+ = 1135
‡¡μ√
π“¡√Ÿª ’ˇÀ≈’ˬ¡¡’æ◊Èπ∑’Ë 625
1135
× = 700625
= 280625
μ“√“߇¡μ√
¥—ßπ—Èπ π“¡√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“π’È¡’æ◊Èπ∑’Ë 280625
μ“√“߇¡μ√ μÕ∫
Page 26
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 226
11. «‘∏’∑” ™”√–‡ß‘πß«¥·√° 311
¢Õß√“§“√∂¬πμå
¬—߇À≈◊Õ à«π∑’ËμâÕß™”√–Õ’° 1111
311
− = 811
¢Õß√“§“√∂¬πμå
™”√–ß«¥∑’Ë Õß 25 ¢Õß∑’ˇÀ≈◊Õ®“°®à“¬ß«¥·√° = 2
58
11× = 16
55 ¢Õß√“§“√∂¬πμå
¬—߇À≈◊Õ à«π∑’ËμâÕß™”√–Õ’° 811
1655
− = 2455
¢Õß√“§“√∂¬πμå
™”√–ß«¥∑’Ë “¡∑—ÈßÀ¡¥‡ªìπ‡ß‘π 192,000 ∫“∑
π—Ëπ§◊Õ 2455
¢Õß√“§“√∂¬πμå §‘¥‡ªìπ‡ß‘π 192,000 ∫“∑
1 ¢Õß√“§“√∂¬πμå §‘¥‡ªìπ‡ß‘π 192000 5524
× = 440,000 ∫“∑
¥—ßπ—Èπ π“¬ÕàÕπ´◊ÈÕ√∂¬πμå√“§“§—π≈– 440,000 ∫“∑ μÕ∫
12. «‘∏’∑” ¡’¢â“«‡ª≈◊Õ° 31212
μ—π
√∂∫√√∑ÿ°‰¥â§√—Èß≈– 115
μ—π
®–μâÕß∫√√∑ÿ°∑—ÈßÀ¡¥ 31212
115
÷ = 6252
56
× §√—Èß
= 312512
§√—Èß
= 2605
12§√—Èß
¥—ßπ—Èπ √∂∫√√∑ÿ°®–μâÕß∫√√∑ÿ°∑—ÈßÀ¡¥ 261 §√—Èß
·≈–§√—Èß ÿ¥∑⓬∫√√∑ÿ°‰¥â 512
65
× = 12 μ—π
π—Ëπ§◊Õ μâÕß∫√√∑ÿ°∑—ÈßÀ¡¥ 261 §√—Èß §√—Èß ÿ¥∑⓬∫√√∑ÿ° 12 μ—π μÕ∫
13. «‘∏’∑” ·À««Õà“πÀπ—ß ◊Õ‚¥¬«—π·√°Õà“π‰¥â 14 ¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥
¬—߇À≈◊Õ∑’ˬ—߉¡à‰¥âÕà“π 44
14
− = 34 ¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥
«—π∑’Ë ÕßÕà“π‰¥â 15 ¢Õß®”π«πÀπâ“∑’ˇÀ≈◊Õ = 1
534
× = 320
¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥
¬—߇À≈◊Õ∑’ˬ—߉¡à‰¥âÕà“π 34
320
− = 1220
¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥
‡π◊ËÕß®“°¬—߇À≈◊Õ∑’ˬ—߉¡à‰¥âÕà“πÕ’° 288 Àπâ“
π—Ëπ§◊Õ 1220
¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥‡∑à“°—∫ 288 Àπâ“
1 ¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥‡∑à“°—∫ 288 2012
× = 480 Àπâ“
¥—ßπ—Èπ Àπ—ß ◊Õ‡≈à¡π’È¡’®”π«πÀπâ“∑—ÈßÀ¡¥ 480 Àπâ“ μÕ∫
Page 27
27·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
14. «‘∏’∑” ‚√߇√’¬π·ÀàßÀπ÷Ëß¡’π—°‡√’¬π™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 Õ¬Ÿà 410
¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥
¡’π—°‡√’¬π™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 2 Õ¬Ÿà 13
¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥
√«¡ Õß™—Èπ‡∑à“°—∫ 410
13
+ = 2230
¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥
‡ªìππ—°‡√’¬π™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 3 Õ¬Ÿà 3030
2230
− = 830
¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥
¡’π—°‡√’¬π™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 3 Õ¬Ÿà 616 §π
π—°‡√’¬π 830
¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥ ‡ªìππ—°‡√’¬π 616 §π
π—°‡√’¬π 1 ¢Õßπ—°‡√’¬π∑—ÈßÀ¡¥ ‡ªìππ—°‡√’¬π 616 308× = 2,310 §π
¥—ßπ—Èπ ‚√߇√’¬ππ’È¡’π—°‡√’¬π∑—ÈßÀ¡¥ 2,310 §π μÕ∫
15. «‘∏’∑” ‚√ßß“π·ÀàßÀπ÷Ëß¡’§πß“π™“¬ 511
¢Õߧπß“π∑—ÈßÀ¡¥
¡’§πß“πÀ≠‘ß 1111
511
− = 611
¢Õߧπß“π∑—ÈßÀ¡¥
¡’§πß“π™“¬∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï Õ¬Ÿà 15 ¢Õߧπß“π™“¬
= 15
511
× = 555
¢Õߧπß“π∑—ÈßÀ¡¥
¡’§πß“πÀ≠‘ß∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï Õ¬Ÿà 25 ¢Õߧπß“πÀ≠‘ß
= 25
611
× = 1255
¢Õߧπß“π∑—ÈßÀ¡¥
√«¡§πß“π∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï Õ¬Ÿà 555
1255
+ = 1755
¢Õߧπß“π∑—ÈßÀ¡¥
§πß“π∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï ¡’Õ¬Ÿà 68 §π
π—Ëπ§◊Õ 1755
¢Õߧπß“π∑—ÈßÀ¡¥ ‡∑à“°—∫ 68 §π
1 ¢Õߧπß“π∑—ÈßÀ¡¥ ‡∑à“°—∫ 685517
× = 220 §π
¥—ßπ—Èπ ‚√ßß“ππ’È¡’§πß“πÕ¬Ÿà∑—ÈßÀ¡¥ 220 §π μÕ∫
·∫∫Ωñ°À—¥ 7
1. (1) 3,214.3 = ( ) ( ) ( ) ( )3 10 2 10 1 10 4 1 31
103 2× + × + × + × + ×⎛
⎝⎞⎠
(2) 625.41 = ( ) ( ) ( )6 10 2 10 5 1 41
101
1
10
22× + × + × + ×⎛
⎝⎞⎠ + ×⎛
⎝⎜⎞⎠⎟
Page 28
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 228
(3) 90.052 = ( )9 10 51
102
1
102 3× + ×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
(4) 0.45678 = 41
105
1
106
1
107
1
108
1
102 3 4 5×⎛
⎝⎞⎠ + ×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
2. (1) μ—«‡≈¢ 2 ¡’§à“ª√–®”À≈—° 1
102 μ—«‡≈¢ 3 ¡’§à“ª√–®”À≈—° 1
103
(2) μ—«‡≈¢ 4 ¡’§à“ª√–®”À≈—° 10 μ—«‡≈¢ 1 ¡’§à“ª√–®”À≈—° 1
(3) μ—«‡≈¢ 3 ¡’§à“ª√–®”À≈—° 110
μ—«‡≈¢ 4 ¡’§à“ª√–®”À≈—° 1
102
(4) μ—«‡≈¢ 3 ¡’§à“ª√–®”À≈—° 110
μ—«‡≈¢ 5 ¡’§à“ª√–®”À≈—° 1
103
3. (1) 51
106
1
107
1
102 3×⎛
⎝⎞⎠ + ×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟ = 0.567
(2) 41
105
1
107
1
103 2 1×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟ = 7
1
105
1
104
1
101 2 3×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟ = 0.754
(3) 11
102
1
103
1
104
1
101 4 3 2×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
= 11
104
1
103
1
102
1
101 2 3 4×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
= 0.1432
(4) 81
107
1
106
1
109
1
101 2 3 4×⎛
⎝⎜⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟
+ ×⎛⎝⎜
⎞⎠⎟ = 0.8769
·∫∫Ωñ°À—¥ 8
1. (1) � (2) � (3) � (4) =
(5) � (6) � (7) � (8) =
2. (1) 0.006, 0.008, 0.05, 0.4, 0.7 (2) 0.0006, 0.003, 0.01, 0.4
(3) �0.82, �0.804, �0.61, �0.072 (4) 0.0001, 0.001, 0.01, 0.1
(5) �5.7, �4.68, �4.2, �3.21
3. (1) 0.65, 0.64, 0.63, 0.61, 0.602 (2) 0.13, 0.12, 0.11, 0.103, 0.102, 0.101
(3) 0.321, 0.312, 0.123, 0.1203, 0.0213 (4) �0.457, �0.671, �0.8012, �5.007, �.5.7
(5) �46.5, �32.71, �4.6, �4.06, �3.271
4. (1) 0.63457457 � 0.63457145 (2) 212.3451341 � 212.3451345
(3) �0.98712727 � �0.9871278278 (4) �0.45678678 � �0.456785678
5. ≈’≈“«‘Ë߇¢â“‡ âπ™—¬‡ªìπ§π·√°
Page 29
29·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
·∫∫Ωñ°À—¥ 9
1. (1) «‘∏’∑” 5 μ“ߧå = 5100
∫“∑ (2) «‘∏’∑” 12 μ“ߧå = 12100
∫“∑
= 0.05 ∫“∑ = 0.12 ∫“∑
¥—ßπ—Èπ 108 ∫“∑ 5 μ“ß§å ¥—ßπ—Èπ 431 ∫“∑ 12 μ“ߧå
= 108.05 ∫“∑ μÕ∫ = 431.12 ∫“∑ μÕ∫
(3) «‘∏’∑” 87 μ“ߧå = 87100
∫“∑ (4) «‘∏’∑” 671 μ“ߧå = 671100
∫“∑
= 0.87 ∫“∑ μÕ∫ = 6.71 ∫“∑
¥—ßπ—Èπ 6,504 ∫“∑ °—∫Õ’° 671 μ“ߧå
= 6,504 + 6.71 ∫“∑
= 6,510.71 ∫“∑ μÕ∫
2. (1) «‘∏’∑” 4 °‘‚≈‡¡μ√ 3 ‡¡μ√ (2) «‘∏’∑” 142 °‘‚≈‡¡μ√ 76 ‡¡μ√
= 43
1000+ °‘‚≈‡¡μ√ = 142
761000
+ °‘‚≈‡¡μ√
= 4.003 °‘‚≈‡¡μ√ μÕ∫ = 142.076 °‘‚≈‡¡μ√ μÕ∫
(3) «‘∏’∑” 37 °‘‚≈‡¡μ√ 124 ‡¡μ√
= 37124
1000+ °‘‚≈‡¡μ√
= 37.124 °‘‚≈‡¡μ√ μÕ∫
3. (1) «‘∏’∑” 7 °‘‚≈°√—¡ 4 °√—¡ (2) «‘∏’∑” 2 °√—¡ = 21000
°‘‚≈°√—¡
= 7
41000
� °‘‚≈°√—¡ = 0.002 °‘‚≈°√—¡ μÕ∫
= 7.004 °‘‚≈°√—¡ μÕ∫
(3) «‘∏’∑” 15 °‘‚≈°√—¡ 153 °√—¡ (4) «‘∏’∑” 1,203 °√—¡ = 12031000
°‘‚≈°√—¡
= 15
1531000
� °‘‚≈°√—¡ = 1.203 °‘‚≈°√—¡ μÕ∫
= 15.153 °‘‚≈°√—¡ μÕ∫
4. (1) «‘∏’∑” 5 ‡¡μ√ 2 ‡´π쑇¡μ√ (2) «‘∏’∑” 18 ‡¡μ√ 41 ‡´π쑇¡μ√
= 52
100+ ‡¡μ√ = 18
41100
+ ‡¡μ√
= 5.02 ‡¡μ√ μÕ∫ = 18.41 ‡¡μ√ μÕ∫
(3) «‘∏’∑” 9 ‡¡μ√ 56 ‡´π쑇¡μ√ 4 ¡‘≈≈‘‡¡μ√ (4) «‘∏’∑” 651 ‡´π쑇¡μ√ 43 ¡‘≈≈‘‡¡μ√
= 956
1004
1000+ + ‡¡μ√ = 651
10043
1000+ ‡¡μ√
= 9 + 0.56 + 0.004 ‡¡μ√ = 6.51 + 0.043 ‡¡μ√
= 9.564 ‡¡μ√ μÕ∫ = 6.553 ‡¡μ√ μÕ∫
Page 30
