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™◊ËÕÀπ—ß ◊Õ : ‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
ºŸâ‡√’¬∫‡√’¬ß : 𓬪√“‚¡∑¬å ¢®√¿—¬»÷°…“π‘‡∑»°å‡™’ˬ«™“≠
”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 1
ISBN : 978-974-650-860-5
æ‘¡æå§√—Èß∑’Ë 1 : ¡‘∂ÿπ“¬π æ.». 2551
®”π«πæ‘¡æå : 4,000 ‡≈à¡
CD : 4,000 ·ºàπ
æ‘¡æå∑’Ë : ‚√ßæ‘¡æå™ÿ¡πÿ¡ À°√≥å°“√‡°…μ√·Ààߪ√–‡∑»‰∑¬ ®”°—¥
79 ∂ππß“¡«ß»å«“π ·¢«ß≈“¥¬“« ‡¢μ®μÿ®—°√ °√ÿ߇∑æ¡À“π§√ 10900‚∑√. 0-2561-4567 ‚∑√ “√ 0-2579-5101𓬂™§¥’ ÕÕ ÿ«√√≥ ºâŸæ‘¡æåºâŸ‚¶…≥“ æ.». 2550
‚∑√. 02 280 5562Website : http://www.inno.obec.go.th
www.ilq.netE-mail : [email protected]
‚§√ß°“√æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ Ÿà “°≈ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“ ”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π°√–∑√«ß»÷°…“∏‘°“√
∫∑π”ç...¥â«¬§«“¡√—°„π∂‘Ëπ
√â“ߧ«“¡‡ªìπ‰∑¬
√ÿ¥Àπâ“°â“«‰°≈ Ÿà “°≈...é§≥‘μ»“ μ√å∂◊Õ‡ªìπ‡§√◊ËÕß¡◊Õ √â“ß°√–∫«π°“√§‘¥
∑’ˇªìπ«‘∑¬“»“ μ√å∑”„À⇰‘¥ “¡Õߧåª√–°Õ∫¢â“ßμâπ
‡Õ° “√π’È...𔇠πÕ°“√‡ √‘¡ √â“ß°“√§‘¥„À⇪ìπ√–∫∫
‰¡àºŸ°μ‘¥°—∫«‘∏’°“√∑’ËÕ“®¡’À≈“°À≈“¬
·μàμâÕßÕ¬Ÿà¿“¬„μâ‡Àμÿº≈§≥‘μ»“ μ√凥’¬«°—π
...ºŸâ„™â ‡≈◊Õ°„™â„Àâ Õ¥§≈âÕß°—∫ ¡Õ߇¥Á°
ª√—∫‡«≈“„Àâ‡À¡“– ¡·≈–°≈¡°≈◊π°—∫«‘∂’™’«‘μ„π∑âÕß∂‘Ëπ
...§ÿ≥¿“æ®–‡°‘¥‡¡◊ËÕ¡’°“√‡√’¬π√Ÿâ√à«¡°—π
√–À«à“ߺŸâ‡√’¬π°—∫ºŸâ®—¥°‘®°√√¡°“√‡√’¬π√Ÿâ (§√Ÿ)
”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π ¢Õ¢Õ∫§ÿ≥§≥–∑”ß“π
‚§√ß°“√æ—≤π“§ÿ≥¿“æ°“√‡√’¬π√Ÿâ Ÿà “°≈ ‚¥¬‡©æ“–Õ¬à“߬‘ËßÕ“®“√¬åª√“‚¡∑¬å ¢®√¿—¬
»÷°…“π‘‡∑»°å‡™’ˬ«™“≠ ”π—°ß“π‡¢μæ◊Èπ∑’Ë°“√»÷°…“°√ÿ߇∑æ¡À“π§√ ‡¢μ 1
™à«¬√“™°“√ ”π—°æ—≤π“π«—μ°√√¡°“√®—¥°“√»÷°…“ ”π—°ß“π§≥–°√√¡°“√
°“√»÷°…“¢—Èπæ◊Èπ∞“π ∑’ˉ¥â§âπ§«â“√«∫√«¡·≈–‡√’¬∫‡√’¬ß ‘Ëß∑’Ë¡’Õ¬Ÿà„π‡Õ° “√‡≈à¡π’È
‡æ◊ËÕ®–‰¥â𔉪„™â„À⇰‘¥ª√–‚¬™πå°—∫‡¥Á°‰∑¬
”π—°ß“π§≥–°√√¡°“√°“√»÷°…“¢—Èπæ◊Èπ∞“π
¡‘∂ÿπ“¬π 2551
“√∫—≠Àπâ“
∫∑π”
‡≈à“ Ÿà°—πøíß 1
™à«¬§‘¥§≥‘μ („Àâ) ‡√Á« 6
°‘®°√√¡°“√§‘¥ 111
§≥‘μ¬Õ¥ §‘¥‡¬’ˬ¡ 127
°≈§≥‘μ §‘¥∑—π‚≈° 135
À“®ÿ¥μà“ß 161
¢’¥ ¢’¥ ‡¢’¬π ‡¢’¬π 167
‡ß“ 171
¡Õ߇ÀÁπ ‡¢’¬π‡À¡◊Õπ 174∫√√≥“πÿ°√¡ 177
1‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
°—πøíßé‡≈à“ Ÿàç
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé2
‡√“‡√‘Ë¡ 9 ‰ªÀ“√ 7 ‰¥â 0
9 736073 ...
