เฉลยแบบฝึกหัดคณิต ม.1ภาคเรียนที่ 2

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

Àπà«¬°“√‡√’¬π√Ÿâ∑’Ë 1

‡»… à«π·≈–∑»π‘¬¡

‡©≈¬

·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

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


(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


®–‰¥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


�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∫


·∫∫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∫


(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∫


(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∫


(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∫


(18) «‘∏’∑”

� � �35

89

713

⎛⎝⎜

⎞⎠⎟

=

��

��

�

��

�

�

3 1175 117

8 659 65

7 4513 45

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

= �351585

�520585

�315585

= �1186585

= � 216585

μÕ∫

2. μÕπ‡™â“π“¬¥”‡π‘π‡¥‘π∑“ß‰¥â 534

°‘‚≈‡¡μ√

μÕπ∫à“¬‡¥‘π∑“ß‰¥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∑“ß‰¥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. «‘∏’∑” π“¬·¢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∫

¬—ß‡À≈◊ÕπÈ”Õ¬Ÿà 81

122

35

− = 52960

ªïö∫

¥—ßπ—Èπ ¬—ß‡À≈◊ÕπÈ”Õ¬Ÿà 52960

ªïö∫ μÕ∫


4. «‘∏’∑” ¡’πÈ”¡—πÕ¬Ÿà 1634

∂—ß

„™âπÈ”¡—π‰ª 525

∂—ß

¬—ß‡À≈◊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. «‘∏’∑” ©“ß¢â“«¡’¢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≈◊Õ¢â“« 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“√«¡°—∫ ‘Ëß¢Õß 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 °‘‚≈°√—¡

«‘∏’∑” ®–μâÕß‡≈◊Õ° ‘Ëß¢Õß A, C, D, E ·≈– F ´÷Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ

351920

634

− = 2915 °‘‚≈°√—¡

¥—ßπ—Èπ ‡≈◊Õ° ‘Ëß¢Õß A, C, D, E ·≈– F μÕ∫

(2) 25 °‘‚≈°√—¡

«‘∏’∑” ®–μâÕß‡≈◊Õ° ‘Ëß¢Õß C, D, E ·≈– F ´÷Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ

351920

47

106

34

− +⎛⎝

⎞⎠ = 24

12 °‘‚≈°√—¡

¥—ßπ—Èπ ‡≈◊Õ° ‘Ëß¢Õß C, D, E ·≈– F μÕ∫


(3) 20 °‘‚≈°√—¡

«‘∏’∑” ®–μâÕß‡≈◊Õ° ‘Ëß¢Õß A, C ·≈– F ÷́Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ

312

47

208

35

315

+ + + = 191320

°‘‚≈°√—¡

¥—ßπ—Èπ ‡≈◊Õ° ‘Ëß¢Õß A, C ·≈– F μÕ∫

(4) 15 °‘‚≈°√—¡

«‘∏’∑” ®–μâÕß‡≈◊Õ° ‘Ëß¢Õß A ·≈– B ÷́Ëß®–¡’πÈ”Àπ—°√«¡°—∫πÈ”Àπ—°°√–‡ªÜ“‡ªìπ

312

47

106

34

+ + = 141920

°‘‚≈°√—¡

¥—ßπ—Èπ ‡≈◊Õ° ‘Ëß¢Õß 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


°‘®°√√¡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


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


°‘®°√√¡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


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


°‘®°√√¡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


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


(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


(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


= 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


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. «‘∏’∑” ·∫àß‰¡â¬“« 8 ‡¡μ√ ÕÕ°‡ªìπ∑àÕπ —ÈπÊ ∑àÕπ≈– 34 ‡¡μ√

®–·∫àß‰¥â 834

÷ = 81

43

× ∑àÕπ

= 1023

∑àÕπ

π—Ëπ§◊Õ ·∫àß‰¥â 10 ∑àÕπ ‡À≈◊Õ‡»…¬“« 23

34

× = 12 ‡¡μ√ μÕ∫

2 2

11


4000

1

39

5

1

1

2. «‘∏’∑” “¬‰ø¢¥Àπ÷Ëß¬“« 14345 ‡¡μ√ ·∫àß‡ªìπ¢¥ —ÈπÊ ¢¥≈– 8

12 ‡¡μ√

®–·∫àß‰¥â 14345

812

÷ = 7195

172

÷

= 7195

217

×

= 143885

= 167885

¥—ßπ—Èπ ®–·∫àß “¬‰ø‰¥â 16 ¢¥ ·≈–‡À≈◊Õ‡»… 7885

172

× = 745 ‡¡μ√ μÕ∫

3. «‘∏’∑” √«¡ “¡«—ππ“¬¥‘π‰∂π“‰¥â 178

213

13

10+ + ‰√à

= 478

13

310

+ + +⎛⎝

⎞⎠ ‰√à

= 4181120

+ ‰√à

= 561

120‰√à

¥—ßπ—Èπ √«¡ “¡«—ππ“¬¥‘π‰∂π“‰¥â 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¡’πÈ”Õ¬Ÿà 35 ¢Õß∂—ß ‡∑à“°—∫ 3