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 230
1. (1) 12.56 + (−7.286) (2) ( − 48.72) +17.295
«‘∏’∑” 12.560 «‘∏’∑” 48.720
7.286−
17.295−
5.274 31.425
¥—ßπ—Èπ 12.56 + (−7.286) ¥—ßπ—Èπ ( − 48.72) +17.295
= 5.274 μÕ∫ = −31.425 μÕ∫
(3) (−14.701) + (−43.036) (4) 28.15 − 10.5
«‘∏’∑” 14.701 «‘∏’∑” 28.15
43.036+
10.50−
57.737 17.65
¥—ßπ—Èπ (−14.701) + (−43.036) ¥—ßπ—Èπ 28.15 −10.5
= −57.737 μÕ∫ = 17.65 μÕ∫
(5) (−42.078) − (−16.29) (6) (−9.56)−46.108
«‘∏’∑” 42.078 «‘∏’∑” 9.560
16.290−
46.108+
25.788 55.668
¥—ßπ—Èπ (−42.078) − (−16.29) ¥—ßπ—Èπ (−9.56) −46.108
= −25.788 μÕ∫ = −55.668 μÕ∫
2. (1) (10.012 − 8.321) + 7.26 (2) (15.04 − 8.0412) − 5.3423
«‘∏’∑” 10.012 «‘∏’∑” 15.0400
8.321−
8.0412−
1.691 6.9988
7.260+
5.3423−
8.951 1.6565
¥—ßπ—Èπ (10.012 − 8.321) + 7.26 ¥—ßπ—Èπ (15.04 − 8.0412) − 5.3423
= 8.951 μÕ∫ = 1.6565 μÕ∫
·∫∫Ωñ°À—¥ 10
Page 31
31·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(3) (8.12 + 6.231) − 12.49 (4) (15 − 4.201) + 8.19
«‘∏’∑” 8.120 «‘∏’∑” 15.000
6.231+
4.201−
14.351 10.799
12.490−
8.190+
1.861 18.989
¥—ßπ—Èπ (8.12 + 6.231) − 12.49 ¥—ßπ—Èπ (15 − 4.201) + 8.19
= 1.861 μÕ∫ = 18.985 μÕ∫
(5) (−13.25) + (14.68−20.07)
«‘∏’∑” 20.07 13.25
14.68−
5.39+
5.39 18.64
¥—ßπ—Èπ (−13.25) + (14.68−20.07) = −18.64 μÕ∫
(6) [(−7.91) − (−45.721)]−16.352
«‘∏’∑” 45.721 37.811
7.910−
16.352−
37.811 21.459
¥—ßπ—Èπ [(−7.91) − (−45.721)]−16.352 = −21.459 μÕ∫
3. π“¬¡’°—∫π“¬¡—Ëß¡’‡ß‘π√«¡°—π 164.75 ∫“∑
π“¬¡—Ëß¡’‡ß‘π¡“°°«à“π“¬¡’Õ¬Ÿà38.75 ∫“∑
‡ß‘π à«π∑’ˇÀ≈◊ÕÀ≈—ß®“°À—° à«π∑’Ë¡“°°«à“ÕÕ°§◊Õ 164.75 − 38.75 = 126.00 ∫“∑
à«π∑’ˇÀ≈◊ÕÕ¬Ÿàπ’ȇ¡◊ËÕ·∫à߇ªìπ Õß à«π‡∑à“°—π®–‰¥â 126 ÷ 2 = 63 ∫“∑
¥—ßπ—Èπ π“¬¡’¡’‡ß‘π 63.00 ∫“∑
π“¬¡—Ëß¡’‡ß‘π 63 + 38.75 = 101.75 ∫“∑
¥—ßπ—Èπ π“¬¡’¡’‡ß‘π63.00 ∫“∑
π“¬¡—Ëß¡’‡ß‘π101.75 ∫“∑ μÕ∫
4. ∑àÕπȔ՗πÀπ÷Ëß¡’§«“¡¬“« 0.93 ‡¡μ√
§«“¡¬“«¢Õ߇ âπºà“π»Ÿπ¬å°≈“ß¿“¬πÕ°‡ªìπ 23 ¢Õߧ«“¡¬“«∑àÕ
= 23
0 93× . ‡¡μ√
= 0.62 ‡¡μ√
Page 32
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 232
¡’§«“¡¬“«‡ âπºà“π»Ÿπ¬å°≈“ß¿“¬„π‡ªìπ 0.435 ‡¡μ√
2 ‡∑à“¢Õߧ«“¡Àπ“ = 0.62 − 0.435 = 0.185 ‡¡μ√
§«“¡Àπ“ = 0 1852
100. × = 9.25 ‡´π쑇¡μ√
¥—ßπ—Èπ ∑àÕπÈ”π’È¡’§«“¡Àπ“ 9.25 ‡´π쑇¡μ√ μÕ∫
5. «‘∏’∑” πÈ”Àπ—° ‘ËߢÕß∑—ÈßÀ¡¥√«¡πÈ”Àπ—°°√–‡ªÜ“Àπ—°
4.47 + 3.25 + 5.18 + 2.39 + 11.47 + 9.63 + 3.71 = 40.10 °‘‚≈°√—¡
¥—ßπ—Èπ ‡¡◊ËÕμâÕß°“√∫√√®ÿ ‘ËߢÕß„À≥⡓°∑’Ë ÿ¥ ·μà¡’πÈ”Àπ—°√«¡°√–‡ªÜ“‰¥â‰¡à‡°‘π 35 °‘‚≈°√—¡
®÷ߧ«√∫√√®ÿ ‘ËߢÕß™‘Èπ∑’Ë 1 ™‘Èπ∑’Ë 2 ™‘Èπ∑’Ë 4 ™‘Èπ∑’Ë 5 ·≈–™‘Èπ∑’Ë 6 ‚¥¬‡Õ“ ‘ËߢÕß™‘Èπ∑’Ë 3 ÕÕ°‰ª ´÷Ëß
®–¡’πÈ”Àπ—°√«¡°√–‡ªÜ“‡ªìπ 40.10 − 5.18 = 34.92 °‘‚≈°√—¡
¥—ßπ—Èπ μâÕ߇Փ™‘Èπ∑’Ë 3 ÕÕ°‰ª ·≈–∑’ˇÀ≈◊Õ¡’πÈ”Àπ—°√«¡¢Õß°√–‡ªÜ“ 34.92 °‘‚≈°√—¡ μÕ∫
6. «‘∏’∑” 𓬙Õ∫‡∑’ˬ«‡¥‘π∑“߇ªìπ√–¬–∑“ß∑—Èß ‘Èπ 10.81 + 11.43 + 15.45 = 37.69 °‘‚≈‡¡μ√
𓬠“¬μ√߇¥‘π∑“ß®“°À“¥æ≈“∂÷ßÀ“¥μ“°«π‡ªìπ√–¬–∑“ß 33.47 °‘‚≈‡¡μ√
¥—ßπ—Èπ 𓬙Õ∫‡∑’ˬ«‡¥‘π∑“ß¡“°°«à“𓬠“¬μ√ß 37.69 − 33.47 = 4.22 °‘‚≈‡¡μ√ μÕ∫
1. (1) 4.17 × 3.2 (2) 5.67 × 2.34
«‘∏’∑” 417 «‘∏’∑” 567
32×
234×
834 2268
12510+
17010+
13344 113400
¥—ßπ—Èπ 4.17 × 3.2 = 13.344 μÕ∫ 13.2678
¥—ßπ—Èπ 5.67 × 2.34 = 13.2678 μÕ∫
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 13
0.435 ¡.
0.62 ¡.
Page 33
33·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(3) 0.421 × 0.56 (4) 7.654 × 0.321
«‘∏’∑” 421 «‘∏’∑” 7654
56×
321×
2526 7654
21050+
153080
23576 2296200
¥—ßπ—Èπ 0.421 × 0.56 = 0.23576 2.456934
μÕ∫ ¥—ßπ—Èπ 7.654 × 0.321 = 2.456934 μÕ∫
(5) (−11.01) × (−1.1) (6) 12.38 ×(−3)
«‘∏’∑” 1101 «‘∏’∑” 1238
11×
3×
1101 3714
11010+
¥—ßπ—Èπ 12.38 ×(−3) = −37.14 μÕ∫
12111
¥—ßπ—Èπ (−11.01) × (−1.1)
= 12.111 μÕ∫
(7) (−42.178) × 0.23 (8) 102.5 ×(−0.02)
«‘∏’∑” 42178 «‘∏’∑” 1025
23×
2×
126534 2050
843560+
¥—ßπ—Èπ 102.5 ×(−0.02) = −2.050μÕ∫
970094
¥—ßπ—Èπ (−42.178) × 0.23
= −9.70094 μÕ∫
2. (1) «‘∏’∑” ºâ“æ—∫Àπ÷Ë߬“« 32.60 ‡¡μ√
¢“¬‡¡μ√≈– 68.50 ∫“∑
®–‰¥â‡ß‘π 32.60 × 68.50 = 2,233.10 ∫“∑
¥—ßπ—Èπ ®–‰¥â‡ß‘π 2,233.10 ∫“∑ μÕ∫
(2) «‘∏’∑” ¢“¬ “¬‰ø§√—Èß·√°‰ª 42.5 ‡¡μ√ ‡¡μ√≈– 18.70 ∫“∑
‰¥â‡ß‘π 42.5 × 18.70 = 794.75 ∫“∑
¢“¬ “¬‰ø§√—Èß∑’Ë Õ߉ª 68.8 ‡¡μ√ ‡¡μ√≈– 18.50 ∫“∑
‰¥â‡ß‘π 68.8 × 18.50 = 1,272.80 ∫“∑
+
Page 34
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 234
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 14
304
0.12875
1. 0.428 ÷ 50.0856
2. 0.5064 ÷ (−4)0.1266
«‘∏’∑” 5 ) 0.4280 «‘∏’∑” 4 ) 0.5064
4000 4000
280 1000
250 800
30 260
30 240
¥—ßπ—Èπ 0.428 ÷ 5 = 0.0856 μÕ∫ 24
24
¥—ßπ—Èπ 0.5064 ÷ (−4) = −0.1266 μÕ∫
3. «‘∏’∑” 0.412 ÷ 3.2 = 0 4123 2..
= 4 1232.
32 ) 4.12000
3.20000
92000
64000
28000
25600
2400
2240
160
160
¥—ßπ—Èπ 0.412 ÷ 3.2 = 0.12875 μÕ∫
¢“¬ “¬‰ø§√—Èß∑’Ë “¡‰ª 50.5 ‡¡μ√ ‡¡μ√≈– 19.50 ∫“∑
‰¥â‡ß‘π 50.5 × 19.50 = 984.75 ∫“∑
®–‰¥â‡ß‘π∑—ÈßÀ¡¥ 794.75 + 1,272.80 + 984.75 = 3,052.30 ∫“∑
¢“¬ “¬‰ø‰ª∑—ÈßÀ¡¥ 42.5 + 68.8 + 50.5 = 161.8 ‡¡μ√
¬—߇À≈◊Õ “¬‰ø¬“« 246.50 − 161.80 = 84.70 ‡¡μ√
¥—ßπ—È𠉥â‡ß‘π∑—ÈßÀ¡¥ 3,052.30 ∫“∑ ·≈–‡À≈◊Õ “¬‰ø 84.70 ‡¡μ√ μÕ∫
4. «‘∏’∑” (−456) ÷ 1.5 = 456015
15 ) 4560 4500
60
60
¥—ßπ—Èπ (−456) ÷ 1.5 = −304 μÕ∫
Page 35
35·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
5. «‘∏’∑” 1.843 ÷ 0.12 = 184 312
.
12 ) 184.3000
12.00000
64.0000
60.0000
4.3000
3.6000
7000
6000
1000
960
40
36
4
¥—ßπ—Èπ 1.843 ÷ 0.12 = 15 3583. ˙ μÕ∫
1. «‘∏’∑” “¬‰ø¢¥Àπ÷Ë߬“« 148.40 ‡¡μ√
μ—¥‡ªìπ‡ â𠬓«‡ âπ≈– 5.75 ‡¡μ√
®–μ—¥‰¥â 148.4 ÷ 5.75 ≈ 25.81 ‡ âπ
π—Ëπ§◊Õ ®–μ—¥‰¥â 25 ‡ âπ μÕ∫
2. «‘∏’∑” ¡’πÈ”ª≈“Õ¬Ÿà 18.45 ≈‘μ√ ∫√√®ÿ„ ࢫ¥¢«¥≈– 0.75 ≈‘μ√
®–∫√√®ÿ‰¥â 18.45 ÷ 0.75 = 184575
¢«¥
= 24.6 ¢«¥
¥—ßπ—Èπ ®–∫√√®ÿ‰¥â 24 ¢«¥ ·≈–‡À≈◊ÕπÈ”ª≈“ 0.6 × 0.75 = 0.45 ≈‘μ√ μÕ∫
3. «‘∏’∑” ‰°à·™à·¢Áßμ—«Àπ÷ËßÀπ—° 2.45 °‘‚≈°√—¡ μ‘¥√“§“‰«â 122.50 ∫“∑
‰°à·™à·¢Áß√“§“°‘‚≈°√—¡≈– 122 502 45
..
= 12250245
∫“∑
= 50.00 ∫“∑
¥—ßπ—Èπ ‰°à·™àß·¢Áß√“§“°‘‚≈°√—¡≈– 50.00 ∫“∑ μÕ∫
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 15
6. «‘∏’∑” (−0.4) ÷(− 0.005) =
�
�
0 4000 005..
= 0 4000 005..
= 4005
= 80
¥—ßπ—Èπ (−0.4) ÷(− 0.005) = 80 μÕ∫
15 3583. ˙
Page 36
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 236
1. (1) 0.41 × 6 = 2.46 (2) 0.7 ×(− 0.3) = −0.21
(3) 4.021 × 6.4 (4) (−65.1) × 3.72
«‘∏’∑” 4021 «‘∏’∑” 651
64×
372×
16084 1302
241260+
45570+
257344 195300
¥—ßπ—Èπ 4.021 × 6.4 = 25.7344 μÕ∫ 242172
¥—ßπ—Èπ 65.1 × 3.72 = 242.172 μÕ∫
(5) 0.48 × 0.12 (6) 100 × 0.01
«‘∏’∑” 48 «‘∏’∑” 100
12×
1×
96 100
48+
¥—ßπ—Èπ 100 × 0.01 = 1 μÕ∫
576
¥—ßπ—Èπ 0.48 × 0.12 = 0.0576 μÕ∫
(7) (−4.32) × (−0.007) (8) (−7.23) × 6.9
«‘∏’∑” 432 «‘∏’∑” 723
7×
69×
3024 6507
¥—ßπ—Èπ (−4.32) × (−0.007) 43380+
= 0.03024 μÕ∫ 49887
¥—ßπ—Èπ (−7.23) × 6.9 = 49.887 μÕ∫
2. (1) «‘∏’∑” (−0.45) ÷ 0.5 = − 0 450 50
.