0
À√◊Õ°“√À“√ μ—«Õ¬à“߇™àπ
9 736
‡¡◊ËÕ‡√“§‘¥∂÷ß«‘∏’°“√¢Õ߇¥Á°‡«’¬¥π“¡°—∫°“√À“§”μÕ∫¥—ßπ’È
®“°‚®∑¬å 11 ≈∫¥â«¬ 8 ‡√“¡—°‡ÀÁπ¿“殓°°“√À“§”μÕ∫¢Õ߇¥Á° Ê ¥—ßπ’È
118
›
1183
›
7 + 3 = 10
§‘¥ 8 ¡“°°«à“ 1 Õ¬Ÿà‡∑à“‰√ (μÕ∫ 7)7 ∫«°°—∫Õ–‰√‡∑à“°—∫ 10 (μÕ∫ 3)
(8 9 10 11)
1 1 1
3
§‘¥01111
83
›
3‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
À√◊Õ‡√‘Ë¡ 9 ‰ªÀ“√ 7 ‰¡à‰¥âμâÕ߉ªÀ“√ 73...¢≥–∑’ˇ¥Á°¢ÕßÕ‘π‡¥’¬À“§”μÕ∫°“√À“√¥—ßπ’È
‡√‘Ë¡§‘¥«à“„Àâæ‘®“√≥“μ—«·√°¢Õßμ—«μ—Èß ∂â“πâÕ¬°«à“μ—«À“√ (7<9) „À⢬—∫‰ª∑“ߢ«“¡◊Õ®π°«à“®–¡’§à“¡“°°«à“μ—«À“√®÷ߧàÕ¬≈ß¡◊ÕÀ“√ (73>9) ·≈–°√–∫«π°“√μàÕ¡“°Á‡À¡◊Õπ°—π°—∫∫â“π‡√“
9 736
9 73672
16 ...97
81
Speed Maths ®÷ß欓¬“¡æŸ¥§ÿ¬‡√◊ËÕß√“«∑’˺Ÿâ‡¢’¬π‡¥‘π∑“߇¢â“√à«¡°“√·¢àߢ—π§≥‘μ»“ μ√åπ“π“™“μ‘ ·≈⫇°Á∫‡°’ˬ«ª√– ∫°“√≥宓°ª√–‡∑»∑’ˉ¥â¡’‚Õ°“ ‡¥‘π∑“߉ª¡“‡≈à“ Ÿà°—πøíß ‡æ◊ËÕª√–‚¬™πå„Àâ§√Ÿπ”‰ª∑¥≈Õß„™â‰ªª√—∫„™â„Àâ‡À¡“– ¡°—∫‡π◊ÈÕÀ“«‘™“√–¥—∫™—Èπ¢Õßμπ
Õ¬à“߉√°Áμ“¡ª√–°“√·√°∑’ˇ√“μâÕߙ૬°—π°Á§◊Õ„À⇥Á°¡’§«“¡ “¡“√∂æ◊Èπ∞“π‡°’Ë¬«°—∫ °“√∑àÕß Ÿμ√§Ÿ≥ ‡æ√“–ªí®®ÿ∫—ππ’ȇ¥Á°®”π«π‰¡àπâÕ¬∑’ˇ«≈“∂Ÿ°∂“¡ ‡™àπ9 Ó 7 ¡’§à“‡∑à“‰√ °Á¡—°®–‡ÀÁπ¿“懥Á°∫√‘°√√¡μ—Èß·μà 9 Ó 1 = 9, 9 Ó 2 = 18‰ª®π∂÷ß 9 Ó 7 = 63 ·≈–À“°„π™à«ßπ’È∂Ÿ°∂“¡¬È”«à“ 9 Ó 7 ¡’§à“‡∑à“‰√ °Á¡—°®–μ—Èßμâπ∑àÕß„À¡à 9 Ó 1 = 9 ‡™àπ°—π ®÷߇°‘¥§«“¡≈à“™â“„π°“√À“§”μÕ∫‰ª¥â«¬¥—ßπ—Èπ À“°∑àÕß Ÿμ√§Ÿ≥·≈– “¡“√∂μÕ∫‰¥âÕ¬à“ßÕ—μ‚π¡—μ‘ §◊Õ ∂“¡ªÿ∫°ÁμÕ∫ªí∫‡™àπ ∂“¡«à“ 9 Ó 7 ¡’§à“‡∑à“‰√ ·≈â«μÕ∫‰¥â‡≈¬«à“ 63 ‚¥¬‰¡à¡’°“√∑àÕ߇√‘Ë¡μâπ®–∑”„Àâ°“√§‘¥§”π«≥¡’§«“¡√«¥‡√Á«¢÷Èπ
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé4
μ—«Õ¬à“߇™àπ
«‘∏’∑”(∑’Ë à«π„À≠à§ÿâπ‡§¬)
À“§”μÕ∫¢Õß (9 + 7 ) ˚ ( + )27
29
57
59
= = 13104063
= ˚ = Ó104063
8063
104063
6380
= ( ) ˚ ( )585 + 45563
45 + 3563
(9 + 7 ) ˚ ( + ) = ( + ) ˚ ( + )27
29
57
59
57
59
657
659
ª√–°“√∂—¥¡“ °“√Ωñ°°“√§‘¥„π„® ∑’Ë查‡™àππ’ȇæ√“–‰ªæ∫‡ÀÁπ∫“ß·Ààß≈Õß∂“¡‡¥Á°«à“ 27 À“√¥â«¬ 6 ‰¥â‡∑à“‰√ ‡¥Á°®–∑”°“√∑¥‡≈Á° Ê
·≈⫧àÕ¬∫Õ°§”μÕ∫«à“ 4 ‡»… 3´÷Ë߇√◊ËÕß√“«‡À≈à“π’È πà“®–§‘¥„π„®‰¥â·≈â« ‡æ√“–§≥‘μ»“ μ√å„π∫“ß°√≥’