510 000× , = 6,000 ≈‘μ√

„™âπÈ”‰ª 14 ¢Õß∂—ß ‡∑à“°—∫ 1

410 000× , = 2,500 ≈‘μ√

®–‡À≈◊ÕπÈ”Õ¬Ÿà 6,000 − 2,500 = 3,500 ≈‘μ√¥—ßπ—Èπ ‡À≈◊ÕπÈ”Õ¬Ÿà 3,500 ≈‘μ√ μÕ∫

6. «‘∏’∑” «—π‡ “√åπ“ß™«π™◊Ëπ„™â‡ß‘π‰ª 58 ¢Õß‡ß‘π∑’Ë¡’Õ¬Ÿà

®–‡À≈◊Õ‡ß‘π 88

58

− = 38 ¢Õß‡ß‘π∑’Ë¡’Õ¬Ÿà

«—πÕ“∑‘μ¬åπ“ß™«π™◊Ëπ„™â‡ß‘π 15 ¢Õß‡ß‘π∑’Ë‡À≈◊Õ

= 15

38

× ¢Õß‡ß‘π∑’Ë¡’Õ¬Ÿà

= 340

¢Õß‡ß‘π∑’Ë¡’Õ¬Ÿà


600

1

31

1

¬—ß‡À≈◊Õ‡ß‘π 38

340

− = 1240

¢Õß‡ß‘π∑’Ë¡’Õ¬Ÿà

π“ß™«π™◊Ëπ¬—ß§ß‡À≈◊Õ‡ß‘π 7,200 ∫“∑

‡ß‘π 1240

¢Õß‡ß‘π∑’Ë¡’Õ¬Ÿà §‘¥‡ªìπ‡ß‘π 7,200 ∫“∑

‡ß‘π 1 ¢Õß‡ß‘π∑’Ë¡’Õ¬Ÿà §‘¥‡ªìπ‡ß‘π 7 2001240

, ÷ = 7200 4012

× = 24,000 ∫“∑

¥—ßπ—Èπ ‡¥‘¡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«‡©≈’Ë¬ 546 523

÷ = 5463

17× °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß

≈ 96.35 °‘‚≈‡¡μ√μàÕ™—Ë«‚¡ß

¥—ßπ—Èπ π“¬πæ¥≈¢—∫√∂¥â«¬Õ—μ√“‡√Á«‡©≈’Ë¬™—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. «‘∏’∑” ‚√ß‡√’¬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 ‚√ß‡√’¬ππ’È¡’π—°‡√’¬π∑—ÈßÀ¡¥ 2,400 §π μÕ∫

10. «‘∏’∑” π“¡√Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“°«â“ß 1225

‡¡μ√

¡’¥â“π¬“«¬“«°«à“ 112 ‡∑à“¢Õß¥â“π°«â“ßÕ¬Ÿà 4 ‡¡μ√

π—Ëπ§◊Õ ¡’¥â“π¬“«‡∑à“°—∫ 32

625

4×⎡⎣⎢

⎤⎦⎥

+ = 1135

‡¡μ√

π“¡√Ÿª ’Ë‡À≈’Ë¬¡¡’æ◊Èπ∑’Ë 625

1135

× = 700625

= 280625

μ“√“ß‡¡μ√

¥—ßπ—Èπ π“¡√Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“π’È¡’æ◊Èπ∑’Ë 280625

μ“√“ß‡¡μ√ μÕ∫


11. «‘∏’∑” ™”√–‡ß‘πß«¥·√° 311

¢Õß√“§“√∂¬πμå

¬—ß‡À≈◊Õ à«π∑’ËμâÕß™”√–Õ’° 1111

311

− = 811

¢Õß√“§“√∂¬πμå

™”√–ß«¥∑’Ë Õß 25 ¢Õß∑’Ë‡À≈◊Õ®“°®à“¬ß«¥·√° = 2

58

11× = 16

55 ¢Õß√“§“√∂¬πμå

¬—ß‡À≈◊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¥∑â“¬∫√√∑ÿ°‰¥â 512

65

× = 12 μ—π

π—Ëπ§◊Õ μâÕß∫√√∑ÿ°∑—ÈßÀ¡¥ 261 §√—Èß §√—Èß ÿ¥∑â“¬∫√√∑ÿ° 12 μ—π μÕ∫

13. «‘∏’∑” ·À««Õà“πÀπ—ß ◊Õ‚¥¬«—π·√°Õà“π‰¥â 14 ¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥

¬—ß‡À≈◊Õ∑’Ë¬—ß‰¡à‰¥âÕà“π 44

14

− = 34 ¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥

«—π∑’Ë ÕßÕà“π‰¥â 15 ¢Õß®”π«πÀπâ“∑’Ë‡À≈◊Õ = 1

534

× = 320

¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥

¬—ß‡À≈◊Õ∑’Ë¬—ß‰¡à‰¥âÕà“π 34

320

− = 1220

¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥

‡π◊ËÕß®“°¬—ß‡À≈◊Õ∑’Ë¬—ß‰¡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∫


14. «‘∏’∑” ‚√ß‡√’¬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


15. «‘∏’∑” ‚√ßß“π·ÀàßÀπ÷Ëß¡’§πß“π™“¬ 511

¢Õß§πß“π∑—ÈßÀ¡¥

¡’§πß“πÀ≠‘ß 1111

511

− = 611


¡’§πß“π™“¬∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï Õ¬Ÿà 15 ¢Õß§πß“π™“¬

= 15

511

× = 555


¡’§πß“πÀ≠‘ß∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï Õ¬Ÿà 25 ¢Õß§πß“πÀ≠‘ß

= 25

611

× = 1255


√«¡§πß“π∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï Õ¬Ÿà 555

1255

+ = 1755


§πß“π∑’Ë¡’Õ“¬ÿ‡°‘π 55 ªï ¡’Õ¬Ÿà 68 §π

π—Ëπ§◊Õ 1755

¢Õß§πß“π∑—ÈßÀ¡¥ ‡∑à“°—∫ 68 §π

1 ¢Õß§πß“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× + × + × + ×⎛

⎝⎞⎠ + ×⎛

⎝⎜⎞⎠⎟


(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. ≈’≈“«‘Ëß‡¢â“‡ âπ™—¬‡ªìπ§π·√°


·∫∫Ωñ°À—¥ 9

1. (1) «‘∏’∑” 5 μ“ß§å = 5100

∫“∑ (2) «‘∏’∑” 12 μ“ß§å = 12100

∫“∑

= 0.05 ∫“∑ = 0.12 ∫“∑

¥—ßπ—Èπ 108 ∫“∑ 5 μ“ß§å ¥—ßπ—Èπ 431 ∫“∑ 12 μ“ß§å

= 108.05 ∫“∑ μÕ∫ = 431.12 ∫“∑ μÕ∫

(3) «‘∏’∑” 87 μ“ß§å = 87100

∫“∑ (4) «‘∏’∑” 671 μ“ß§å = 671100

∫“∑

= 0.87 ∫“∑ μÕ∫ = 6.71 ∫“∑

¥—ßπ—Èπ 6,504 ∫“∑ °—∫Õ’° 671 μ“ß§å

= 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 ‡´πμ‘‡¡μ√ (2) «‘∏’∑” 18 ‡¡μ√ 41 ‡´πμ‘‡¡μ√