.(2) 0.604 ÷ 4
0.151 = − 45
50«‘∏’∑” 4 ) 0.604
50 ) 45.0 400
00 200
45.0 200
45.0 04
¥—ßπ—Èπ (−0.45) ÷ 0.5 = −0.9 μÕ∫ 4
¥—ßπ—Èπ 0.604 ÷ 4 = 0.151 μÕ∫
·∫∫Ωñ°À—¥ 11
0.9
Page 37
37·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
0.25
(3) «‘∏’∑” 0.428 ÷ (−0.02) = 0 4280 020
.
.(4) «‘∏’∑” 0.0609 ÷ 0.003 = 0 0609
0 003..
= 42 82. = 60 9
3.
= 21.4 = 20.3
¥—ßπ—Èπ 0.428 ÷ 0.02 = 21.4 μÕ∫ ¥—ßπ—Èπ 0.0609 ÷ 0.003 = 20.3 μÕ∫
(5) «‘∏’∑” (−0.234) ÷ (−0.0008) = 0 23400 0008..
(6) «‘∏’∑” 0.0625 ÷ 0.25 = 6 2525.
= 23408
25 ) 6.25
= 292.5 5.00
¥—ßπ—Èπ (−0.234) ÷ (−0.0008) 1.25
= 292.5 μÕ∫ 1.25
¥—ßπ—Èπ 0.0625 ÷ 0.25 = 0.25 μÕ∫
(7) «‘∏’∑” (−63.862) ÷ (−0.74) = 63 8620 74..
(8) «‘∏’∑” (−3.2768) ÷ (−0.32) =
�
�
3 27680 32..
= − 6386 274
. = 327 6832
.
= −86.3 μÕ∫ = 10.24 μÕ∫
3. «‘∏’∑” ‡™◊Õ°‡ âπÀπ÷Ë߬“« 15.48 ‡¡μ√ μ—¥ÕÕ°‡ªìπ 2 ∑àÕπ ‚¥¬∑àÕπÀπ÷Ë߬“«°«à“Õ’°∑àÕπÀπ÷Ëß 2.32
‡¡μ√
¥—ßπ—Èπ À—° à«π∑’ˬ“«ÕÕ°‰ª°àÕπ®–‡À≈◊Õ
15.48 − 2.32 = 13.16 ‡¡μ√
¥—ßπ—Èπ ∑àÕπ —Èπ¡’§«“¡¬“« 13 162. = 6.58 ‡¡μ√
·≈–∑àÕ𬓫¡’§«“¡¬“« 6.58 + 2.32 = 8.90 ‡¡μ√ μÕ∫
4. «‘∏’∑” √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“°«â“ß 4.7 ‡´π쑇¡μ√ ¬“« 12.6 ‡´π쑇¡μ√
¡’æ◊Èπ∑’Ë 4.7 × 12.6 = 59.22 μ“√“߇´π쑇¡μ√
¥—ßπ—Èπ √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“¡’æ◊Èπ∑’Ë 59.22 μ“√“߇´π쑇¡μ√ μÕ∫
5. «‘∏’∑” „Àâ®”π«π®”π«ππ—Èπ§◊Õ x
x ÷ 8.4 = 16.5
À√◊Õ x8 4.
= 16.5
π” 8.4 §Ÿ≥∑—Èß Õߢâ“ߢÕß ¡°“√ ®–‰¥â
x = 8.4 × 16.5 = 138.6
¥—ßπ—Èπ ®”π«π®”π«ππ—Èπ§◊Õ 138.6 μÕ∫
Page 38
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 238
6. «‘∏’∑” ´◊Èբ⓫ “√ 4.3 °‘‚≈°√—¡ √“§“°‘‚≈°√—¡≈– 12.50 ∫“∑
®–μâÕß®à“¬‡ß‘π 4.3 × 12.50 = 53.75 ∫“∑
¥—ßπ—Èπ ®–μâÕß®à“¬‡ß‘π 53.75 ∫“∑ μÕ∫
7. «‘∏’∑” √∂∫√√∑ÿ°∫√√∑ÿ°¥‘π‰¥â‡μÁ¡∑’ˇ∑’ˬ«≈– 2.32 μ—π
μâÕß°“√∫√√∑ÿ°¥‘π 45 μ—π
μâÕß∫√√∑ÿ°∑—ÈßÀ¡¥ 452 32.
= 4500232
‡∑’ˬ«
≈ 1992232
‡∑’ˬ«
π—Ëπ§◊Õ μâÕß∫√√∑ÿ° 20 ‡∑’ˬ« ‚¥¬‡∑’ˬ«∑’Ë 20 ∫√√∑ÿ°¥‘π‰¥â 92232
2 32× . ≈ 0.92 μ—π μÕ∫
8. «‘∏’∑” ·¥ß¡’‡ß‘π 57.25 ∫“∑
¥”¡’‡ß‘π 171.75 ∫“∑
·¥ß¡’‡ß‘π‡ªìπ 57 25171 75
..
‡∑à“¢Õߥ”
= 572517175
‡∑à“¢Õߥ”
= 13
‡∑à“¢Õߥ”
¥”¡’‡ß‘π‡ªìπ 171 7557 25
..
෈Ңͧᴧ
= 3 ෈Ңͧᴧ
¥—ßπ—Èπ ·¥ß¡’‡ß‘π‡ªìπ 13 ‡∑à“¢Õߥ” ·≈–¥”¡’‡ß‘π 3 ‡∑à“¢Õß·¥ß μÕ∫
1. (1) 0.4 (2) 0.8 (3) −0.7 (4) 0.1 (5) 0.3
(6) 0.6 (7) −0.9 (8) 0.2 (9) 0.5 (10) 23.2
(11) −18.6 (12) 91.7 (13) 138.3 (14) −475.2
(15) 12345
= 1238
10
= 123.8
2. (1) 0.2 (2) 0.8 (3) 0.1
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 16
Page 39
39·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1. (1) 0.04 (2) −0.17 (3) 0.23 (4) 14.56
(5) −432.03 (6) 432.001 (7) −4,320.13 (8) 4,320.09
2 (1)5
100(2)
64100
(3) − 93100
(4) 12
100
(5) −1521
100(6) 145
9100
(7) 22299
100(8) −100
1100
1. (1) 0.234 (2) −0.435 (3) 0.002 (4) 0.004
(5) −32.9 (6) 8.52 (7) −49.993 (8) 83.005
(9) 496.40 (10) 888.034 (11) 2,341.900 (12) 1,876.003
2. (1)4
10(2)
6100
(3) − 71000
(4)189
1000
(5) 694
1000(6) − 8
510
(7) 3650
1000(8) 39
71000
(9) − 6783
100(10) 82
3871000
(11) − 90482
1000(12) 700
51000
·∫∫Ωñ°À—¥ 12
1. (1)9
10(2) − 87
100(3)
9100
(4)930
1000(5) − 931
1000
(6)7
1000(7)
400510000
(8) − 542100
(9)6210141000
(10)12345678
10000
2. (1) 0.8 (2) −8.1 (3) 0.34 (4) 9.81 (5) −38.10
(6) 4.561 (7) −321.4 (8) 42.356 (9) −7.456 (10) 31.024
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 17
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 18
Page 40
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 240
1.23
= 0.6666... 2.43
= 1.333...
= 0.6̇ = 1.3̇
3.56
= 0.83333... 4.79
= 0.777...
= 0.83̇ = 0.7̇
5.239
= 2.555... 6.790
0.0777...0.= 2.5̇ «‘∏’∑” 90 ) 7.0000000
6.3000000
7000000
6300000
700000
¥—ßπ—Èπ 790
= 0.07̇ μÕ∫
7.3790
0.4111...0.8.
899
0.0808....«‘∏’∑” 90 ) 37.0000000 «‘∏’∑” 99 ) 8.000000
36.0000000 7.920000
1.0000000 80000
9000000 79200
1000000 800
900000 ¥—ßπ—Èπ 899
= 0.08˙ ˙ μÕ∫
100000
¥—ßπ—Èπ 3790
= 0.41̇ μÕ∫
9.1999
0.1919....10.
41990
0.04141...«‘∏’∑” 99 ) 19.000000 «‘∏’∑” 990 ) 41.000000
9.900000 39.600000
9.100000 1.400000
8.910000 990000
·∫∫Ωñ°À—¥ 13
Page 41
41·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
190000 410000
99000 396000
91000 14000
89100 9900
1900 ¥—ßπ—Èπ 41990
= 0.041˙ ˙ μÕ∫
¥—ßπ—Èπ 1999
= 0.19˙ ˙ μÕ∫
11.627
= 29
= 0.2222... 12.411
= 0.3636...
= 0.2̇ = 0.36˙ ˙
13.27
0.2857142014.
127
1.7142857«‘∏’∑” 7 ) 2.00000000 «‘∏’∑” 7 ) 12.0000000
1.40000000 70.000000
60000000 5.0000000
56000000 4.9000000
4000000 1000000
3500000 700000
500000 300000
490000 280000
10000 20000
7000 14000
3000 6000
2800 5600
200 400
140 350
60 500
¥—ßπ—Èπ 27 = 0.285714˙ ˙ μÕ∫ 490
10
¥—ßπ—Èπ 127
= 1.714285˙ ˙ μÕ∫
Page 42
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 242
15. «‘∏’∑”18110
= 955
0.1636316.
1115
0.733055 ) 9.00000 «‘∏’∑” 15 ) 11.0000
5.50000 10.5000
3.50000 5000
3.30000 4500
20000 500
16500 450
3500 50
3300 ¥—ßπ—Èπ 1115
= 0.73̇ μÕ∫
200
165
35
¥—ßπ—Èπ 18110
= 0.163˙ ˙ μÕ∫
17. «‘∏’∑”1320
= 13 520 5
×× 18.
1633
0.48480= 65
100«‘∏’∑” 33 ) 16.00000
= 0.65 13.20000
= 0.650̇ μÕ∫ 2.80000
2.64000
16000
13200
2800
2640
160
¥—ßπ—Èπ 1633
= 0.48˙ ˙ μÕ∫
19.1445
0.3111020.
1740
0.425«‘∏’∑” 45 ) 14.00000 «‘∏’∑” 40 ) 17.000
13.50000 16.000
50000 1.000
45000 800
5000 200
4500 200
500 ¥—ßπ—Èπ 1740
= 0.4250̇ μÕ∫
¥—ßπ—Èπ 1445
= 0.31̇ μÕ∫
Page 43
1. 20 2. 70 3. 100 4. 340
5. 560 6. 850 7. 1 8. 3
9. 12 10. 24 11. 6 12. 47
μÕ∫§”∂“¡Àπâ“ 90-91 ‡√◊ËÕß°“√«—¥§«“¡¬“«‡ªì‡¡μ√ (¡.) ·≈–°‘‚≈‡¡μ√ (°¡.)
1. ¡‘≈≈‘‡¡μ√ 2. ‡¡μ√, ‡´π쑇¡μ√ 3. ‡¡μ√
4. °‘‚≈‡¡μ√ 5. ‡´π쑇¡μ√ 6. ‡¡μ√
7. °‘‚≈‡¡μ√
1. 4 2. 26 3. 56 4. 4,500
5. 1,200 6. 2,400 7. 3 8. 1.4
9. 24,000 10. 12,000 11. 7 12. 26
13. 14,000 14. 57,000
1. (1) 35, 3.5 (2) 46, 4.6 (3) 60, 6 (4) 14, 1.4 (5) 27, 2.7
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 1
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 2
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 3
Àπ૬°“√‡√’¬π√Ÿâ∑’Ë 2
°“√ª√–¡“≥§à“
‡©≈¬
Page 44
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 244
2. (1) 27 ‡¡μ√ + 34 ‡´π쑇¡μ√ = 2,700 + 34 = 2,734 ´¡.
(2) 3 ‡¡μ√ + 45 ¡‘≈≈‘‡¡μ√ + 3 ‡´π쑇¡μ√ = 300 + 4.5 + 3 = 307.5 ´¡.
(3) 3 ‡¡μ√ 130 ¡‘≈≈‘‡¡μ√ + 3 ‡´π쑇¡μ√ = 300 + 13 + 3 = 316 ´¡.
(4) 364 ‡¡μ√ = 3641000
= 0.364 °¡.
(5) 36,784 ‡¡μ√ = 367841000
= 36.784 °¡.
(6) 438 ‡´π쑇¡μ√ = 438100
= 4.38 ¡.
(7) 0.27 °‘‚≈‡¡μ√ = 0.27 × 1,000 = 270 ¡.
(8) 35 ‡¡μ√ = 35 × 100 = 3,500 ´¡.
(9) 345 ¡‘≈≈‘‡¡μ√ = 3451000
= 0.345 ¡.
(10) 349 ¡‘≈≈‘‡¡μ√ = 34910
= 34.9 ´¡.