À“°¡’°“√∑¥¡“°‡∑à“‰√ °Á®–¬‘Ëߙⓡ“°‡∑à“π—ÈπÕ’°ª√–°“√∑’Ë®–°≈à“«∂÷ߧ◊Õ§√ŸμâÕß欓¬“¡°√–μÿâπ À√◊Õ‡ πÕ·π–§«“¡
À≈“°À≈“¬¢Õß°“√‰ª Ÿà§”μÕ∫‡À¡◊Õπ∑’Ë°≈à“«‰«â¢â“ßμâπ À√◊Õ∫“ߧ√—Èß°ÁÕ“®π”‡ πÕ‚®∑¬åªí≠À“∑’Ë¡’¡“°°«à“ 1 §”μÕ∫°Á‰¥â ¥â«¬§«“¡À≈“°À≈“¬π’È ‡¥Á°®–¡’‚Õ°“ ‡≈◊Õ°«‘∏’∑’ˇ¢“§‘¥«à“¥’„π¢≥–π—Èπ À√◊Õ„Àâ‡Àμÿº≈„π§”μÕ∫∑’ˇ¢’¬πÕÕ°¡“·≈â«¡’§«“¡ ÿ¢ ∫“¬„®„π°“√·°âªí≠À“À√◊Õ‡√’¬π√Ÿâ·μà≈–§√—Èß
6 2724
3
4
5‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
®–æ∫«à“μâÕ߇º™‘≠°—∫μ—«‡≈¢À√◊Õ®”π«π∑’Ë¡’§à“¡“° ‚Õ°“ æ≈“¥μàÕ°“√§”π«≥®÷ß¡’¡“°μ“¡‰ª¥â«¬
≈Õßæ‘®“√≥“°—∫«‘∏’π’È
= 65 ˚ 5= 13
= ( + ) ˚ ( + )57
59
657
659
(9 + 7 ) ˚ ( + )27
29
57
59
= [65 Ó ( + )] ˚ [5 Ó ( + )]17
19
17
19
‡√◊ËÕß√“«¿“¬„π‡≈à¡ Speed Maths ®÷ß欓¬“¡§âπÀ“·≈â«π”‡ πÕ‚®∑¬å∑’ˇªìπ‡√◊ËÕß√“«·≈–·π«∑“ß°“√·°âªí≠À“ ‘Ëß≈–Õ—πæ—π≈–πâÕ¬¡“𔇠πÕ ‡æ◊ËÕ„À⇰‘¥ª√–‚¬™πåμàÕ°“√𔉪„™â„π°“√®—¥°“√‡√’¬π√Ÿâ§≥‘μ»“ μ√å √–¥—∫ª√–∂¡»÷°…“ª√–°“√ ÿ¥∑⓬‡™‘≠™«π∑à“πºŸâª°§√Õß §√Ÿ ‡«≈“ Õπ查§ÿ¬§≥‘μ»“ μ√å°—∫‡¥Á°Õ¬à“μ‘¥°√Õ∫À≈—° Ÿμ√π–§√—∫ π—Ëπ§◊Õ §«√查À√◊Õ‡ªî¥‚Õ°“ „À⇥Á°‰¥â‡√’¬π√Ÿâ·≈–¢≥–‡¥’¬«°—πÀ“°μâÕß°“√„À⇥Á°‰∑¬ “¡“√∂¬◊π·π«Àπâ“√à«¡ª√–™“§¡‚≈°§«√®—¥°‘®°√√¡°“√‡√’¬π°“√ Õπ∑’ˇ¥Á°‰¥â≈ß¡◊Õ°√–∑”¡“°∑’Ë ÿ¥ Õ’°∑—È߇ªî¥°«â“ß°√Õ∫À≈—° Ÿμ√·≈–¡Õß “√–À≈—° Ÿμ√μ“¡¡“μ√∞“π “°≈¥â«¬
¢Õ∫§ÿ≥∑’Ë∑à“ππ”‡√◊ËÕß√“«¿“¬„π‡≈à¡π’ȉª„™â
ª√“‚¡∑¬å ¢®√¿—¬»÷°…“π‘‡∑»°å‡™’ˬ«™“≠
65 Ó ( + ) = 1317
19
655
5 Ó ( + ) =17
19
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé6
™à«¬§‘¥§≥‘μ („Àâ) ‡√Á«™à«¬§‘¥§≥‘μ („Àâ) ‡√Á«
‡ªìπμÕπ∑’Ë𔇠πÕ‡∑§π‘§°“√§”π«≥∑’Ë„™âÀ≈—°°“√∑’Ëπà“ π„®‰¡à«à“®–„™â°“√§√∫√âÕ¬
À√◊Õ„™â§«“¡√Ÿâ¢Õߧ«“¡§‘¥√«∫¬Õ¥ (Concepts) æ◊Èπ∞“π∫“ߪ√–°“√¡“ª√–¬ÿ°μå„™âÕ¬à“ß™“≠©≈“¥
7‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 1438 + 96 =
«‘∏’∑”438 + 96 = (438 + 100) › 4
= 538 › 4= 534
⨷Œ 2698 + 834 =
«‘∏’∑”698 + 834 = (700 + 834) › 2
= 1,534 › 2= 1,532
‚®∑¬å 3537 › 397 =
«‘∏’∑”537 › 397 = (537 › 400) + 3
= 137 + 3= 140
‚®∑¬å 4307 + 298 › 99 › 202 =
«‘∏’∑”307 + 298 › 99 › 202 = (300 + 7) + (300 › 2) › (100 › 1) › (200 + 2)