= 52

100+ ‡¡μ√ = 18

41100

+ ‡¡μ√

= 5.02 ‡¡μ√ μÕ∫ = 18.41 ‡¡μ√ μÕ∫

(3) «‘∏’∑” 9 ‡¡μ√ 56 ‡´πμ‘‡¡μ√ 4 ¡‘≈≈‘‡¡μ√ (4) «‘∏’∑” 651 ‡´πμ‘‡¡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∫


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


(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∑’Ë‡À≈◊ÕÀ≈—ß®“°À—° à«π∑’Ë¡“°°«à“ÕÕ°§◊Õ 164.75 − 38.75 = 126.00 ∫“∑

à«π∑’Ë‡À≈◊ÕÕ¬Ÿàπ’È‡¡◊ËÕ·∫àß‡ªìπ 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. ∑àÕπÈ”Õ—πÀπ÷Ëß¡’§«“¡¬“« 0.93 ‡¡μ√

§«“¡¬“«¢Õß‡ âπºà“π»Ÿπ¬å°≈“ß¿“¬πÕ°‡ªìπ 23 ¢Õß§«“¡¬“«∑àÕ

= 23

0 93× . ‡¡μ√

= 0.62 ‡¡μ√


¡’§«“¡¬“«‡ âπºà“π»Ÿπ¬å°≈“ß¿“¬„π‡ªìπ 0.435 ‡¡μ√

2 ‡∑à“¢Õß§«“¡Àπ“ = 0.62 − 0.435 = 0.185 ‡¡μ√

§«“¡Àπ“ = 0 1852

100. × = 9.25 ‡´πμ‘‡¡μ√

¥—ßπ—Èπ ∑àÕπÈ”π’È¡’§«“¡Àπ“ 9.25 ‡´πμ‘‡¡μ√ μÕ∫

5. «‘∏’∑” πÈ”Àπ—° ‘Ëß¢Õß∑—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 ‘Ëß¢Õß„Àâ‰¥â¡“°∑’Ë ÿ¥ ·μà¡’πÈ”Àπ—°√«¡°√–‡ªÜ“‰¥â‰¡à‡°‘π 35 °‘‚≈°√—¡

®÷ß§«√∫√√®ÿ ‘Ëß¢Õß™‘Èπ∑’Ë 1 ™‘Èπ∑’Ë 2 ™‘Èπ∑’Ë 4 ™‘Èπ∑’Ë 5 ·≈–™‘Èπ∑’Ë 6 ‚¥¬‡Õ“ ‘Ëß¢Õß™‘Èπ∑’Ë 3 ÕÕ°‰ª ´÷Ëß

®–¡’πÈ”Àπ—°√«¡°√–‡ªÜ“‡ªìπ 40.10 − 5.18 = 34.92 °‘‚≈°√—¡

¥—ßπ—Èπ μâÕß‡Õ“™‘Èπ∑’Ë 3 ÕÕ°‰ª ·≈–∑’Ë‡À≈◊Õ¡’πÈ”Àπ—°√«¡¢Õß°√–‡ªÜ“ 34.92 °‘‚≈°√—¡ μÕ∫

6. «‘∏’∑” π“¬™Õ∫‡∑’Ë¬«‡¥‘π∑“ß‡ªìπ√–¬–∑“ß∑—Èß ‘Èπ 10.81 + 11.43 + 15.45 = 37.69 °‘‚≈‡¡μ√

π“¬ “¬μ√ß‡¥‘π∑“ß®“°À“¥æ≈“∂÷ßÀ“¥μ“°«π‡ªìπ√–¬–∑“ß 33.47 °‘‚≈‡¡μ√

¥—ßπ—Èπ π“¬™Õ∫‡∑’Ë¬«‡¥‘π∑“ß¡“°°«à“π“¬ “¬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 ¡.


(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÷Ëß¬“« 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 Õß‰ª 68.8 ‡¡μ√ ‡¡μ√≈– 18.50 ∫“∑

‰¥â‡ß‘π 68.8 × 18.50 = 1,272.80 ∫“∑

+


°‘®°√√¡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√

¬—ß‡À≈◊Õ “¬‰ø¬“« 246.50 − 161.80 = 84.70 ‡¡μ√

¥—ßπ—Èπ ‰¥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∫


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÷Ëß¬“« 148.40 ‡¡μ√

μ—¥‡ªìπ‡ âπ ¬“«‡ 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„ à¢«¥¢«¥≈– 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. ˙


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


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÷Ëß¬“« 15.48 ‡¡μ√ μ—¥ÕÕ°‡ªìπ 2 ∑àÕπ ‚¥¬∑àÕπÀπ÷Ëß¬“«°«à“Õ’°∑àÕπÀπ÷Ëß 2.32

‡¡μ√

¥—ßπ—Èπ À—° à«π∑’Ë¬“«ÕÕ°‰ª°àÕπ®–‡À≈◊Õ

15.48 − 2.32 = 13.16 ‡¡μ√

¥—ßπ—Èπ ∑àÕπ —Èπ¡’§«“¡¬“« 13 162. = 6.58 ‡¡μ√

·≈–∑àÕπ¬“«¡’§«“¡¬“« 6.58 + 2.32 = 8.90 ‡¡μ√ μÕ∫

4. «‘∏’∑” √Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“°«â“ß 4.7 ‡´πμ‘‡¡μ√ ¬“« 12.6 ‡´πμ‘‡¡μ√

¡’æ◊Èπ∑’Ë 4.7 × 12.6 = 59.22 μ“√“ß‡´πμ‘‡¡μ√

¥—ßπ—Èπ √Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“¡’æ◊Èπ∑’Ë 59.22 μ“√“ß‡´πμ‘‡¡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 Õß¢â“ß¢Õß ¡°“√ ®–‰¥â

x = 8.4 × 16.5 = 138.6

¥—ßπ—Èπ ®”π«π®”π«ππ—Èπ§◊Õ 138.6 μÕ∫


6. «‘∏’∑” ´◊ÈÕ¢â“« “√ 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¡∑’Ë‡∑’Ë¬«≈– 2.32 μ—π

μâÕß°“√∫√√∑ÿ°¥‘π 45 μ—π

μâÕß∫√√∑ÿ°∑—ÈßÀ¡¥ 452 32.