·∫∫Ωñ°À—¥ 1
1. AB = 3 π‘È«, CD = 4 π‘È«
2. AB = 3.7 π‘È«, CD = 3.2 π‘È«
3. (1) 5+3+6 = 14 ‡´π쑇¡μ√
(2) 2.0+1.3+2.3 = 5.6 π‘È«
4. «‘∏’∑” ≈«¥¢¥∑’ËÀπ÷Ë߬“« 8.45+
‡¡μ√
≈«¥¢¥∑’Ë Õ߬“« 4.32 ‡¡μ√
π”¡“μàÕ°—π®–¬“« 12.77 ‡¡μ√
¥—ßπ—Èπ ≈«¥®–¬“«ª√–¡“≥ 12.8 ‡¡μ√ μÕ∫
5. «‘∏’∑” æ◊Èπ∑’˺π—ßÀâÕß = 2(°«â“ß� Ÿß)�2(¬“«� Ÿß)
= 2(4.75�2.85)�2(5.25�2.85)
= 27.075�29.925
= 57 μ“√“߇¡μ√
æ◊Èπ∑’ˇ楓π = °«â“ß�¬“«
= 4.75�5.25
= 24.9375 μ“√“߇¡μ√
æ◊Èπ∑’˺‘«ÀâÕß�æ◊Èπ∑’ˇ楓π = 57�24.9375
= 81.9375 μ“√“߇¡μ√
¥—ßπ—Èπ æ◊Èπ∑’˺π—ßÀâÕß·≈–‡æ¥“πª√–¡“≥ 81.9 μ“√“߇¡μ√ μÕ∫
Page 45
45·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1. (1) 91,546,380 (2) 91,546,400 (3) 91,546,000 (4) 91,550,000 (5) 91,500,000
2. (1) 631 (2) 631.5 (3) 631.49 (4) 631.494
μÕ∫§”∂“¡Àπâ“ 100 ‡√◊ËÕß°“√ª√–¡“≥§à“·≈–°“√π”§à“ª√–¡“≥‰ª„™â
1. °“√ª√–¡“≥√–¬–∑“ß
2. Õ“®®–‡®ÕÀ√◊Õ‰¡à‡®Õ°Á‰¥â ¢÷ÈπÕ¬Ÿà°—∫∑—°…–°“√ª√–¡“≥√–¬–∑“ߢÕߺŸâ‡≈àπ‡°¡ ·≈–°“√ —߇°μ
3. Õ“®„™â°“√ª√–¡“≥√“§“„À⇪ìπ®”π«π‡μÁ¡∑’Ëßà“¬μàÕ°“√ª√–¡“≥·≈–‰¡à‡°‘¥§«“¡º‘¥æ≈“¥¡“°π—°
4. Õ“®„™â°“√ª√–¡“≥‡∑’¬∫°—∫‡°≥±å∑’Ë¡’ª√– ∫°“√≥å¡“°àÕπ À√◊ÕÕ“®„™â¿“™π–¢π“¥‡≈Á°μ«ß ·≈â«¥Ÿ
«à“¿“™π–μ«ß‡¡≈Á¥∂—Ë«‰¥âª√–¡“≥°’ˇ¡≈Á¥ ·≈⫧”π«≥À“§à“‚¥¬ª√–¡“≥‰¥â®“°®”π«π§√—Èß∑’Ë„™â
¿“™π–¢π“¥‡≈Á°μ«ß‰¥â
·π«∑“ß°“√Õ¿‘ª√“¬ªí≠À“Àπâ“ 100-101
1. π—°‡√’¬πÕ“®‡∑’¬∫‡§’¬ß√–¬–∑“ßμàÕ‡«≈“∑’Ëπ—°‡√’¬π‡§¬‡¥‘πÕ¬Ÿà‡ªìπª√–®”¡“„™â„π°“√Õ¿‘ª√“¬μÕ∫§”∂“¡
À√◊Õ„π°√≥’∑’Ëπ—°‡√’¬π‰¡à‡§¬ —߇°μ√–¬–∑“ß∑’ˇ¥‘π‰¥âμàÕ‡«≈“∑’Ë„™â‰ª °ÁÕ“®‰ªμ√«® Õ∫®“°°“√
ªØ‘∫—μ‘¥Ÿ°àÕπ·≈â«®÷ßμÕ∫§”∂“¡°Á‰¥â
2. ¢âÕπ’ÈÕ“®™à«¬°—𧑥·≈–À“·π«∑“ß„π°“√À“§”μÕ∫‰¥âÀ≈“¬«‘∏’ ‡™àπ π—°‡√’¬πÕ“®≈Õß®—∫‡«≈“¥Ÿ«à“„π
1 π“∑’ π—°‡√’¬π®–√ÕßπÈ”‰¥â°’Ë≈‘μ√ ·≈â«¡“„™â§‘¥§”π«≥®“°‡«≈“∑’Ë„™âÕ“∫πÈ”´÷Ëßπ—°‡√’¬π‡§¬ªØ‘∫—μ‘Õ¬Ÿà
·≈–„Àâ𔉪∑¥≈Õߪؑ∫—μ‘·≈–À“§”μÕ∫¡“𔇠πÕ„π™—Èπ‡√’¬π
1. (1) 80 (2) 12 (3) 30 (4) 0.2 (5) 4,500
(6) 32 (7) 4 (8) 0.3 (9) 100 (10) 12
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 4
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 5
Page 46
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 246
2.
ª√‘¡“≥πÈ”Ωπ
®”π«π‡¡◊ËÕªí¥‡»…‡ªìπ®”π«π‡μÁ¡∑’Ë„°≈⇧’¬ß∑’Ë ÿ¥
®—ßÀ«—¥(¡‘≈≈‘‡¡μ√)
®”π«π ®”π«π ®”π«π
‡μÁ¡Àπ૬ ‡μÁ¡√âÕ¬ ‡μÁ¡æ—π
¢Õπ·°àπ 1,402.6 1,403 1,400 1,000
π§√æπ¡ 2,995.5 2,996 3,000 3,000
¡ÿ°¥“À“√ 1,901.7 1,902 1,900 2,000
√âÕ¬‡ÕÁ¥ 1,357.2 1,357 1,400 1,000
‡≈¬ 1,414.8 1,415 1,400 1,000
°≈π§√ 1,888.6 1,889 1,900 2,000
ÿ√‘π∑√å 1,857.9 1,858 1,900 2,000
·∫∫Ωñ°À—¥ 2
1. (1) «‘∏’∑” 31.86 + 17.75 + 22.05 (2) «‘∏’∑” 2.32 + 1.60 + 0.773
≈ 31.9 + 17.8 + 22.1 ≈ 2.3 + 1.6 + 0.8
≈ 71.8 μÕ∫ ≈ 4.7 μÕ∫
(3) «‘∏’∑” 20.66 − 13.95 + 4.94 (4) «‘∏’∑” 0.072 + 0.0390 + 0.05741
≈ (20.7 − 14.0) + 4.9 ≈ 0.1 + 0.0 + 0.1
≈ 6.7 + 4.9 ≈ 0.2 μÕ∫
≈ 11.6 μÕ∫
(5) «‘∏’∑” 838.06 − 347.45 (6) «‘∏’∑” 82.36 + 75.98 + 19.9
≈ 838.1 − 347.5 ≈ 82.4 + 76.0 + 19.9
≈ 490.6 μÕ∫ ≈ 178.3 μÕ∫
2. (1) «‘∏’∑” 0.0774 − 0.0253 (2) «‘∏’∑” 0.718 + 0.893 + 0.415
≈ 0.08 − 0.03 ≈ 0.72 + 0.89 + 0.42
≈ 0.05 μÕ∫ ≈ 2.03 μÕ∫
(3) «‘∏’∑” 5.837 + 4.293 − 3.307 (4) «‘∏’∑” (6.428 − 4.644) + 1.037
≈ (5.84 + 4.29) − 3.31 ≈ (6.43 − 4.64) + 1.04
≈ 10.13 − 3.31 ≈ 1.79 + 1.04
≈ 6.82 μÕ∫ ≈ 2.83 μÕ∫
Page 47
47·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
3. AB ¬“«‰¥âμ—Èß·μà 8.5 Àπ૬¢÷Èπ‰ª ·μଓ«‰¡à∂÷ß 9.5 Àπ૬ π—Ëπ§◊Õ 8.5 � AB � 9.5
4.
5. À¡Ÿà∫â“π∫—«·¥ßÀà“ß®“°À¡Ÿà∫â“π√à«¡√ࡇ¬Áπ‡ªìπ√–¬–∑“ß°√≥’∂â“„™â°“√ªí¥‡»…°àÕπ®–‰¥â
27 + 45 ≈ 30 + 50 °‘‚≈‡¡μ√≈ 80 °‘‚≈‡¡μ√
´÷Ëß°“√ª√–¡“≥‡™àππ’È®–§≈“¥‡§≈◊ËÕπ¡“°·μà∂â“æ‘®“√≥“«à“ 27 ¡’§à“„°≈â 25 ®–‰¥â§”μÕ∫∑’ˇÀ¡“– ¡°«à“§◊Õ
27 + 45 ≈ 25 + 45 °‘‚≈‡¡μ√≈ 70 °‘‚≈‡¡μ√
6. „™â°“√ª√–¡“≥™à«¬§‘¥«à“‡¥Á°·μà≈–§π¡’‡ß‘πæÕ´◊ÈÕ ‘Ëß∑’ˇ¢“μâÕß°“√À√◊Õ‰¡à(1) «‘∏’∑” ·¥ß¡’‡ß‘π 250 ∫“∑ μâÕß°“√´◊ÈÕ‰¡â¥Õ° 3 °√–∂“ß
‰¡â¥Õ°°√–∂“ß≈– 79 ∫“∑ ª√–¡“≥‡ªìπ 80 ∫“∑‰¡â¥Õ° 3 °√–∂“ß ‡ªìπ‡ß‘πª√–¡“≥ 3 × 80 ≈ 240 ∫“∑¥—ßπ—Èπ ®÷ß¡’‡ß‘πæÕ´◊ÈÕ μÕ∫
(2) «‘∏’∑” ¥”¡’‡ß‘π 250 ∫“∑ μâÕß°“√´◊ÈÕ°âÕπ°√«¥ 3 ∂ÿß°âÕπ°√«¥∂ÿß≈– 59 ∫“∑ 3 ∂ÿß ‡ªìπ‡ß‘πª√–¡“≥ 3 × 60 ≈ 180 ∫“∑¥—ßπ—Èπ ®÷ß¡’‡ß‘πæÕ´◊ÈÕ μÕ∫
(3) «‘∏’∑” ¢“«¡’‡ß‘π 500 ∫“∑ μâÕß°“√´◊ÈÕ¡Õ 7 ™ÿ¥¡Õ å™ÿ¥≈– 69 ∫“∑ ª√–¡“≥‡ªìπ 70 ∫“∑´◊ÈÕ¡Õ å 7 ™ÿ¥ ‡ªìπ‡ß‘πª√–¡“≥ 7 × 70 ∫“∑ ≈ 490 ∫“∑¥—ßπ—Èπ ®÷ß¡’‡ß‘πæÕ´◊ÈÕ μÕ∫
(4) «‘∏’∑” ‡¢’¬«¡’‡ß‘π 325 ∫“∑ μâÕß°“√´◊ÈÕ‰¡â¥Õ° 5 °√–∂“ß
‰¡â¥Õ°°√–∂“ß≈– 79 ∫“∑ ª√–¡“≥‡ªìπ 80 ∫“∑
‰¡â¥Õ° 5 °√–∂“ß ‡ªìπ‡ß‘πª√–¡“≥ 5 × 80 ∫“∑ ≈ 400 ∫“∑
¥—ßπ—Èπ ®÷ß¡’‡ß‘π‰¡àæÕ´◊ÈÕ μÕ∫
®”π«π
À“®”π«π‡μÁ¡∑’Ë„°≈â∑’Ë ÿ¥
®”π«π‡μÁ¡≈â“π ®”π«π‡μÁ¡· π ®”π«π‡μÁ¡À¡◊Ëπ ®”π«π‡μÁ¡æ—π
9,645,637 10,000,000 9,600,000 9,650,000 9,646,000
45,345,678 45,000,000 45,300,000 45,350,000 45,346,000
81,726,354 82,000,000 81,700,000 81,730,000 81,726,000
473,645,619 474,000,000 473,600,000 473,650,000 473,646,000
Page 48
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 248
(5) «‘∏’∑” ‡À≈◊Õß¡’‡ß‘π 175 ∫“∑ μâÕß°“√´◊ÈÕ°âÕπ°√«¥ 3 ∂ÿß
°âÕπ°√«¥∂ÿß≈– 59 ∫“∑ ª√–¡“≥‡ªìπ 60 ∫“∑
°âÕπ°√«¥ 3 ∂ÿß √“§“ª√–¡“≥ 3 × 60 ≈ 180 ∫“∑
¥—ßπ—Èπ ®÷ß¡’‡ß‘π‰¡àæÕ´◊ÈÕ μÕ∫
À¡“¬‡Àμÿ °√≥’À“§à“ª√–¡“≥‰¥â„°≈⇧’¬ß°—∫‡ß‘π∑’Ë¡’Õ¬Ÿà¡“° Õ“® √ÿªº‘¥æ≈“¥°Á‰¥â °√≥’π’ÈÕ“®μâÕß
μ√«® Õ∫®“°®”π«π∑’ˇªìπ§à“®√‘ßÕ’°§√—Èß
(6) «‘∏’∑” øÑ“¡’‡ß‘π 200 ∫“∑ μâÕß°“√´◊ÈÕ¥‘πº ¡ 2 ∂ÿß
¥‘πº ¡∂ÿß≈– 98 ∫“∑ ª√–¡“≥‡ªìπ 100 ∫“∑
¥‘πº ¡ 2 ∂ÿß √“§“ª√–¡“≥ 2 × 100 ≈ 200 ∫“∑
¥—ßπ—Èπ ®÷ß¡’‡ß‘πæÕ´◊ÈÕ μÕ∫
(7) «‘∏’∑” æ‘ߧ塒‡ß‘π 575 ∫“∑ μâÕß°“√´◊ÈÕ‡øîπ 4 ™ÿ¥
‡øîπ™ÿ¥≈– 150 ∫“∑ 4 ™ÿ¥ ‡ªìπ‡ß‘π 150 × 4 = 600 ∫“∑
¥—ßπ—Èπ ®÷ß¡’‡ß‘π‰¡àæÕ´◊ÈÕ μÕ∫