= 300 + 7 + 300 › 2 + 1 › 100 › 2 › 200= 300 + 300 › 100 › 200 + 7 › 2 + 1 › 2= 304
307 = 300 + 7298 = 300 › 299 = 100 › 1
202 = 200 + 2
∴∴∴∴∴
397 = 400 › 3537 › 397 = 537 › (400 › 3)
= (537 › 400) + 3∴∴∴∴∴
∴∴∴∴∴
698 = 700 › 2698 + 834= 700 › 2 + 834= (700 + 834) › 2
∴∴∴∴∴
∴∴∴∴∴
96 = 100 › 4498 + (100 › 4)= (498 + 100) › 4
∴∴∴∴∴
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé8
‚®∑¬å 7632 › 156 › 232 =
«‘∏’∑”632 › 156 › 232 = 632 › 232 › 156
= 400 › 156= 244
‚®∑¬å 8128 + 186 + 72 › 86 =
«‘∏’∑”128 + 186 + 72 › 86 = 128 + 72 + 186 › 86
= 200 + 100= 300
‚®∑¬å 51,009 + 196 › 505 › 97=
«‘∏’∑”1,009 + 196 › 505 › 97 = (1,000 + 9) + (200 › 4) › (500 + 5) › (100 › 3)
= 1,000 + 9 + 200 › 4 › 500 › 5 › 100 + 3= (1,000 + 200 › 500 › 100) + (9 › 4 › 5 + 3)= 603
⨷Œ 657 + 238 + 44 =
«‘∏’∑”57 + 238 + 44 = 57 + 238 + 43 + 1
= (57 + 43) + 238 + 1= 100 + 239= 339
44 = 43 + 1∴∴∴∴∴
9‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 1025 Ó 5 Ó 64 Ó 125 =
«‘∏’∑”25 Ó 5 Ó 64 Ó 125 = 25 Ó 5 Ó 2 Ó 4 Ó 8 Ó 125
= (25 Ó 4) Ó (5 Ó 2) Ó (8 Ó 125)= 100 Ó 10 Ó 1,000= 1,000,000 = 106
‚®∑¬å 9812 › 593 + 193 =
«‘∏’∑”812 › 593 + 193 = 812 › (593 › 193)
= 812 › 400= 412
‚®∑¬å 1175 Ó 16 =
«‘∏’∑”75 Ó 16 = 25 Ó 3 Ó 4 Ó 4
= (25 Ó 4) Ó (3 Ó 4)= 100 Ó 12= 1,200
64 = 2 Ó 4 Ó 8
75 = 25 Ó 316 = 4 Ó 4
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé10
‚®∑¬å 13291 + 47 Ó 97 =
«‘∏’∑”291 + 47 Ó 97 = (97 Ó 3) + 47 Ó 97
= 97 (3 + 47)= 97 Ó 50= 4,850
‚®∑¬å 1452 Ó 62 › 124 =
«‘∏’∑”52 Ó 62 › 124 = 52 Ó 62 › 62 Ó 2
= 62 (52 › 2)= 62 Ó 50= 3,100
‚®∑¬å 12101 Ó 9,999 =
«‘∏’∑”101 Ó 9,999 = (100 + 1) Ó 9,999
= 999,900 + 9,999= 1,009,899
291 = 97 Ó 3∴∴∴∴∴
11‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 158 Ó 109 › 78 Ó 9 =
«‘∏’∑”8 Ó 109 › 78 Ó 9 = 8 Ó 109 › (8 + 70) Ó 9
= 8 Ó 109 › 8 Ó 9 › 70 Ó 9= 8 (109 › 9) › 630= 800 › 630= 170