= 4500232

‡∑’Ë¬«

≈ 1992232

‡∑’Ë¬«

π—Ëπ§◊Õ μâÕß∫√√∑ÿ° 20 ‡∑’Ë¬« ‚¥¬‡∑’Ë¬«∑’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“¢Õß¥”

= 572517175

‡∑à“¢Õß¥”

= 13

‡∑à“¢Õß¥”

¥”¡’‡ß‘π‡ªìπ 171 7557 25

..

‡∑à“¢Õß·¥ß

= 3 ‡∑à“¢Õß·¥ß

¥—ßπ—Èπ ·¥ß¡’‡ß‘π‡ªìπ 13 ‡∑à“¢Õß¥” ·≈–¥”¡’‡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


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


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


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∫


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∫

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√, ‡´πμ‘‡¡μ√ 3. ‡¡μ√

4. °‘‚≈‡¡μ√ 5. ‡´πμ‘‡¡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

Àπà«¬°“√‡√’¬π√Ÿâ∑’Ë 2

°“√ª√–¡“≥§à“

‡©≈¬


2. (1) 27 ‡¡μ√ + 34 ‡´πμ‘‡¡μ√ = 2,700 + 34 = 2,734 ´¡.

(2) 3 ‡¡μ√ + 45 ¡‘≈≈‘‡¡μ√ + 3 ‡´πμ‘‡¡μ√ = 300 + 4.5 + 3 = 307.5 ´¡.

(3) 3 ‡¡μ√ 130 ¡‘≈≈‘‡¡μ√ + 3 ‡´πμ‘‡¡μ√ = 300 + 13 + 3 = 316 ´¡.

(4) 364 ‡¡μ√ = 3641000

= 0.364 °¡.

(5) 36,784 ‡¡μ√ = 367841000

= 36.784 °¡.

(6) 438 ‡´πμ‘‡¡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 ‡´πμ‘‡¡μ√

(2) 2.0+1.3+2.3 = 5.6 π‘È«

4. «‘∏’∑” ≈«¥¢¥∑’ËÀπ÷Ëß¬“« 8.45+

‡¡μ√

≈«¥¢¥∑’Ë Õß¬“« 4.32 ‡¡μ√

π”¡“μàÕ°—π®–¬“« 12.77 ‡¡μ√

¥—ßπ—Èπ ≈«¥®–¬“«ª√–¡“≥ 12.8 ‡¡μ√ μÕ∫

5. «‘∏’∑” æ◊Èπ∑’Ëºπ—ßÀâÕß = 2(°«â“ß� Ÿß)�2(¬“«� Ÿß)

= 2(4.75�2.85)�2(5.25�2.85)

= 27.075�29.925

= 57 μ“√“ß‡¡μ√

æ◊Èπ∑’Ë‡æ¥“π = °«â“ß�¬“«

= 4.75�5.25

= 24.9375 μ“√“ß‡¡μ√

æ◊Èπ∑’Ëº‘«ÀâÕß�æ◊Èπ∑’Ë‡æ¥“π = 57�24.9375

= 81.9375 μ“√“ß‡¡μ√

¥—ßπ—Èπ æ◊Èπ∑’Ëºπ—ßÀâÕß·≈–‡æ¥“πª√–¡“≥ 81.9 μ“√“ß‡¡μ√ μÕ∫


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√–¡“≥√–¬–∑“ß¢ÕßºŸâ‡≈àπ‡°¡ ·≈–°“√ —ß‡°μ

3. Õ“®„™â°“√ª√–¡“≥√“§“„Àâ‡ªìπ®”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√–¡“≥°’Ë‡¡≈Á¥ ·≈â«§”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‡§¬ —ß‡°μ√–¬–∑“ß∑’Ë‡¥‘π‰¥âμàÕ‡«≈“∑’Ë„™â‰ª °ÁÕ“®‰ªμ√«® Õ∫®“°°“√

ªØ‘∫—μ‘¥Ÿ°àÕπ·≈â«®÷ßμÕ∫§”∂“¡°Á‰¥â

2. ¢âÕπ’ÈÕ“®™à«¬°—π§‘¥·≈–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

·≈–„Àâπ”‰ª∑¥≈ÕßªØ‘∫—μ‘·≈–À“§”μÕ∫¡“π”‡ πÕ„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


2.

ª√‘¡“≥πÈ”Ωπ

®”π«π‡¡◊ËÕªí¥‡»…‡ªìπ®”π«π‡μÁ¡∑’Ë„°≈â‡§’¬ß∑’Ë ÿ¥

®—ßÀ«—¥(¡‘≈≈‘‡¡μ√)

®”π«π ®”π«π ®”π«π

‡μÁ¡Àπà«¬ ‡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∫


3. AB ¬“«‰¥âμ—Èß·μà 8.5 Àπà«¬¢÷Èπ‰ª ·μà¬“«‰¡à∂÷ß 9.5 Àπà«¬ π—Ëπ§◊Õ 8.5 � AB � 9.5

4.

5. À¡Ÿà∫â“π∫—«·¥ßÀà“ß®“°À¡Ÿà∫â“π√à«¡√à¡‡¬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∫∑’Ë‡À¡“– ¡°«à“§◊Õ

27 + 45 ≈ 25 + 45 °‘‚≈‡¡μ√≈ 70 °‘‚≈‡¡μ√

6. „™â°“√ª√–¡“≥™à«¬§‘¥«à“‡¥Á°·μà≈–§π¡’‡ß‘πæÕ´◊ÈÕ ‘Ëß∑’Ë‡¢“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


(5) «‘∏’∑” ‡À≈◊Õß¡’‡ß‘π 175 ∫“∑ μâÕß°“√´◊ÈÕ°âÕπ°√«¥ 3 ∂ÿß

°âÕπ°√«¥∂ÿß≈– 59 ∫“∑ ª√–¡“≥‡ªìπ 60 ∫“∑

°âÕπ°√«¥ 3 ∂ÿß √“§“ª√–¡“≥ 3 × 60 ≈ 180 ∫“∑


À¡“¬‡Àμÿ °√≥’À“§à“ª√–¡“≥‰¥â„°≈â‡§’¬ß°—∫‡ß‘π∑’Ë¡’Õ¬Ÿà¡“° Õ“® √ÿªº‘¥æ≈“¥°Á‰¥â °√≥’π’ÈÕ“®μâÕß