(8) «‘∏’∑” ¡◊¥¡’‡ß‘π 150 ∫“∑ μâÕß°“√´◊ÈÕºß∂à“π 3 ∂ÿß
ºß∂à“π∂ÿß≈– 64 ∫“∑ ª√–¡“≥‡ªìπ 60 ∫“∑
ºß∂à“π 3 ∂ÿß √“§“ª√–¡“≥ 3 × 60 ≈ 180 ∫“∑
¥—ßπ—Èπ ®÷ß¡’‡ß‘π‰¡àæÕ´◊ÈÕ μÕ∫
(9) «‘∏’∑” π«≈¡’‡ß‘π 500 ∫“∑ μâÕß°“√´◊ÈÕÕà“ß·°â« 1 „∫ ‡øîπ 1 ™ÿ¥ ·≈–¡Õ 1 ™ÿ¥
Õà“ß·°â«„∫≈– 225 ∫“∑
‡øîπ™ÿ¥≈– 150 ∫“∑
¡Õ ™ÿ¥≈– 69 ∫“∑
®–μâÕß®à“¬‡ß‘πª√–¡“≥ 230 + 150 + 70 ≈ 450 ∫“∑
¥—ßπ—Èπ ®÷ß¡’‡ß‘πæÕ´◊ÈÕ μÕ∫
(10) «‘∏’∑” · ¥¡’‡ß‘π 300 ∫“∑ μâÕß°“√´◊ÈÕºß∂à“π 1 ∂ÿß ¥‘πº ¡ 1 ∂ÿß °âÕπ°√«¥ 1 ∂ÿß ·≈–‰¡â¥Õ°
1 °√–∂“ß
ºß∂à“π∂ÿß≈– 64 ∫“∑ ¥‘πº ¡∂ÿß≈– 98 ∫“∑
°âÕπ°√«¥∂ÿß≈– 59 ∫“∑ ‰¡â¥Õ°°√–∂“ß≈– 79 ∫“∑
®–μâÕß®à“¬‡ß‘πª√–¡“≥ 60 + 100 + 60 + 80 ≈ 300 ∫“∑
°√≥’π’ÈμâÕßμ√«® Õ∫®“°§à“®√‘ß ®–æ∫«à“μâÕß®à“¬‡ß‘π 64 + 98 + 59 + 79 = 300 ∫“∑ ´÷Ëß
¡’‡ß‘πæÕ¥’∑’Ë®–´◊ÈÕ μÕ∫
Page 49
49·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
7. «‘∏’∑” ·ºàπ∫—π∑÷°¢âÕ¡Ÿ≈ 3.24 ¥Õ≈≈“√å
ºâ“À¡÷° 6.93 ¥Õ≈≈“√å
«ß≈âÕ‡§√◊ËÕßæ‘¡æå 9.76 ¥Õ≈≈“√å
°√–¥“…æ‘¡æå 8.21 ¥Õ≈≈“√å
À“º≈∫«°¢Õß∑»π‘¬¡ ª√–¡“≥ 0.2 + 0.9 + 0.7 + 0.2 ≈ 2.0 ¥Õ≈≈“√å
º≈∫«°®”π«π‡μÁ¡‰¥â 26 ¥Õ≈≈“√å
¥—ßπ—Èπ √“§“ª√–¡“≥ 2 + 26 ≈ 28 ¥Õ≈≈“√å μÕ∫
8. «‘∏’∑” ºâ“º◊πÀπ÷Ëß°«â“ß 48 ‡´π쑇¡μ√
ª√–¡“≥‡ªìπ 50 ‡´π쑇¡μ√
¬“« 2.4 ¢Õߧ«“¡°«â“ß ≈ 2.5 × 50 ‡´π쑇¡μ√ ≈ 125 ‡´π쑇¡μ√
¥—ßπ—Èπ ¡’æ◊Èπ∑’˪√–¡“≥ 50 × 125 ≈ 6,250 μ“√“߇´π쑇¡μ√ μÕ∫
9. «‘∏’∑” ÿπ—¢¢Õß≈ÿß·´¡ Ÿß 37.8 ‡´π쑇¡μ√ ª√–¡“≥‡ªìπ 40 ‡´π쑇¡μ√
≈ÿß·´¡ Ÿß‡ªìπ 4.2 ‡∑à“¢Õß ÿπ—¢ ª√–¡“≥§«“¡ Ÿß‡ªìπ 4 ‡∑à“¢Õß ÿπ—¢
¥—ßπ—Èπ ≈ÿß·´¡ Ÿßª√–¡“≥ 4 × 40 ≈ 160 ‡´π쑇¡μ√ μÕ∫
10. «‘∏’∑” ¢—∫√∂®“°°√ÿ߇∑æœ ‰ª®—ßÀ«—¥μ√“¥¥â«¬§«“¡‡√Á«‡©≈’ˬ 79.46 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß ª√–¡“≥
‡ªìπ 80 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
„π‡«≈“ 63
10 ™—Ë«‚¡ß À√◊Õª√–¡“≥ 6
14 ≈ 25
4 ™—Ë«‚¡ß
®–¢—∫‰¥â√–¬–∑“ߪ√–¡“≥ 254
80× ≈ 500 °‘‚≈‡¡μ√
¥—ßπ—Èπ ¢—∫√∂‰¥â√–¬–∑“ߪ√–¡“≥ 500 °‘‚≈‡¡μ√ μÕ∫
11. «‘∏’∑” √Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“¡’¥â“π°«â“ß°«â“ß 12.4 ≈ 12.5 ‡¡μ√
¥â“𬓫¬“«°«à“ 2 ‡∑à“¢Õߥâ“π°«â“ßÕ¬Ÿà 5 ‡¡μ√
¥â“𬓫ª√–¡“≥ 2(12.5) + 5 ≈ 30 ‡¡μ√
¡’æ◊Èπ∑’˪√–¡“≥ 12.5 × 30 ≈ 375 μ“√“߇¡μ√
¥—ßπ—Èπ √Ÿª ’ˇÀ≈’ˬ¡π’È¡’æ◊Èπ∑’˪√–¡“≥ 375 μ“√“߇¡μ√ μÕ∫
12. «‘∏’∑” 𓬠¬“¡¡’‡ß‘π¥Õ≈≈“√å 834.54 ¥Õ≈≈“√å
ª√–¡“≥‡ªìπ 830 ¥Õ≈≈“√å
‡ß‘π 1 ¥Õ≈≈“√å ·≈°‡ß‘π∫“∑‰¥â 33.44 ∫“∑ ª√–¡“≥‡ªìπ 30 ∫“∑
𓬠¬“¡·≈°‡ß‘π∫“∑‰¥âª√–¡“≥ 830 × 30 ≈ 24,900 ∫“∑
¥—ßπ—Èπ 𓬠¬“¡·≈°‡ß‘π∫“∑‰¥âª√–¡“≥ 24,900 ∫“∑ μÕ∫
°“√§Ÿ≥‡™àππ’ȉ¡à‡ªìπ‰ªμ“¡°Æ°“√ªí¥‡»… ·μà®–‰¥â§à“„°≈⇧’¬ß§à“®√‘ß¡“°°«à“ ‡π◊ËÕß®“°®”π«π
Àπ÷Ëߪí¥≈ßÕ’°®”π«πÀπ÷Ëßªí¥¢÷Èπ
20
1
Page 50
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 250
1. «‘∏’∑” ≈¥√“§“ 117 − 86 = 31 ∫“∑
ª√–¡“≥‡ªìπ 30 ∫“∑
√â“π§â“μ‘¥√“§“°“߇°ß 117 ∫“∑ ª√–¡“≥‡ªìπ 120 ∫“∑
¥—ßπ—Èπ ≈¥√“§“ª√–¡“≥√âÕ¬≈– 30120
100× ≈ 25
¥—ßπ—Èπ √â“π§â“≈¥√“§“„Àâª√–¡“≥√âÕ¬≈– 25 μÕ∫
2. «‘∏’∑” ¢—∫√∂√–¬–∑“ß 753 °‘‚≈‡¡μ√ ª√–¡“≥ 750 °‘‚≈‡¡μ√
„™âπÈ”¡—π‰ª 58.7 ≈‘μ√ ª√–¡“≥ 60 ≈‘μ√
¥—ßπ—Èπ πÈ”¡—π 1 ≈‘μ√ ®–¢—∫√∂‰¥âª√–¡“≥ 75060
≈ 12.5 °‘‚≈‡¡μ√
¥—ßπ—Èπ ‚¥¬‡©≈’ˬπÈ”¡—π 1 ≈‘μ√ ®–¢—∫√∂‰¥âª√–¡“≥ 12.5 °‘‚≈‡¡μ√ μÕ∫
3. «‘∏’∑” Õ“¿“æ√¢—∫√∂Õ¬ŸàÀà“ß®“°æ‘…≥ÿ‚≈° 135 °‘‚≈‡¡μ√
·≈–¢—∫√∂¥â«¬§«“¡‡√Á«‡©≈’ˬ 75 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß
¥—ßπ—Èπ ®–„™â‡«≈“ª√–¡“≥ 13575
≈ 2 ™—Ë«‚¡ß
®“°‡«≈“ 13.15 π. „™â‡«≈“Õ’°ª√–¡“≥ 2 ™—Ë«‚¡ß ®–‡ªìπ‡«≈“ 15.15 π“Ãî°“
¥—ßπ—Èπ Õ“¿“æ√®–‰ª∑—π‡ªî¥ß“π∑’Ëæ‘…≥ÿ‚≈°‡«≈“ 17.00 π. μÕ∫
4. «‘∏’∑” ‡ ◊ÈÕμ—«≈– 129 ∫“∑ ´◊ÈÕ 2 μ—« √“§“ª√–¡“≥ 2 × 130 ∫“∑ À√◊Õª√–¡“≥ 260 ∫“∑
ª“°°“ 1 ¥â“¡ √“§“ 98 ∫“∑ ª√–¡“≥‡ªìπ 100 ∫“∑
¥—ßπ—Èπ ´◊ÈÕ‡ ◊ÈÕ·≈–ª“°°“‡ªìπ‡ß‘πª√–¡“≥ 260 + 100 ∫“∑ À√◊Õª√–¡“≥ 360 ∫“∑
¿“…’¡Ÿ≈§à“‡æ‘Ë¡Õ’°ª√–¡“≥ 36 ∫“∑
¥—ßπ—Èπ μâÕß®à“¬‡ß‘πª√–¡“≥ 360 + 36 ∫“∑ À√◊Õª√–¡“≥ 396 ∫“∑
¥—ßπ—Èπ √“§“∑’Ëæπ—°ß“π∫Õ°§◊Õ 479 ∫“∑ ®÷߇ªìπ‰ª‰¡à‰¥â μÕ∫
5. «‘∏’∑” »√’‰æ√´◊ÈÕ ‘π§â“Õ¬à“ßÀπ÷Ëß≈¥√“§“®“° 11,895 ∫“∑ ‡ªìπ 8,900 ∫“∑
¥—ßπ—Èπ ≈¥√“§“ª√–¡“≥ 12,000 − 9,000 ≈ 3,000 ∫“∑
≈¥√“§“ ≈ 3000 10012000
%
×
≈ 25%
¥—ßπ—Èπ ‘π§â“≈¥√“§“ª√–¡“≥√âÕ¬≈– 25 μÕ∫
·∫∫Ωñ°À—¥ 3
Page 51
51·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
6. Õ“√¬“‰ª À√—∞Õ‡¡√‘°“
(1) «‘∏’∑” Õ“√¬“√—∫ª√–∑“πÕ“À“√ 37.35 ¥Õ≈≈“√å
‡ ’¬¿“…’√âÕ¬≈– 5 ·≈–‡ ’¬§à“∫√‘°“√√âÕ¬≈– 15 ¢Õß√“§“
¥—ßπ—Èπ μâÕ߇ ’¬¿“…’·≈–§à“∫√‘°“√ 20100
37 35× . ¥Õ≈≈“√å
≈ 20100
40× ¥Õ≈≈“√å
≈ 8 ¥Õ≈≈“√å
¥—ßπ—Èπ Õ“√¬“μâÕß™”√–‡ß‘πª√–¡“≥ 37 + 8 ¥Õ≈≈“√å
≈ 45 ¥Õ≈≈“√å μÕ∫
(2) «‘∏’∑” §à“ —́°ºâ“™—Ë«‚¡ß≈– 3.65 ¥Õ≈≈“√å
Õ“√¬“¡’‡À√’¬≠Õ¬Ÿà 25 ¥Õ≈≈“√å
Õ“√¬“μâÕß°“√´—°ºâ“ 8 ™—Ë«‚¡ß
∂â“μâÕß°“√μ√«® Õ∫‚¥¬§‘¥„π„®«à“¡’‡ß‘πæÕÀ√◊Õ‰¡à «‘∏’∑’˧‘¥ßà“¬§◊Õ§‘¥®“°À“§à“ª√–¡“≥
¢Õß 3.65 × 8
‡π◊ËÕß®“° 8 × 3.65 ¥Õ≈≈“√å ≈ 8 × 3.7 ¥Õ≈≈“√å
≈ 29.6 ¥Õ≈≈“√å
¥—ßπ—Èπ Õ“√¬“¡’‡ß‘π‰¡àæÕ μÕ∫
7. (1) «‘∏’∑” ¡»√’´◊ÈÕÀßÕπ‰°à 3 μâπ ·≈–¡–¡à«ß 1 μâπ
®–μâÕß®à“¬‡ß‘π (3 × 75) + 143 ≈ (3 × 80) + 140 ∫“∑
≈ 380 ∫“∑ μÕ∫
(2) «‘∏’∑” ‡¥à𥓫¡’‡ß‘π 1,000 ∫“∑
∂â“´◊ÈÕ¡–¡à«ß 6 μâπ ·≈–ÀßÕπ‰°à 1 μâπ
®–μâÕß®à“¬‡ß‘π (6 × 143) + 75 ≈ (6 × 140) + 80 ∫“∑
≈ 840 + 80 ∫“∑
≈ 920 ∫“∑ μÕ∫
(3) «‘∏’∑” Õ¥ÿ≈¬å´◊ÈÕ®”ªï 3 μâ𠇪ìπ‡ß‘π 3 × 99 ≈ 3 × 100 ∫“∑
≈ 300 ∫“∑
„Àâ∏π∫—μ√„∫≈– 500 ∫“∑ ®÷߉¥â‡ß‘π∑Õπ
≈ 500 − 300 ∫“∑
≈ 200 ∫“∑
¥—ßπ—È𠉥â√—∫‡ß‘π∑Õπª√–¡“≥ 200 ∫“∑ μÕ∫
Page 52
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 252
(4) «‘∏’∑” ª√–™“ ª√’¥“ ·≈–‡ª√¡ √«¡‡ß‘π°—π´◊ÈÕμâπ‰¡â∑—Èß 4 ™π‘¥ ™π‘¥≈– 1 μâ𠇪ìπ‡ß‘π
53 + 75 + 99 + 143 ∫“∑
„™â°“√ªí¥‡»… ®–‰¥â 50 + 80 + 100 + 140 = 370 ∫“∑
¥—ßπ—Èπ ®–μâÕß®à“¬‡ß‘π§π≈– 3703
≈ 123 ∫“∑
¥—ßπ—Èπ μâÕß®à“¬‡ß‘π§π≈–ª√–¡“≥ 123 ∫“∑ μÕ∫