‚®∑¬å 16(360 + 108) ˚ 36 =
«‘∏’∑”(360 + 108) ˚ 36 = 360 ˚ 36 + 108 ˚ 36
= 10 + 3= 13
⨷Œ 17
«‘∏’∑”
4 › 9 + (8 › 2 ) = 4 + 8 › (9 › 2 )34
711
14
411
34
14
711
411
4 › 9 + (8 › 2 ) = 34
711
14
411
= 13 › 12= 1
= 13 › (9 + 2 )711
411
(360 + 108) ˚ 36
= +
= 10 + 3
36036
10836
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé12
‚®∑¬å 18 Ó 37 = 4445
«‘∏’∑”
= 37 › 3745
Ó 37 = (1 › ) Ó 374445
145
= 36 845
⨷Œ 19
«‘∏’∑”
27 Ó = 1526
27 Ó = (26 + 1) Ó1526
1526
= (26 Ó ) + (1 Ó )1526
1526
= 15 1526
= 36 + ›4545
3745
13‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 20
«‘∏’∑”
73 Ó = 115
18
73 Ó = (72 + ) Ó115
18
1615
18
= 72 Ó + Ó1615
18
18
= 9 + 215
= 9 215
⨷Œ 21
«‘∏’∑”
Ó 27 + Ó 41 = 15
35
= 30
= Ó 5035
= Ó (9 + 41)35
Ó 27 + Ó 41 = Ó 9 + Ó 4115
35
35
35
73 = 1 + 72115
115
= 72 + 1 115
= 72 + 1615
Ó 27 = Ó 3 Ó 915
= Ó 9
1535
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé14
⨷Œ 23
«‘∏’∑”
= 4 120
= 4 + 120
= (164 ˚ 41) + Ó4120
141
166 ˚ 41 = (164 + 2 ) ˚ 41120
120
166 ˚ 41 = 120
⨷Œ 22
«‘∏’∑”
Ó + Ó + Ó56
113
59
213
518
613
= Ó + Ó + Ó16
513
29
513
618
513
= ( + + )513
16
29
618
= ( + + )513
318
418
618
= Ó513
1318
= 518
Ó + Ó + Ó = 56
113
59
213
518
613
= (164 + ) ˚ 414120
Ó =56
5 Ó 16 Ó 13
113
=1 Ó 56 Ó 13
= Ó16
513
15‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 24
«‘∏’∑”
2,006 ˚ 2,006 = 2,0062,007
2,006 ˚ 2,006 = 2,006 ˚2,0062,007
2,006 Ó 2,007 + 2,0062,007
= 2,006 ˚ 2,006 Ó 2,0082,007
= 2,006 Ó 2,0072,006 Ó 2,008
= 2,0072,008
⨷Œ 25
«‘∏’∑”
9,039,030 ˚ 43,043 =
9,039,030 = 903 Ó 1,001 Ó 1043,043 = 43 Ó 1,001
= 210
=9,039,03043,043
903 Ó 1,001 Ó 1043 Ó 1,001
= 9,03043
∴∴∴∴∴
= 2,006 ˚ 2,006 Ó (2,007+1)2,007
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé16
⨷Œ 26
«‘∏’∑”
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 888,888 Ó 888,888
= 8 Ó 8888,888 Ó 888,888
= 1111,111 Ó 111,111
= 112,345,654,321
⨷Œ 27