μ√«® Õ∫®“°®”π«π∑’Ë‡ªìπ§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‘ß§å¡’‡ß‘π 575 ∫“∑ μâÕß°“√´◊ÈÕ‡øîπ 4 ™ÿ¥

‡øîπ™ÿ¥≈– 150 ∫“∑ 4 ™ÿ¥ ‡ªìπ‡ß‘π 150 × 4 = 600 ∫“∑


(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 ∫“∑


(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∫


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 ‡´πμ‘‡¡μ√

ª√–¡“≥‡ªìπ 50 ‡´πμ‘‡¡μ√

¬“« 2.4 ¢Õß§«“¡°«â“ß ≈ 2.5 × 50 ‡´πμ‘‡¡μ√ ≈ 125 ‡´πμ‘‡¡μ√

¥—ßπ—Èπ ¡’æ◊Èπ∑’Ëª√–¡“≥ 50 × 125 ≈ 6,250 μ“√“ß‡´πμ‘‡¡μ√ μÕ∫

9. «‘∏’∑” ÿπ—¢¢Õß≈ÿß·´¡ Ÿß 37.8 ‡´πμ‘‡¡μ√ ª√–¡“≥‡ªìπ 40 ‡´πμ‘‡¡μ√

≈ÿß·´¡ Ÿß‡ªìπ 4.2 ‡∑à“¢Õß ÿπ—¢ ª√–¡“≥§«“¡ Ÿß‡ªìπ 4 ‡∑à“¢Õß ÿπ—¢

¥—ßπ—Èπ ≈ÿß·´¡ Ÿßª√–¡“≥ 4 × 40 ≈ 160 ‡´πμ‘‡¡μ√ μÕ∫

10. «‘∏’∑” ¢—∫√∂®“°°√ÿß‡∑æœ ‰ª®—ßÀ«—¥μ√“¥¥â«¬§«“¡‡√Á«‡©≈’Ë¬ 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√–¬–∑“ßª√–¡“≥ 254

80× ≈ 500 °‘‚≈‡¡μ√

¥—ßπ—Èπ ¢—∫√∂‰¥â√–¬–∑“ßª√–¡“≥ 500 °‘‚≈‡¡μ√ μÕ∫

11. «‘∏’∑” √Ÿª ’Ë‡À≈’Ë¬¡º◊πºâ“¡’¥â“π°«â“ß°«â“ß 12.4 ≈ 12.5 ‡¡μ√

¥â“π¬“«¬“«°«à“ 2 ‡∑à“¢Õß¥â“π°«â“ßÕ¬Ÿà 5 ‡¡μ√

¥â“π¬“«ª√–¡“≥ 2(12.5) + 5 ≈ 30 ‡¡μ√

¡’æ◊Èπ∑’Ëª√–¡“≥ 12.5 × 30 ≈ 375 μ“√“ß‡¡μ√

¥—ßπ—Èπ √Ÿª ’Ë‡À≈’Ë¬¡π’È¡’æ◊Èπ∑’Ëª√–¡“≥ 375 μ“√“ß‡¡μ√ μÕ∫

12. «‘∏’∑” π“¬ ¬“¡¡’‡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 ∫“∑

π“¬ ¬“¡·≈°‡ß‘π∫“∑‰¥âª√–¡“≥ 830 × 30 ≈ 24,900 ∫“∑

¥—ßπ—Èπ π“¬ ¬“¡·≈°‡ß‘π∫“∑‰¥âª√–¡“≥ 24,900 ∫“∑ μÕ∫

°“√§Ÿ≥‡™àππ’È‰¡à‡ªìπ‰ªμ“¡°Æ°“√ªí¥‡»… ·μà®–‰¥â§à“„°≈â‡§’¬ß§à“®√‘ß¡“°°«à“ ‡π◊ËÕß®“°®”π«π

Àπ÷Ëßªí¥≈ßÕ’°®”π«πÀπ÷Ëßªí¥¢÷Èπ

20

1


1. «‘∏’∑” ≈¥√“§“ 117 − 86 = 31 ∫“∑

ª√–¡“≥‡ªìπ 30 ∫“∑

√â“π§â“μ‘¥√“§“°“ß‡°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 ‚¥¬‡©≈’Ë¬πÈ”¡—π 1 ≈‘μ√ ®–¢—∫√∂‰¥âª√–¡“≥ 12.5 °‘‚≈‡¡μ√ μÕ∫

3. «‘∏’∑” Õ“¿“æ√¢—∫√∂Õ¬ŸàÀà“ß®“°æ‘…≥ÿ‚≈° 135 °‘‚≈‡¡μ√

·≈–¢—∫√∂¥â«¬§«“¡‡√Á«‡©≈’Ë¬ 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 ∫“∑ ®÷ß‡ªìπ‰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


6. Õ“√¬“‰ª À√—∞Õ‡¡√‘°“

(1) «‘∏’∑” Õ“√¬“√—∫ª√–∑“πÕ“À“√ 37.35 ¥Õ≈≈“√å

‡ ’¬¿“…’√âÕ¬≈– 5 ·≈–‡ ’¬§à“∫√‘°“√√âÕ¬≈– 15 ¢Õß√“§“

¥—ßπ—Èπ μâÕß‡ ’¬¿“…’·≈–§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 «‘∏’∑’Ë§‘¥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) «‘∏’∑” ‡¥àπ¥“«¡’‡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 μâπ ‡ªìπ‡ß‘π 3 × 99 ≈ 3 × 100 ∫“∑

≈ 300 ∫“∑

„Àâ∏π∫—μ√„∫≈– 500 ∫“∑ ®÷ß‰¥â‡ß‘π∑Õπ

≈ 500 − 300 ∫“∑

≈ 200 ∫“∑

¥—ßπ—Èπ ‰¥â√—∫‡ß‘π∑Õπª√–¡“≥ 200 ∫“∑ μÕ∫


(4) «‘∏’∑” ª√–™“ ª√’¥“ ·≈–‡ª√¡ √«¡‡ß‘π°—π´◊ÈÕμâπ‰¡â∑—Èß 4 ™π‘¥ ™π‘¥≈– 1 μâπ ‡ªìπ‡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