8. (1) «‘∏’∑” §√◊ËÕßμ—¥À≠â“√“§“ 8,650 ∫“∑
≈¥√“§“ 35%
‡π◊ËÕß®“° 10% ‡∑à“°—∫ 865 ∫“∑ ≈ 870 ∫“∑
¥—ßπ—Èπ 30% ª√–¡“≥ 2,610 ∫“∑
·≈– 5% ª√–¡“≥ 435 ∫“∑
√«¡ 35% ‰¥â à«π≈¥ ≈ 2,610 + 435 ∫“∑
≈ 2,600 + 400 ∫“∑
≈ 3,000 ∫“∑ μÕ∫
(2) «‘∏’∑” ¡ÿ¥∫—π∑÷°√“§“ 289 ≈ 300 ∫“∑
≈¥√“§“ 15% ª√–¡“≥ 15100
300× ≈ 45 ∫“∑ μÕ∫
(3) «‘∏’∑” ≈Ÿ°∫“ ‡°μ∫Õ≈√“§“ 859 ∫“∑ ≈¥√“§“ 40%
≈¥√“§“ 40100
859× ∫“∑ ≈ 410
860× ∫“∑
≈ 340 ∫“∑
π—Ëπ§◊Õ ≈¥√“§“ª√–¡“≥ 340 ∫“∑ μÕ∫
(4) «‘∏’∑” ‡ ◊ÈÕ°—πÀπ“«√“§“ 425 ∫“∑ ≈¥√“§“ 33%
≈¥√“§“ 33100
425× ∫“∑
≈ 33 400100× ∫“∑
≈ 135 ∫“∑
¥—ßπ—Èπ ≈¥√“§“ª√–¡“≥ 135 ∫“∑ μÕ∫
4
Page 53
1. (1, 6), (2, 12), (3, 18), (4, 24), (5, 30), (6, 36) ·≈– (7, 42)
2. ®“°¢âÕ 1
(6, 36) À¡“¬∂÷ß „™âπÈ”¡–π“« 6 ∂⫬ „™âπÈ”μ“≈ 36 °âÕπ
(6, 36) ·≈– (36, 6) ¡’§«“¡À¡“¬‰¡à‡À¡◊Õπ°—π ‡æ√“–«à“
(6, 36) À¡“¬∂÷ß „™âπÈ”¡–π“« 6 ∂⫬ „™âπÈ”μ“≈ 36 °âÕπ
(36, 6) À¡“¬∂÷ß „™âπÈ”¡–π“« 36 ∂⫬ „™âπÈ”μ“≈ 6 °âÕπ
3. 72 °âÕπ 4. 9 ∂⫬
1. (1, 10), (2, 20), (3, 30), (4, 40), (5, 50), (6, 60) 2. ‚μä– 1 μ—« ¡’‡°â“Õ’È 10 μ—«
3. ‚μä– x μ—« ¡’‡°â“Õ’È y μ—«
1. (0, 8) 2. (9, 9) 3. (1, 6) 4. (6, 3) 5. (4, 0) 6. (0, 0)
μÕ∫§”∂“¡®“°μ—«Õ¬à“ß∑’Ë 1 Àπâ“ 118
1. 49 ∫“∑ 2. 300 ∫“∑ 3. Õ¬Ÿà„π·π«‡ âπμ√߇¥’¬«°—π
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 1
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 2
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 3
Àπ૬°“√‡√’¬π√Ÿâ∑’Ë 3
§ŸàÕ—π¥—∫·≈–°√“ø
‡©≈¬
Page 54
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 254
(2, 440), (3, 660), (4, 880), (5, 1,100), (6, 1,320), (7, 1,540)
1. (1) (1, 6), (2, 7), (3, 8), (4, 9)
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 4
·∫∫Ωñ°À—¥ 1
Y ª√‘¡“≥πÈ”π¡ (¡‘≈≈‘≈‘μ√)
X ®”π«π (¢«¥)
1,760
1,540
1,320
1,100
880
660
440
220
1 2 3 4 5 6 7 8 9 10 11 12 13
(1, 220)
(2, 440)
(3, 660)
(4, 880)
(5, 1,100)
(6, 1,320)
(7, 1,540)
O
1
1 2 3 4
2
3
4
5
6
7
8
9
(1, 6)
(2, 7)
(3, 8)
(4, 9)
O
Page 55
55·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(2) (3, 1), (3, 2), (3, 3), (3, 4)
(3) (1, 2), (2, 2), (3, 2), (4, 2), (5, 2)
(4) (1, 6), (2, 12), (3, 18), (4, 24), (5, 30)
2. (3, 1), (6, 2), (9, 3), (12, 4), (15, 5), (18, 6)
3. (1) (8, 3) (2) (10, 0) (3) (3, 5) (4) (0, 6)
4. A(3, 5), B(8, 3), C(10, 0), D(6, 0), E(3, 8), F(5, 4), G(0, 9) ·≈– H(0, 6)
5. (1) B(7, 8), C(6, 6), D(8, 5) (2) A(2, 8), C(3, 7), D(5, 6)
O
1
1 2 3 4 5
2
3
4 (3, 4)
(3, 3)
(3, 2)
(3, 1)
O
1
1 2 3 4 5
2
3
4
(1, 2) (2, 2) (3, 2) (4, 2) (5, 2)
O
6
1 2 3 4 5
12
18
24
30
(4, 24)
(5, 30)
(3, 18)
(2, 12)
(1, 6)
Page 56
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 256
6. (1) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 10 Àπ૬ Àà“ß®“°·°π X ‡ªìπ√–¬– 7 Àπ૬ ¡’æ‘°—¥ (10, 7)
(2) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 7 Àπ૬ Àà“ß®“°·°π X ‡ªìπ√–¬– 2 Àπ૬ ¡’æ‘°—¥ (7, 2)
(3) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 0 Àπ૬ Àà“ß®“°·°π X ‡ªìπ√–¬– 7 Àπ૬ ¡’æ‘°—¥ (0, 7)
(4) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 2 Àπ૬ Àà“ß®“°·°π X ‡ªìπ√–¬– 3 Àπ૬ ¡’æ‘°—¥ (2, 3)
(5) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 4 Àπ૬ Àà“ß®“°·°π X ‡ªìπ√–¬– 0 Àπ૬ ¡’æ‘°—¥ (4, 0)
(6) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 4 Àπ૬ Àà“ß®“°·°π X ‡ªìπ√–¬– 6 Àπ૬ ¡’æ‘°—¥ (4, 6)
7. (1) (0, 7) ‡ªìπæ‘°—¥¢Õß√Ÿª (2) (2, 3) ‡ªìπæ‘°—¥¢Õß√Ÿª
(3) (4, 0) ‡ªìπæ‘°—¥¢Õß√Ÿª (4) (4, 6) ‡ªìπæ‘°—¥¢Õß√Ÿª
(5) (7, 2) ‡ªìπæ‘°—¥¢Õß√Ÿª (6) (10, 7) ‡ªìπæ‘°—¥¢Õß√Ÿª
8.
‡¢’¬π§ŸàÕ—π¥—∫¢Õߧ«“¡ —¡æ—π∏剥⇪ìπ (0, 3), (1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9), (7, 10)
μ—«μ—Èß º≈≈—æ∏å
0 3
1 4
2 5
3 6
4 7
5 8
6 9
7 10
.......
.......
.......
.......
.......
.......
.... .... ....... ....... ....... ....... .........
Page 57
57·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1.
2.
·∫∫Ωñ°À—¥ 2
μ—«μ—Èß º≈≈—æ∏å
4 6
3 5
2 4
1 3
0 2
− 1 1
− 2 0
− 3 − 1
− 4 − 2
......
.......
.......
.......
.......
.......
11
10
9
8
7
6
5
4
3
2
10
− 1
− 2
− 3
− 4
− 5
− 6
− 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 7
Y
X
Page 58
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 258
3.
‡¢’¬π§ŸàÕ—π¥—∫‰¥â¥—ßπ’È (4, 2), (3, 1), (2, 0), (1, − 1), (0, − 2), (− 1, − 3), (− 2, − 4), (− 3, − 5), (− 4, − 6)
μ—«μ—Èß º≈≈—æ∏å
4 2
3 1
2 1
1 −1
0 −2
− 1 −3
− 2 −4
− 3 − 5
− 4 − 6
......
.......
.......
.......
O
(3, 1)
(4, 2)
(2, 0)
(1, �1)
(0, �2)
(�2, �4)
(�3, �5)
(�4, �6)
(�1, �3)
‡¢’¬π§ŸàÕ—π¥—∫‰¥â¥—ßπ’È (4, 6), (3, 5), (2, 4), (1, 3), (0, 2), (− 1, 1), (− 2, 0), (− 3, − 1), (− 4, − 2)
O
(3, 5)
(4, 6)
(2, 4)
(0, 2)
(1, 3)
(�4, �2)
(�2, 0)
(�1, 1)
(�3, �1)
Y
X
Y
X
.... .... .... .... ...... ......
...... ...... ...... ......
Page 59
59·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
1. (1, 10), (2, 7), (3, 4), (4, 2), (5, 1), (6, 2), (7, 4), (8, 7), (9, 10)
2.
(1) 35 Õß»“‡´≈‡ ’́¬
(2) 100 Õß»“‡´≈‡ ’́¬
·∫∫Ωñ°À—¥ 3
O
1
1 2 3 4 5 6 7 8 9
2
3
4
5
6
7
8
9
10
(3, 4)
(4, 2) (6, 2)
(5, 1)
(7, 4)
(2, 7)(8, 7)
(1, 10) (9, 10)
O
10
4 8 12 16 20 24 28 32 36 40
20
30
40
50
60
70
80
90
100
(4, 20)
(8, 30)
(12, 40)
(16, 60)
(20, 80)
(25, 90)
(40, 100)
(30, 100)
‡«≈“ (π“∑’)
Õÿ≥À¿Ÿ¡‘ (Õß»“‡´≈‡´’¬ )
Page 60
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 260
3.