«‘∏’∑”2,005 Ó 2,006 › 1
2,005 + 2,004 Ó 2,006
= 1
=(2,004 + 1) Ó 2,006 › 12,005 + 2,004 Ó 2,006
= 2,005 Ó 2,006 › 1
2,005 + 2,004 Ó 2,006
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
8 8 8
8 888 8
=2,005 + 2,004 Ó 2,0062,005 + 2,004 Ó 2,006
= =2,004 Ó 2,006 + 2,006 › 1
2,005 + 2,004 Ó 2,0062,004 Ó 2,006 + 2,0052,005 + 2,004 Ó 2,006
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1888,888 Ó 888,888
17‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 28
«‘∏’∑”
+ + = b12
13
14
°”Àπ¥„Àâ 1 + + + = a12
13
14
A = a Ó ( b + ) › ( a + ) Ó b15
15
= ab + a › ab › b15
15
= (a › b)15
= a › b15
15
A = (1 + + + ) Ó ( + + + ) › (1 + + + + ) Ó ( + + )12
13
14
12
13
14
15
12
13
14
15
12
13
14
À“§à“ A
= 15
a › b = (1 + + + ) › ( + + )= 1
12
13
14
12
13
14
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé18
= 99100
= 1 › 1100
= 1 › + › + › + ... + ›12
12
13
13
14
199
1100
⨷Œ 29
„ÀâÀ“§à“ + + + ... +11 Ó 2
12 Ó 3
13 Ó 4
199 Ó 100
«‘∏’∑”
= › ...13 Ó 4
14
13
= ›12 Ó 3
13
12
= 1 ›11 Ó 2
12
∴∴∴∴∴ + + +11 Ó 2
12 Ó 3
13 Ó 4
= (1 › ) + ( › ) + ( › ) + ...12
12
13
13
14
19‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 30
«‘∏’∑”
= 625
= ( › ) Ó12
150
12
= [ ( › ) + ( › ) + ... + ( › ) ] Ó 12
12
14
14
16
148
150
( + + + ... + ) Ó 12
22 Ó 4
24 Ó 6
26 Ó 8
248 Ó 50
+ + + ... + = 12 Ó 4
14 Ó 6
16 Ó 8
148 Ó 50
⨷Œ 31
«‘∏’∑”
∂⓺≈≈—æ∏å¢Õß 10024 › 24 ∂Ÿ°‡¢’¬π„π√Ÿª°“√°√–®“¬ „ÀâÀ“º≈√«¡¢Õßμ—«‡≈¢‚¥¥®“°º≈≈—æ∏åπ’È
1002 = 10,0001003 = 1,000,00010024 › 24 = 999...976
º≈√«¡μ—«‡≈¢‚¥¥ (9 Ó 46) + 7 + 6 = 427
46 μ—«
∴∴∴∴∴
∴∴∴∴∴
= ( ) Ó12 Ó 4
22 Ó 4
12
= Ó14
12
= ( › ) Ó 12
12
14 + + + ... +1
2 Ó 41
4 Ó 61
6 Ó 81
48 Ó 50
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé20
‚®∑¬å 320.0495 Ó 2,500 + 495 Ó 0.24 + 51 Ó 4.95 =
«‘∏’∑”0.0495 Ó 2,500 + 495 Ó 0.24 + 51 Ó 4.95= 495 Ó 0.25 + 495 Ó 0.24 + 0.51 Ó 495= 495 Ó (0.25 + 0.24 + 0.51)= 495 Ó 1= 495