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 ∂â«¬ „™âπÈ”μ“≈ 36 °âÕπ

(6, 36) ·≈– (36, 6) ¡’§«“¡À¡“¬‰¡à‡À¡◊Õπ°—π ‡æ√“–«à“

(6, 36) À¡“¬∂÷ß „™âπÈ”¡–π“« 6 ∂â«¬ „™âπÈ”μ“≈ 36 °âÕπ

(36, 6) À¡“¬∂÷ß „™âπÈ”¡–π“« 36 ∂â«¬ „™âπÈ”μ“≈ 6 °âÕπ

3. 72 °âÕπ 4. 9 ∂â«¬

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√ß‡¥’¬«°—π

°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 1

°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 2

°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 3

Àπà«¬°“√‡√’¬π√Ÿâ∑’Ë 3

§ŸàÕ—π¥—∫·≈–°√“ø

‡©≈¬


(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


(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)


6. (1) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 10 Àπà«¬ Àà“ß®“°·°π X ‡ªìπ√–¬– 7 Àπà«¬ ¡’æ‘°—¥ (10, 7)

(2) Õ¬ŸàÀà“ß®“°·°π Y ‡ªìπ√–¬– 7 Àπà«¬ Àà“ß®“°·°π X ‡ªìπ√–¬– 2 Àπà«¬ ¡’æ‘°—¥ (7, 2)





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¥—∫¢Õß§«“¡ —¡æ—π∏å‰¥â‡ªìπ (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

.......

.......

.......

.......

.......

.......

.... .... ....... ....... ....... ....... .........


1.

2.

·∫∫Ωñ°À—¥ 2


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


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)


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

.... .... .... .... ...... ......

...... ...... ...... ......


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»“‡´≈‡´’¬ )


3.

(1) 4 «—π (2) 7.5 ‡´πμ‘‡¡μ√ (3) 0.86 ‡´πμ‘‡¡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¢Õß‡ 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 (‡´πμ‘‡¡μ√)

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

Àπà«¬°“√‡√’¬π√Ÿâ∑’Ë 4

¡°“√‡™‘ß‡ âπμ—«·ª√‡¥’¬«

‡©≈¬

°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 1

°‘®°√√¡μ√«® Õ∫§«“¡‡¢â“„® 2


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


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


(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∫


(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


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“ß¢Õß ¡°“√ 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∫


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“ß¢Õß ¡°“√

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“ß¢Õß ¡°“√ 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∫


25. «‘∏’∑” 48.1 + 0.2x = 52.5 26. «‘∏’∑” 31 55

. +x

= 56.5

π” 10 §Ÿ≥∑—Èß Õß¢â“ß¢Õß ¡°“√ 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∫ÀâÕß‡√’¬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


μÕπ∑’Ë 1

1. «‘∏’∑” π“¬¡¥‰¥â√—∫∑’Ë∑”°‘π®“°√—∞∫“≈ k ‰√à

π“¬¥â«ß‰¥â√—∫∑’Ë∑”°‘π®“°√—∞∫“≈ 168 ‰√à

π“¬¥â«ß‰¥â√—∫∑’Ë∑”°‘π®“°√—∞∫“≈‡ªìπ 3 ‡∑à“¢Õßπ“¬¡¥

®–‰¥â ¡°“√ 3k = 168 μÕ∫

2. «‘∏’∑” π“¬‚√®πå∑”π“‰¥â¢â“«‡ª≈◊Õ° y ‡°«’¬π

π“¬√ÿàß∑”π“‰¥â¢â“«‡ª≈◊Õ° 12.7 ‡°«’¬π

π“¬√ÿàß∑”π“‰¥â¢â“«‡ª≈◊Õ° 15 ¢Õßπ“¬‚√®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∫‡≈’È¬ßª≈“∑ÕßÕ¬Ÿà q μ—«

™‘¥‡≈’È¬ßª≈“∑ÕßÕ¬Ÿà 15 μ—«

™‘¥‡≈’È¬ßª≈“∑ÕßπâÕ¬°«à“ 23 ‡∑à“¢Õß™Õ∫Õ¬Ÿà 5 μ—«

®–‰¥â ¡°“√ 23

5q − = 15 μÕ∫

5. «‘∏’∑” 3 ‡∑à“¢Õßº≈∫«°¢Õß a °—∫ 7 ‡ªìπ 210

‡¢’¬π ¡°“√‰¥â‡ªìπ 3(a + 7) = 210 μÕ∫

6. «‘∏’∑”34 ‡∑à“¢Õßº≈∫«°¢Õß b °—∫ 10 ‡ªìπ 18

‡¢’¬π ¡°“√‰¥â‡ªìπ 34

10( )b + = 18 μÕ∫

7. «‘∏’∑”56 ‡∑à“¢Õßº≈μà“ß¢Õß 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


μÕπ∑’Ë 2

1. «‘∏’∑” „Àâ‡®’Í¬∫¡’‡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—Èπ ‡¥‘¡‡®’Í¬∫¡’‡ß‘πÕ¬Ÿà 328 ∫“∑ μÕ∫

2. «‘∏’∑” „Àâ‡¥‘¡‡¥àπ¡’¥‘π ÕÕ¬Ÿà x ·∑àß

‡¥àππ”¥‘π Õ®”π«ππ’È·∫àß„Àâ‡¥Á° 36 §π §π≈– 4 ·∑àß

ª√“°Ø«à“¬—ß‡À≈◊Õ¥‘π ÕÕ¬ŸàÕ’° 6 ·∑àß

®–‰¥â ¡°“√ x = (36 × 4) + 6

x = 150

¥—ßπ—Èπ ‡¥‘¡‡¥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. «‘∏’∑” „Àâ·¥ß¡’≈Ÿ°À‘πÕ¬Ÿà x ≈Ÿ°