(1) 4 «—π (2) 7.5 ‡´π쑇¡μ√ (3) 0.86 ‡´π쑇¡μ√
4. (1) 03.00 π. ¡’Õÿ≥À¿Ÿ¡‘ 20 Õß»“‡´≈‡´’¬
(2) 12.00 π. ¡’Õÿ≥À¿Ÿ¡‘ 32 Õß»“‡´≈‡´’¬
(3) 12 Õß»“‡´≈‡´’¬
(4) 26 Õß»“‡´≈‡´’¬
(5) 03.00 ∂÷ß 12.00 π. μ—Èß·μà 20 Õß»“‡´≈‡´’¬ ∂÷ß 32 Õß»“‡´≈‡´’¬
5. (1) 70,000 ∫“∑ (2) 10 ‡¥◊Õπ (3) 18 ‡¥◊Õπ (4) 71,250 ∫“∑
6. (1) 3.5 «‘π“∑’ (2) 5.33 ‡¡μ√μàÕ«‘π“∑’
(3) ‰¡à∑—π ‡æ√“–°√“ø¢Õ߇ âπ‚§âß®–‰¡àμ—¥°—πÕ’°
O
3
3 6 9 12 15 18 21 24 27 30
6
9
12
15
18
21
24
27
(1, 0.7)
(3, 2.0)
(7, 3.5)
(10, 5.4)
(15, 10.5)
(18, 15.2)
(20, 20.0)(25, 21.5)
‡«≈“ («—π)
§«“¡ Ÿß (‡´π쑇¡μ√)
Page 61
1. (1) 20 (2) 30 (3) 21 (4) 55
(5) 47 (6) 262 (7) 82 (8) 1,023
2. (1) 3n−2 (2) 3n−1 (3) 4n−2 (4) n2�1
(5) 2n
(6) 3n
(7) 4n (8) 5n
3. (1) 51 (2) 4n+3
(3) ·°â‚®∑¬å‡ªìπ ∂â“®”π«π„π≈”¥—∫∑’Ë n ‡∑à“°—∫ 63 ®ßÀ“ n
μÕ∫ 15
1. ‡ªìπ 2. ‰¡à‡ªìπ 3. ‡ªìπ 4. ‰¡à‡ªìπ 5. ‰¡à‡ªìπ 6. ‡ªìπ
1. 8 2. 5 3. 2 4. 10 − 2
5. 8 6. 5 + 8 7. 10 − 3 8. 7 + 5
·∫∫Ωñ°À—¥ 1
Àπ૬°“√‡√’¬π√Ÿâ∑’Ë 4
¡°“√‡™‘߇ âπμ—«·ª√‡¥’¬«
‡©≈¬
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 1
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 2
Page 62
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 262
1. «‘∏’∑” x + 15 = 37 2. «‘∏’∑” y + 18 = 42
x + 15 − 15 = 37 − 15 y + 18 − 18 = 42 − 18
x = 22 μÕ∫ y = 24 μÕ∫
3. «‘∏’∑” z − 16 = 11 4. «‘∏’∑” a − 13 = 14
z − 16 + 16 = 11 + 16 a − 13 + 13 = 14 + 13
z = 27 μÕ∫ a = 27 μÕ∫
5. «‘∏’∑” q + 37 = 64 6. «‘∏’∑” 13 = 18 − c
q + 37 − 37 = 64 − 37 13 + c = 18 − c + c
q = 27 μÕ∫ 13 + c = 18
13 + c − 13 = 18 − 13
c = 5 μÕ∫
7. «‘∏’∑” 15 + b = 48 8. «‘∏’∑” p − 19 = 37
15 + b − 15 = 48 − 15 p − 19 + 19 = 37 + 19
b = 33 μÕ∫ p = 56 μÕ∫
1. 2 2. 5 3. 9 4. 3 5. 2
6. 3 7.54
1. «‘∏’∑”18
x = 12 2. «‘∏’∑”19
y = 7
818
x⎛⎝
⎞⎠ = 8(12) 9
19
y⎛⎝
⎞⎠ = 9(7)
x = 96 μÕ∫ y = 63 μÕ∫
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 3
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 4
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 5
Page 63
63·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
3. «‘∏’∑” 5z = 105 4. «‘∏’∑” 12w = 84
15
5( )z = 15
105( )1
1212( )w = 1
1284( )
z = 21 μÕ∫ w = 7 μÕ∫
5. «‘∏’∑” 8a = 46 6. «‘∏’∑”c2
= 37
18
8( )a = 18
46( ) 22c⎛
⎝⎞⎠ = 2(37)
a = 468
c = 74 μÕ∫
= 234
μÕ∫
7. «‘∏’∑” 4 × p = 104 8. «‘∏’∑” b ÷ 6 = 49
14
4( )p = 14
104( )b6
= 49
p = 26 μÕ∫ 66b⎛
⎝⎞⎠ = 6(49)
b = 294 μÕ∫
1. (1) 6 (2) 6 (3) 3 (4) 15
(5) 20 − 16 (6) 20 − 16 (7) 8 (8) 27 + 8
2. (1) 5 (2) 6 (3) 7 (4) 5 (5) 3
(6) 7 (7)147
(8) 3 (9) 5 × 3
3. (1) «‘∏’∑” x + 8 = 47 (2) «‘∏’∑” a − 9 = 34
x + 8 − 8 = 47 − 8 a − 9 + 9 = 34 + 9
x = 39 μÕ∫ a = 43 μÕ∫
(3) «‘∏’∑” y + 5 = 108 (4) «‘∏’∑” z − 46 = 59
y + 5 − 5 = 108 − 5 z − 46 + 46 = 59 + 46
y = 103 μÕ∫ z = 105 μÕ∫
·∫∫Ωñ°À—¥ 2
Page 64
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 264
(5) «‘∏’∑” 45 + a = 142 (6) «‘∏’∑” 23 = 38 − b
45 + a − 45 = 142 − 45 23 + b = 38 − b + b
a = 97 μÕ∫ 23 + b = 38
23 + b − 23 = 38 − 23
b = 15 μÕ∫
(7) «‘∏’∑” 34 + a = 132 (8) «‘∏’∑” 15 = 37 − x
34 + a − 34 = 132 − 34 15 + x = 37 − x + x
a = 98 μÕ∫ 15 + x = 37
15 + x − 15 = 37 − 15
x = 22 μÕ∫
(9) «‘∏’∑” 35 + c = 67 (10) «‘∏’∑” 36 = 45 − a
35 + c − 35 = 67 − 35 36 + a = 45 − a + a
c = 32 μÕ∫ 36 + a = 45
36 + a − 36 = 45 − 36
a = 9 μÕ∫
(11) «‘∏’∑” 5x = 400 (12) «‘∏’∑” 6y = 165
55x = 400
566y = 165
6
x = 80 μÕ∫ y = 552
μÕ∫
(13) «‘∏’∑” 3z = 39 (14) «‘∏’∑” 7c = 42
33z = 39
377c = 42
7
z = 13 μÕ∫ c = 6 μÕ∫
(15) «‘∏’∑”a5
= 13 (16) «‘∏’∑”a8
= 13
55a⎛
⎝⎞⎠ = 5(13) 8
8a⎛
⎝⎞⎠ = 8(13)
a = 65 μÕ∫ a = 104 μÕ∫
(17) «‘∏’∑” 3 = 15a
(18) «‘∏’∑”b4
= 30
3 × a = 15a
a× 44
× b = 4 × 30
3a = 15 b = 120 μÕ∫
33a = 15
3
a = 5 μÕ∫
Page 65
65·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
(19) «‘∏’∑” 7 = 42c
(20) «‘∏’∑”25
x = 14
7 × c = 42c
c× 52
25
× x = 52
14×
77c = 42
7x = 35 μÕ∫
c = 6 μÕ∫
1. «‘∏’∑” 3x − 1 = 11 2. «‘∏’∑” 12x + 5 = 20
3x − 1 + 1 = 11 + 1 12x + 5 − 5 = 20 − 5
3x = 12 12x = 15
33x = 12
31212
x = 1512
x = 4 μÕ∫ x = 54
μÕ∫
3. «‘∏’∑” 6y + 13 = 49 4. «‘∏’∑” 2x − 8 = 26
6y + 13 − 13 = 49 − 13 2x − 8 + 8 = 26 + 8
6y = 36 2x = 34
66y = 36
622x = 34
2
y = 6 μÕ∫ x = 17 μÕ∫
5. «‘∏’∑” 18 = 3x − 6 6. «‘∏’∑” 29 = 14 + 5a
¥—ßπ—Èπ 3x − 6 = 18 ¥—ßπ—Èπ 14 + 5a = 29
3x − 6 + 6 = 18 + 6 14 + 5a − 14 = 29 − 14
3x = 24 5a = 15
33x = 24
355a = 15
5
x = 8 μÕ∫ a = 3 μÕ∫
7. «‘∏’∑”34
3b + = 33 8. «‘∏’∑” 8
13
+ a = 75
34
3 3b + − = 33 − 3 8
13
8+ −a = 75 − 8
34b = 30
13
a = 67
43
34
× b = 43
30× 313
a⎛⎝
⎞⎠ = 3 × 67
b = 40 μÕ∫ a = 201 μÕ∫
·∫∫Ωñ°À—¥ 3
Page 66
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 266
9. «‘∏’∑”23
5x − = 11 10. «‘∏’∑”32
5k + = 17
23
5 5x − + = 11 + 532
5 5k + − = 17 − 5
23
x = 1632k = 12
32
23
× x = 32
16× 23
32
× k = 23
12×
x = 24 μÕ∫ k = 8 μÕ∫
11. «‘∏’∑”1
1223
a + = 56
12. «‘∏’∑” 635
a − = 125
π” 12 §Ÿ≥∑—Èß 2 ¢â“ߢÕß ¡°“√ 635
35
a − + = 125
35
+
121
1212
23
a⎛⎝
⎞⎠ + ⎛
⎝⎞⎠ = 12
56
⎛⎝
⎞⎠ 6a = 3
a + 8 = 1016
6× a = 16
3×
a + 8 − 8 = 10 − 8 a = 12
μÕ∫
a = 2 μÕ∫
13. «‘∏’∑” 4(x + 2) = 16 14. «‘∏’∑”3
6+ x = 5
14
4 2× +( )x = 14
16× 63
6× +⎛
⎝⎞⎠
x = 6 × 5
x + 2 = 4 3 + x = 30
x + 2 − 2 = 4 − 2 3 + x − 3 = 30 − 3
x = 2 μÕ∫ x = 27 μÕ∫
15. «‘∏’∑”3
105( )p − = 6 16. «‘∏’∑” 6(5 + 2c) = 42
103
310
5× −( )p = 103
6× 16
6 5 2× +( )c = 16
42×
(p − 5) = 20 5 + 2c = 7
p − 5 + 5 = 20 + 5 5 + 2c − 5 = 7 − 5
p = 25 μÕ∫ 2c = 2
c = 1 μÕ∫
17. «‘∏’∑”x − 2
3= 4 18. «‘∏’∑”
23
1( )y + = 16
32
3x −⎛
⎝⎞⎠ = 3 × 4
32
23
1× +( )y = 32
16×
x − 2 = 12 y + 1 = 24
x = 14 μÕ∫ y = 23 μÕ∫
Page 67
67·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
19. «‘∏’∑”2 7
4x − = 8 20. «‘∏’∑” 4 = 2
312( )x −
42 7
4x−⎛
⎝⎜⎞⎠⎟
= 4 × 8 ¥—ßπ—Èπ 23
12( )x − = 4
2x − 7 = 3232
23
12× −( )x = 32
4×
2x − 7 + 7 = 32 + 7 x − 12 = 6
2x = 39 x − 12 + 12 = 6 + 12
x = 392
μÕ∫ x = 18 μÕ∫
21. «‘∏’∑” 12 − 6x = 4 22. «‘∏’∑” 1712
− a = 5
12 − 6x + 6x = 4 + 6x 1712
12
− +a a = 512
+ a
12 = 4 + 6x 17 = 512
+ a
12 − 4 = 4 + 6x − 4 17 − 5 = 512
5+ −a
8 = 6x 12 = 12
a
86
= 66x
2 × 12 = 212
× a
43
= x 24 = a
À√◊Õ x = 43
μÕ∫ À√◊Õ a = 24 μÕ∫
23. «‘∏’∑” 112
12
− a = 34
24. «‘∏’∑” 27.1 = 4 + 0.3b
32
12
− a = 34
π” 10 §Ÿ≥∑—Èß 2 ¢â“ߢÕß ¡°“√
32
12
12
− +a a = 34
12
+ a 271 = 40 + 3b
32
= 34
12
+ a 271 − 40 = 40 + 3b − 40
π” 4 §Ÿ≥∑—Èß 2 ¢â“ߢÕß ¡°“√ 231 = 3b
6 = 3 + 2a2313
= 33b
6 − 3 = 3 + 2a − 3 77 = b
3 = 2a À√◊Õ b = 77 μÕ∫
32
= a
À√◊Õ a = 32
μÕ∫
Page 68
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 268
25. «‘∏’∑” 48.1 + 0.2x = 52.5 26. «‘∏’∑” 31 55
. +x
= 56.5
π” 10 §Ÿ≥∑—Èß Õߢâ“ߢÕß ¡°“√ 31 55
31 5. .+ −x
= 56.5 − 31.5
481 + 2x = 5255x
= 25
481 + 2x − 481 = 525 − 4815x
x× = 25x
2x = 44 5 = 25x
x = 22 μÕ∫525
= 2525
x
15
= x
À√◊Õ x = 15
μÕ∫
1. (1) 3x + 5 = 41 (2) 3(x + 7) = 51 (3) 3x + 4 = 100 (4)23