‚®∑¬å 3350 Ó 96 + 57 Ó 4 + 7 Ó 66 =
«‘∏’∑”50 Ó 96 + 57 Ó 4 + 7 Ó 66= 50 Ó 96 + (50 + 7) Ó 4 + 7 Ó 66= 50 Ó 96 + 50 Ó 4 + 7 Ó 4 + 7 Ó 66= 50 Ó (96 + 4) + 7 Ó (4 + 66)= 5,000 + 490= 5,490
21‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’π’È®–‡À¡“–À“°¡’°“√À“º≈√«¡∑’ˉ¡à¡“°π—°
12 + 32 + 52 + 72 = 1 + 9 + 25 + 49= 84
À“°‰¡à„™â Ÿμ√°Á∑”«‘∏’ª°μ‘°Á‰¥â
‡¡◊ËÕ L ‡ªìπ®”π«π§’Ë∑’Ë¡’§à“¡“°∑’Ë ÿ¥®“°‚®∑¬å L = 7
6º≈√«¡ =
L Ó (L + 1) (L + 2)
º≈√«¡ = 7 Ó (7 + 1) (7 + 2)
6
= 7 Ó 8 Ó 9 = 84
6
‚®∑¬å 35„ÀâÀ“º≈∫«°¢Õß 12 + 32 + 52 + 72
«‘∏’∑”‚®∑¬å≈—°…≥–π’È„™â Ÿμ√
‚®∑¬å 349,999 Ó 7,778 + 3,333 Ó 6,666 =
«‘∏’∑”9,999 Ó 7,778 + 3,333 Ó 6,666= 9,999 Ó 7,778 + 3,333 Ó 3 Ó 2,222= 9,999 Ó 7,778 + 9,999 Ó 2,222= (7,778 + 2,222) Ó 9,999= 10,000 Ó 9,999= 99,990,000
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé22
= 1,330
‚®∑¬å 36„ÀâÀ“º≈∫«°¢Õß 12 + 32 + 52 + 72 + 92 + 112 + 132 + 152 + 172 + 192
«‘∏’∑”À“°¡’À“º≈√«¡∑’Ë¡’À≈“¬®”π«π °“√„™â Ÿμ√πà“®–‡À¡“– ¡°«à“ L = 9
®“° Ÿμ√ º≈√«¡ = L Ó (L + 1) Ó (L + 2)
6
= 19 Ó 20 Ó 21
6
12 + 32 + 52 + 72+ 92 + 112 + 132 + 152 + 172 + 192
= 1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + 289 + 361= 1,330
À“°‰¡à„™â Ÿμ√°Á∑”«‘∏’ª°μ‘‰¥â
23‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
Ÿμ√
‚®∑¬å 37„ÀâÀ“º≈∫«°¢Õß 22 + 42 + 62
«‘∏’∑”‚®∑¬å≈—°…≥–π’È„™â Ÿμ√‡™àπ‡¥’¬«°—∫°“√À“º≈∫«°¢Õß®”π«π§’ˬ°°”≈—ß Õß
º≈∫«° = L Ó (L + 1) (L + 2)
6
‡¡◊ËÕ L = ®”π«π§Ÿà∑’Ë¡’§à“¡“°∑’Ë ÿ¥= 6
= 56
º≈∫«° = 6 Ó (7) Ó (8)
6
À“°‰¡à„™â Ÿμ√°Á„™â«‘∏’ª°μ‘‰¥â
22 + 42 + 62 = 4 + 16 + 36= 56
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé24
22 + 42 + 62 + ... + 202 + 222
= 22 + 42 + 62 + 82 + 102 + 122 + 142 + 162 + 182 + 202 + 222
= 4 + 16 + 36 + 64 + 100 + 144 + 196 + 256 + 324 + 400 + 488= 2,024
L = 22
= 2,024
À“°‰¡à„™â Ÿμ√°Á„™â«‘∏’ª°μ‘‰¥â
‚®∑¬å 38„ÀâÀ“º≈∫«°¢Õß 22 + 42 + 62 + 82 + ... + 202 + 222
«‘∏’∑”