¥”¡’≈Ÿ°À‘π¡“°°«à“ Õß‡∑à“¢Õß·¥ßÕ¬Ÿà 5 ≈Ÿ°

¥—ßπ—Èπ ¥”¡’≈Ÿ°À‘π 2x + 5

·μà¥”¡’≈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∫


5. «‘∏’∑” „Àâ™Õ∫‡≈’È¬ßª≈“∑Õß x μ—«

™‘¥‡≈’È¬ßª≈“∑ÕßπâÕ¬°«à“ 23 ‡∑à“¢Õß™Õ∫ 5 μ—«

¥—ßπ—Èπ ™‘¥‡≈’È¬ßª≈“∑Õß 23

5x − μ—«

·μà™‘¥‡≈’È¬ßª≈“∑Õß 15 μ—«

®–‰¥â ¡°“√ 23

5x − = 15

23

x = 15 + 5

23

x = 20

x = 32

20×

= 30

¥—ßπ—Èπ ™Õ∫‡≈’È¬ßª≈“∑Õß 30 μ—« μÕ∫

6. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëß§◊Õ x

3 ‡∑à“¢Õßº≈∫«°¢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÷Ëß§◊Õ x

∂â“π” 5 ¡“≈∫ÕÕ°®“°®”π«π®”π«πÀπ÷Ëß·≈â« “¡‡∑à“¢Õßº≈μà“ßπ—Èπ§◊Õ 69

®–‰¥â ¡°“√ 3(x − 5) = 69

x − 5 = 693

x − 5 = 23

x = 23 + 5

= 28


8. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëß§◊Õ x

‡»… “¡ à«π ’Ë¢Õß®”π«π®”π«πÀπ÷Ëß∫«°°—∫ 17 ·≈â«¡’§à“ 59

®–‰¥â ¡°“√ 34

17x + = 59

34

x = 59 − 17


34

x = 42

x = 43

42×

= 56


9. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëß§◊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


10. «‘∏’∑” º≈§Ÿ≥¢Õß®”π«π Õß®”π«π‡ªìπ 114

∂â“®”π«πÀπ÷Ëß‡ªìπ 47

„ÀâÕ’°®”π«πÀπ÷Ëß‡ªìπ x

®–‰¥â 47

x = 114

x = 74

114

×

= 18

¥—ßπ—Èπ ®”π«πÕ’°®”π«πÀπ÷Ëß§◊Õ 18

μÕ∫

11. (1) «‘∏’∑” „Àâ‡§√◊ËÕß´—°ºâ“√“§“ x ∫“∑

®—°√¬“π¬πμå√“§“§—π≈– 28,273 ∫“∑

√“§“®—°√¬“π¬πμå‡ªìπ 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¬—ß‡À≈◊Õ‡ß‘π 32,525 ∫“∑

‡¥‘¡π“¬·¥ß¡’‡ß‘π 32,525 + 28,273 + 16,156 ∫“∑

= 76,954 ∫“∑

¥—ßπ—Èπ ‡¥‘¡π“¬·¥ß¡’‡ß‘π 76,954 ∫“∑ μÕ∫


12. «‘∏’∑” „Àâ‚√ß‡√’¬ππ’È¡’π—°‡√’¬π∑—ÈßÀ¡¥ x §π

¡’π—°°’Ã“™“¬ 148 §π

¡’π—°°’Ã“À≠‘ß 144 §π

¡’π—°°’Ã“™“¬§‘¥‡ªìπ 225

¢Õßπ—°‡√’¬π™“¬∑—ÈßÀ¡¥

π—Ëπ§◊Õ ¡’π—°‡√’¬π™“¬∑—ÈßÀ¡¥ = 148252

×

= 1,850 §π

¡’π—°°’Ã“À≠‘ß§‘¥‡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


13. «‘∏’∑” 5 ‡∑à“¢Õßº≈∫«°¢Õß®”π«π‡ªì¥∑’Ëπ“¬·¥ß·≈–π“¬¥”‡≈’È¬ß®–‡∑à“°—∫‡ªì¥¢Õßπ“¬‡¢’¬«

π“¬¥”¡’‡ªì¥ 48 μ—« π“¬‡¢’¬«¡’‡ªì¥ 500 μ—«

„Àâπ“¬·¥ß¡’‡ªì¥ x μ—«

®–‰¥â ¡°“√ 5(x + 48) = 500

5 485

( )x + = 5005

x + 48 = 100

x = 100 − 48

x = 52

¥—ßπ—Èπ π“¬·¥ß¡’‡ªì¥ 52 μ—« μÕ∫

14. «‘∏’∑” „Àâ®”π«π®”π«πÀπ÷Ëß§◊Õ x

7 ‡∑à“¢Õßº≈μà“ß¢Õß®”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¡¥


°√≥’∑’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∑’Ë‰¥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∫

°‘®°√√¡∑’Ë 1 ∂÷ß°‘®°√√¡∑’Ë 6 ‡ªìπ°‘®°√√¡∑’Ëπ—°‡√’¬πμâÕß≈ß¡◊ÕªØ‘∫—μ‘ ®÷ß®–∑√“∫‰¥â«à“·μà≈–§πæ∫

ªí≠À“À√◊ÕÕÿª √√§Õ¬à“ß‰√

®ß«“¥√Ÿª§≈’Ë¢Õß∑√ß “¡¡‘μ‘μàÕ‰ªπ’È

1. 2.

3. 4.

·∫∫Ωñ°À—¥ 1

15 ¡¡.

15 ¡¡.

40 ¡¡.

3 ´¡.5 ´¡. 4 ´¡.

10 ´¡.

5 ´¡.

18 ´¡.9

´¡.9

´¡.9

´¡.6

´¡.

9 ´¡.

9 ´¡. 9 ´¡.6 ´¡.

3´¡.

30 ´¡.10 ´¡.

18 ´¡. 18 ´¡.

Àπà«¬°“√‡√’¬π√Ÿâ∑’Ë 5

°‘®°√√¡°“√ ”√«®√Ÿª‡√¢“§≥‘μ “¡¡‘μ‘

‡©≈¬


5.

°‘®°√√¡∑’Ë 2 (Àπâ“ 186)

°‘®°√√¡°“√ ”√«®√Ÿª‡√¢“§≥‘μ “¡¡‘μ‘

5 ´¡.

5 ´¡.

15 ´¡.