5x − = 7
(5) 6x − 7 = 65
2. «‘∏’∑” „ÀâÀ¬Õ¥‡À√’¬≠„ à°√–ªÿ° °√–ªÿ°≈– x ‡À√’¬≠
®–‰¥â ¡°“√ 4x + 3 = 59 μÕ∫
3. «‘∏’∑” „Àâ∫ÿμ√§π‡≈Á°¡’Õ“¬ÿ x ªï
∫ÿμ√§π‚μÕ“¬ÿ¡“°°«à“∫ÿμ√§π‡≈Á°Õ¬Ÿà 4 ªï
¥—ßπ—Èπ ∫ÿμ√§π‚μ¡’Õ“¬ÿ x + 4 ªï
∫ÿμ√§π‚μ·≈–∫ÿμ√§π‡≈Á°¡’Õ“¬ÿ√«¡°—π (x + 4) + x = 2x + 4 ªï
∫‘¥“Õ“¬ÿ 48 ªï ∫ÿμ√§π‚μ·≈–∫ÿμ√§π‡≈Á°¡’Õ“¬ÿ√«¡°—π‡ªìπ 13 ¢ÕßÕ“¬ÿ∫‘¥“
®–‰¥â ¡°“√ 2x + 4 = 13
48× μÕ∫
4. «‘∏’∑” §«“¡¬“«√Õ∫ÀâÕ߇√’¬π‡ªìπ 40 ‡¡μ√
ÀâÕß¡’§«“¡°«â“ß 9 ‡¡μ√
„ÀâÀâÕß¡’§«“¡¬“« x ‡¡μ√
®–‰¥â ¡°“√ 2(x + 9) = 40 μÕ∫
5. «‘∏’∑” „Àâª√’™“¡’πÈ”Àπ—° x °‘‚≈°√—¡
∫—ßÕ√¡’πÈ”Àπ—°‡ªìπ 37 ‡∑à“¢ÕßπÈ”Àπ—°ª√’™“ = 3
7x °‘‚≈°√—¡
·μà∫—ßÕ√¡’πÈ”Àπ—° 28 °‘‚≈°√—¡
®–‰¥â ¡°“√ 37
x = 28 μÕ∫
°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 6
Page 69
69·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
μÕπ∑’Ë 1
1. «‘∏’∑” π“¬¡¥‰¥â√—∫∑’Ë∑”°‘π®“°√—∞∫“≈ k ‰√à
𓬥â«ß‰¥â√—∫∑’Ë∑”°‘π®“°√—∞∫“≈ 168 ‰√à
𓬥â«ß‰¥â√—∫∑’Ë∑”°‘π®“°√—∞∫“≈‡ªìπ 3 ‡∑à“¢Õßπ“¬¡¥
®–‰¥â ¡°“√ 3k = 168 μÕ∫
2. «‘∏’∑” 𓬂√®πå∑”𓉥â¢â“«‡ª≈◊Õ° y ‡°«’¬π
π“¬√ÿàß∑”𓉥â¢â“«‡ª≈◊Õ° 12.7 ‡°«’¬π
π“¬√ÿàß∑”𓉥â¢â“«‡ª≈◊Õ° 15 ¢Õß𓬂√®πå
®–‰¥â ¡°“√ 15
y = 12.7 μÕ∫
3. «‘∏’∑” ·¥ß¡’≈Ÿ°À‘πÕ¬Ÿà p ≈Ÿ°
¥”¡’≈Ÿ°À‘πÕ¬Ÿà 61 ≈Ÿ°
¥”¡’≈Ÿ°À‘π¡“°°«à“ 2 ‡∑à“¢Õß·¥ßÕ¬Ÿà 5 ≈Ÿ°
®–‰¥â ¡°“√ 2p + 5 = 61 μÕ∫
4. «‘∏’∑” ™Õ∫‡≈’Ȭߪ≈“∑ÕßÕ¬Ÿà q μ—«
™‘¥‡≈’Ȭߪ≈“∑ÕßÕ¬Ÿà 15 μ—«
™‘¥‡≈’Ȭߪ≈“∑ÕßπâÕ¬°«à“ 23 ‡∑à“¢Õß™Õ∫Õ¬Ÿà 5 μ—«
®–‰¥â ¡°“√ 23
5q − = 15 μÕ∫
5. «‘∏’∑” 3 ‡∑à“¢Õߺ≈∫«°¢Õß a °—∫ 7 ‡ªìπ 210
‡¢’¬π ¡°“√‰¥â‡ªìπ 3(a + 7) = 210 μÕ∫
6. «‘∏’∑”34 ‡∑à“¢Õߺ≈∫«°¢Õß b °—∫ 10 ‡ªìπ 18
‡¢’¬π ¡°“√‰¥â‡ªìπ 34
10( )b + = 18 μÕ∫
7. «‘∏’∑”56 ‡∑à“¢Õߺ≈μà“ߢÕß t °—∫ 17 ‡ªìπ 25 ‚¥¬∑’Ë t ¡’§à“¡“°°«à“ 17
‡¢’¬π ¡°“√‰¥â‡ªìπ 56
17( )t − = 25 μÕ∫
8. «‘∏’∑” 5 ‡∑à“¢Õß z ¡“°°«à“ 130 Õ¬Ÿà 45
‡¢’¬π ¡°“√‰¥â‡ªìπ 5z − 130 = 45 μÕ∫
9. «‘∏’∑” º≈§Ÿ≥¢Õß a ·≈– 47 ‡ªìπ 1
14
‡¢’¬π ¡°“√‰¥â‡ªìπ 47
a = 114
μÕ∫
·∫∫Ωñ°À—¥ 4
Page 70
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 270
μÕπ∑’Ë 2
1. «‘∏’∑” „À⇮’ͬ∫¡’‡ß‘πÕ¬Ÿà x ∫“∑
·¡à„Àâ¡“Õ’° 350 ∫“∑
‡¡◊ËÕπ”‡ß‘π∑’Ë¡’Õ¬Ÿà‡¥‘¡¡“π—∫√«¡°—∫‡ß‘π∑’Ë·¡à„Àâ®–π—∫‰¥â 678 ∫“∑
®–‰¥â ¡°“√ x + 350 = 678 μÕ∫
x = 678 − 350
= 328
¥—ßπ—È𠇥‘¡‡®’ͬ∫¡’‡ß‘πÕ¬Ÿà 328 ∫“∑ μÕ∫
2. «‘∏’∑” „À⇥‘¡‡¥àπ¡’¥‘π ÕÕ¬Ÿà x ·∑àß
‡¥àπ𔥑π Õ®”π«ππ’È·∫àß„À⇥Á° 36 §π §π≈– 4 ·∑àß
ª√“°Ø«à“¬—߇À≈◊Õ¥‘π ÕÕ¬ŸàÕ’° 6 ·∑àß
®–‰¥â ¡°“√ x = (36 × 4) + 6
x = 150
¥—ßπ—È𠇥‘¡‡¥àπ¡’¥‘π ÕÕ¬Ÿà 150 ·∑àß μÕ∫
3. «‘∏’∑” „ÀâπÈ”º÷Èß´◊ÈÕ ¡ÿ¥¡“ x ‡≈à¡
π”¡“√«¡°—∫∑’Ë¡’Õ¬Ÿà‡¥‘¡Õ’° 148 ‡≈à¡
·®°„Àâπ—°‡√’¬π 43 §π §π≈– 6 ‡≈à¡ ·®°‰¥âæÕ¥’
®–‰¥â ¡°“√ x + 14843
= 6
43 14843
( )x + = 43 × 6
x + 148 = 258
x + 148 − 148 = 258 − 148
x = 110
¥—ßπ—Èπ πÈ”º÷Èß´◊ÈÕ ¡ÿ¥¡“ 110 ‡≈à¡ μÕ∫
4. «‘∏’∑” „Àⷥߡ’≈Ÿ°À‘πÕ¬Ÿà x ≈Ÿ°
¥”¡’≈Ÿ°À‘π¡“°°«à“ Õ߇∑à“¢Õß·¥ßÕ¬Ÿà 5 ≈Ÿ°
¥—ßπ—Èπ ¥”¡’≈Ÿ°À‘π 2x + 5
·μ॔¡’≈Ÿ°À‘π 61 ≈Ÿ°
®–‰¥â ¡°“√ 2x + 5 = 61
2x + 5 − 5 = 61 − 5
2x = 56
x = 562
= 28
¥—ßπ—Èπ ·¥ß¡’≈Ÿ°À‘πÕ¬Ÿà 28 ≈Ÿ° μÕ∫
Page 71
71·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
5. «‘∏’∑” „Àâ™Õ∫‡≈’Ȭߪ≈“∑Õß x μ—«
™‘¥‡≈’Ȭߪ≈“∑ÕßπâÕ¬°«à“ 23 ‡∑à“¢Õß™Õ∫ 5 μ—«
¥—ßπ—Èπ ™‘¥‡≈’Ȭߪ≈“∑Õß 23
5x − μ—«
·μà™‘¥‡≈’Ȭߪ≈“∑Õß 15 μ—«
®–‰¥â ¡°“√ 23
5x − = 15
23
x = 15 + 5
23
x = 20
x = 32
20×
= 30
¥—ßπ—Èπ ™Õ∫‡≈’Ȭߪ≈“∑Õß 30 μ—« μÕ∫
6. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëߧ◊Õ x
3 ‡∑à“¢Õߺ≈∫«°¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ 7 ‡∑à“°—∫ 210
®–‰¥â ¡°“√ 3(x + 7) = 210
x + 7 = 2103
x + 7 = 70
x = 70 − 7
= 63
¥—ßπ—Èπ ®”π«π®”π«ππ—Èπ§◊Õ 63 μÕ∫
7. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëߧ◊Õ x
∂â“π” 5 ¡“≈∫ÕÕ°®“°®”π«π®”π«πÀπ÷Ëß·≈â« “¡‡∑à“¢Õߺ≈μà“ßπ—Èπ§◊Õ 69
®–‰¥â ¡°“√ 3(x − 5) = 69
x − 5 = 693
x − 5 = 23
x = 23 + 5
= 28
¥—ßπ—Èπ ®”π«π®”π«ππ—Èπ§◊Õ 28 μÕ∫
8. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëߧ◊Õ x
‡»… “¡ à«π ’Ë¢Õß®”π«π®”π«πÀπ÷Ëß∫«°°—∫ 17 ·≈â«¡’§à“ 59
®–‰¥â ¡°“√ 34
17x + = 59
34
x = 59 − 17
Page 72
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 272
34
x = 42
x = 43
42×
= 56
¥—ßπ—Èπ ®”π«π®”π«ππ—Èπ§◊Õ 56 μÕ∫
9. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëߧ◊Õ x
5 ‡∑à“¢Õß®”π«π®”π«πÀπ÷Ëß¡“°°«à“ 130 Õ¬Ÿà 45
®–‰¥â ¡°“√ 5x − 130 = 45
5x = 45 + 130
5x = 175
x = 1755
= 35
¥—ßπ—Èπ ®”π«π®”π«ππ—Èπ§◊Õ 35 μÕ∫
10. «‘∏’∑” º≈§Ÿ≥¢Õß®”π«π Õß®”π«π‡ªìπ 114
∂â“®”π«πÀπ÷Ë߇ªìπ 47
„ÀâÕ’°®”π«πÀπ÷Ë߇ªìπ x
®–‰¥â 47
x = 114
x = 74
114
×
= 18
¥—ßπ—Èπ ®”π«πÕ’°®”π«πÀπ÷Ëߧ◊Õ 18
μÕ∫
11. (1) «‘∏’∑” „À⇧√◊ËÕß´—°ºâ“√“§“ x ∫“∑
®—°√¬“π¬πμå√“§“§—π≈– 28,273 ∫“∑
√“§“®—°√¬“π¬πμ凪ìπ 74 ‡∑à“¢Õß√“§“‡§√◊ËÕß´—°ºâ“
®–‰¥â ¡°“√ 74
x = 28,273
x = 47
28 273× ,
= 16,156
π—Ëπ§◊Õ ‡§√◊ËÕß´—°ºâ“√“§“ 16,156 ∫“∑ μÕ∫
(2) «‘∏’∑” ‡¡◊ËÕ´◊ÈÕ‡§√◊ËÕß´—°ºâ“·≈–®—°√¬“π¬πμå·≈â«π“¬·¥ß¬—߇À≈◊Շߑπ 32,525 ∫“∑
‡¥‘¡π“¬·¥ß¡’‡ß‘π 32,525 + 28,273 + 16,156 ∫“∑
= 76,954 ∫“∑
¥—ßπ—È𠇥‘¡π“¬·¥ß¡’‡ß‘π 76,954 ∫“∑ μÕ∫
Page 73
73·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
12. «‘∏’∑” „Àâ‚√߇√’¬ππ’È¡’π—°‡√’¬π∑—ÈßÀ¡¥ x §π
¡’π—°°’Ó™“¬ 148 §π
¡’π—°°’ÓÀ≠‘ß 144 §π
¡’π—°°’Ó™“¬§‘¥‡ªìπ 225
¢Õßπ—°‡√’¬π™“¬∑—ÈßÀ¡¥
π—Ëπ§◊Õ ¡’π—°‡√’¬π™“¬∑—ÈßÀ¡¥ = 148252
×
= 1,850 §π
¡’π—°°’ÓÀ≠‘ߧ‘¥‡ªìπ 350
¢Õßπ—°‡√’¬πÀ≠‘ß∑—ÈßÀ¡¥
π—Ëπ§◊Õ ¡’π—°‡√’¬πÀ≠‘ß∑—ÈßÀ¡¥ = 144503
×
= 2,400 §π
¥—ßπ—Èπ x = 1,850 + 2,400
= 4,250
¥—ßπ—Èπ ‚√߇√’¬ππ’È¡’π—°‡√’¬π∑—ÈßÀ¡¥ 4,250 §π μÕ∫
13. «‘∏’∑” 5 ‡∑à“¢Õߺ≈∫«°¢Õß®”π«π‡ªì¥∑’Ë𓬷¥ß·≈–𓬥”‡≈’Ȭ߮–‡∑à“°—∫‡ªì¥¢Õß𓬇¢’¬«
𓬥”¡’‡ªì¥ 48 μ—« 𓬇¢’¬«¡’‡ªì¥ 500 μ—«
„Àâ𓬷¥ß¡’‡ªì¥ x μ—«
®–‰¥â ¡°“√ 5(x + 48) = 500
5 485
( )x + = 5005
x + 48 = 100
x = 100 − 48
x = 52
¥—ßπ—Èπ 𓬷¥ß¡’‡ªì¥ 52 μ—« μÕ∫
14. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëߧ◊Õ x
7 ‡∑à“¢Õߺ≈μà“ߢÕß®”π«π®”π«πÀπ÷Ëß°—∫ 6 ¡’§à“‡∑à“°—∫ 82
°√≥’∑’Ë x ¡“°°«à“ 6 ®–‰¥â
7(x − 6) = 82
x − 6 = 827
x = 827
6+
= 1247
= 1757
144 = 3
50× π—°‡√’¬πÀ≠‘ß∑—ÈßÀ¡¥
148 = 2
25× π—°‡√’¬π™“¬∑—ÈßÀ¡¥
Page 74
·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 274
°√≥’∑’Ë x πâÕ¬°«à“ 6 ®–‰¥â
7(6 − x) = 82
6 − x = 827
6 = 827
+ x
À√◊Õ 827
+ x = 6
x = 6827
−
= − 407
= −557
¥—ßπ—Èπ ®”π«π®”π«ππ—Èπ§◊Õ 1757 À√◊Õ −5
57
μÕ∫
15. «‘∏’∑” „Àâ «ππ’È¡’μâπ¡–¡à«ß x μâπ
¡’μâπ¢πÿπ 117 μâπ
¡’μâπ¡–æ√â“« 401 μâπ
®”π«πμâπ¢πÿπ„π «πÀ—°ÕÕ°®“° 23 ¢Õß®”π«πμâπ¡–¡à«ß„π «π º≈≈—æ∏å∑’ˉ¥â®–‡∑à“°—∫
®”π«πμâπ¡–æ√â“«„π «π
®–‰¥â ¡°“√ 23
117x − = 401
23
x = 401 + 117
23
x = 518
x = 32
518×
= 3 × 259
= 777
¥—ßπ—Èπ «ππ’È¡’μâπ¡–¡à«ß 777 μâπ μÕ∫
Page 75
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77·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
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79·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
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·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 280
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81·ºπ°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√åæ◊Èπ∞“π ™—Èπ¡—∏¬¡»÷°…“ªï∑’Ë 1 ¿“§‡√’¬π∑’Ë 2
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