¢âÕ ∑√ß “¡¡‘μ‘ ¡ÿ¡¬Õ¥ Àπâ“ ¢Õ∫

1. æ’√–¡‘¥∞“π ’Ë‡À≈’Ë¬¡®—μÿ√— 5 5 8

2. ª√‘´÷¡∞“π “¡‡À≈’Ë¬¡ 6 5 9

3. ª√‘´÷¡∞“π ’Ë‡À≈’Ë¬¡®—μÿ√— 8 6 12

4. ª√‘´÷¡∞“πÀâ“‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“ 10 7 15

5. ª√‘´÷¡∞“πÀ°‡À≈’Ë¬¡¥â“π‡∑à“¡ÿ¡‡∑à“ 12 8 18


°‘®°√√¡∑’Ë 3 (Àπâ“ 186)

§«“¡ —¡æ—π∏å√–À«à“ß V, F ·≈– E §◊Õ V + F = E + 2

°‘®°√√¡∑’Ë 1 (Àπâ“ 188)

1. (1) √Ÿª«ß°≈¡ (2) √Ÿª«ß√’ (3) √Ÿª “¡‡À≈’Ë¬¡2. (1) √Ÿª ’Ë‡À≈’Ë¬¡¡ÿ¡©“° (2) √Ÿª ’Ë‡À≈’Ë¬¡¡ÿ¡©“° (3) √Ÿª “¡‡À≈’Ë¬¡3. (1) √Ÿª ’Ë‡À≈’Ë¬¡®—μÿ√— (2) √Ÿª“¡‡À≈’Ë¬¡

°‘®°√√¡∑’Ë 1 (Àπâ“ 190−191)

∑√ßÀ≈“¬Àπâ“ ®”π«π®ÿ¥¬Õ¥¡ÿ¡ (V) ®”π«πÀπâ“ (F) ®”π«π —π (E)

≈Ÿ°∫“»°å 8 6 12

∑√ß ’ËÀπâ“ 4 4 6

æ’√–¡‘¥∞“π ’Ë‡À≈’Ë¬¡®—μÿ√— 5 5 8

ª√‘´÷¡∞“πÀâ“‡À≈’Ë¬¡ 10 7 15

ª√‘´÷¡∞“πÀ°‡À≈’Ë¬¡ 12 8 18

¿“æ ¥â“π∫π ¥â“πÀπâ“ ¥â“π¢â“ß


°‘®°√√¡∑’Ë 1 (Àπâ“ 192)

1.

2.

3.

4.




°‘®°√√¡∑’Ë 2 (Àπâ“ 195)

1. 2.

3. 4.

·∫∫Ωñ°À—¥ 2

1.

(1)

(2)

(3)

(4)

√Ÿª‡√¢“§≥‘μ “¡¡‘μ‘ ¢π“π°—∫∞“π μ—Èß©“°°—∫∞“π


(5)

2.

(1)

(2)

(3)

(4)

(5)

√Ÿª‡√¢“§≥‘μ “¡¡‘μ‘ ¢π“π°—∫∞“π μ—Èß©“°°—∫∞“π



3. (1) (2)

(3)

1. (1) ’·¥ß (2) ’øÑ“ (3) ∂ÿß„∫∑’Ë 1 (4) ∂ÿß„∫∑’Ë 2

2. (1) °≈àÕß„∫∑’Ë 2 ‡æ√“–¡’®”π«π≈Ÿ°∫Õ≈ ’¡à«ßμàÕ≈Ÿ°∫Õ≈∑—ÈßÀ¡¥¡“°°«à“°≈àÕß„∫∑’Ë 1

(2) °≈àÕß„∫∑’Ë 2 ‡æ√“–¡’®”π«π≈Ÿ°∫Õ≈ ’ â¡μàÕ≈Ÿ°∫Õ≈∑—ÈßÀ¡¥¡“°°«à“°≈àÕß„∫∑’Ë 1

3. (1) ‰¡à¡’ ‡æ√“–‰¡à¡’≈Ÿ°Õ¡√ μ√Õ«å‡∫Õ√å√’Õ¬Ÿà‡≈¬

(2) ¡’‚Õ°“ ‰¥â≈Ÿ°Õ¡√ ‡¥’¬«°—π

(3) ≈Ÿ°Õ¡√ °“·ø·≈–≈Ÿ°Õ¡√ Õßÿàπ ‡æ√“–¡’®”π«π‡∑à“°—π

(4) ≈Ÿ°Õ¡√ ™ÁÕ°‚°·≈μ ‡æ√“–¡’®”π«π¡“°∑’Ë ÿ¥

(5) ≈Ÿ°Õ¡√ °“·øÀ√◊Õ≈Ÿ°Õ¡√ Õßÿàπ ‡æ√“–¡’®”π«ππâÕ¬∑’Ë ÿ¥

4. (1) ‡≈¢ 3 ‡æ√“–¡’Õ¬Ÿà 6 à«π ´÷Ëß‡ªìπ∫√‘‡«≥∑’Ë¡“°∑’Ë ÿ¥

(2) ‡≈¢ 2 ‡æ√“–¡’Õ¬Ÿà 2 à«π ´÷Ëß‡ªìπ∫√‘‡«≥∑’ËπâÕ¬∑’Ë ÿ¥

(3) ‰¡à¡’ ‡æ√“–„π·ªÑπÀ¡ÿπ‰¡à¡’μ—«‡≈¢ 4

5. (1) ¡’‚Õ°“ ‡∑à“°—π

(2) ‡ªìπ‰ª‰¡à‰¥â ‡æ√“–≈Ÿ°‡μã“‰¡à¡’·μâ¡ 7

(3) ‡ªìπ‰ª‰¥â ‡æ√“–≈Ÿ°‡μã“¡’‡©æ“–·μâ¡ 1 ∂÷ß 6

(4) ‚Õ°“ ∑’Ë≈Ÿ°‡μã“¢÷Èπ·μâ¡‰¡à‡°‘π 4

6. ‚Õ°“ ∑’Ë®–∂Ÿ°√“ß«—≈‡≈¢∑â“¬ 2 μ—« ¡’¡“°°«à“

·∫∫Ωñ°À—¥

Àπà«¬°“√‡√’¬π√Ÿâ∑’Ë 6

‚Õ°“ ¢Õß‡Àμÿ°“√≥å

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