Ÿμ√ º≈∫«° = L Ó (L + 1) Ó (L + 2)
6
º≈∫«° = 22 Ó 23 Ó 24
6∴∴∴∴∴
25‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 40
π—Ëπ§◊Õ ®–‡À≈◊Õμ—«‡»…¢Õߧ”μÕ∫‡ªìπμ—«‡»…¢Õ߇»… à«π·√°·≈–μ—« à«π¢Õߧ”μÕ∫‡ªìπμ—« à«π¢Õ߇»… à«πμ—« ÿ¥∑⓬
„ÀâÀ“º≈§Ÿ≥¢Õß Ó Ó Ó Ó12
23
34
45
56
«‘∏’∑”12
23
34
45
56
Ó Ó Ó Ó = 16
‚®∑¬å 39„ÀâÀ“º≈∫«°¢Õß 13 + 23 + 33
«‘∏’∑”‚®∑¬å≈—°…≥–π’È„™â Ÿμ√
º≈∫«° = M Ó M
‡¡◊ËÕ M =
L Ó (L + 1) ‚¥¬ L = ®”π«π∑’Ë¡’§à“¡“°∑’Ë ÿ¥ 2
L = 3 ; M = 3 Ó 4 = 6
2
º≈∫«° = 6 Ó 6= 36
À“°‰¡à„™â Ÿμ√°Á„™â«‘∏’ª°μ‘‰¥â
13 + 23 + 33 = 1 + 8 +27= 36
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé26
⨷Œ 41
«‘∏’∑”12
23
34
3031
Ó Ó Ó ... Ó = 131
„ÀâÀ“º≈§Ÿ≥¢Õß Ó Ó Ó ... Ó12
23
34
3031
⨷Œ 42
«‘∏’∑”
= 5099
= 50 Ó 199
„ÀâÀ“§à“¢Õß (1 + ) (1 › ) (1 + ) (1 › ) ... (1 + ) (1 › )12
12
13
13
199
199
(1 + ) (1 › ) (1 + ) (1 › ) ... (1 + ) (1 › )12
12
13
13
199
199
= ( Ó Ó Ó ... ) ( Ó Ó ... Ó )10099
32
43
54
9899
12
23
34
27‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 43
«‘∏’∑”
À√◊Õ
„ÀâÀ“§à“¢Õß 2,005 Ó 2,0032,004
= 2,003 2,0032,004
= 2,003 + 2,0032,004
= 2,004 Ó + 1 Ó2,0032,004
2,0032,004
= 2,005 › 2,0052,004
= 2,005 (1 › )12,004
= 2,005 Ó 1 › 2,005 Ó 12,004
2,005 Ó = 2,005 Ó (1 › )2,0032,004
12,004
2,005 Ó = (2,004 + 1) Ó 2,0032,004
2,0032,004
2,005 › 2,0052,004
= 2,005 › (1 + )12,004
= 2,005 › 1 › 12,004
= 2,004 › 12,004
= 2,003 + 1 › 12,004
= 2,003 + 2,0032,004
= 2,003 2,0032,004
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé28
= ( ˚ ) Ó (1 ˚ ) Ó ( ˚ )47
27
19
59
411
211
⨷Œ 44
«‘∏’∑”
= 1
„ÀâÀ“§à“¢Õß 12 Ó 8 ˚ 9 ˚ 8 Ó 9 ˚ 1214
13
45
13
45
14
12 Ó 8 ˚ 9 ˚ 8 Ó 9 ˚ 1214
13
45
13
45
14
= 12 ˚ 12 Ó 8 ˚ 8 Ó 9 ˚ 914
14
13
13
45
45
⨷Œ 45
= 8
= 2 Ó 2 Ó 2
«‘∏’∑”
= Ó 1 Ó ˚ ˚ ˚47
19
411
211
27
59
( Ó 1 Ó ) ˚ ( Ó Ó )47
19
411
211
27
59
„ÀâÀ“§à“¢Õß ( Ó 1 Ó ) ˚ ( Ó Ó )47
19
411
211
27
59
= 1 Ó 1 Ó 1 = 1
12 ˚ 12 Ó 8 ˚ 8 Ó 9 ˚ 914
14
13
13
45
45
= Ó Ó
14
12
14
12
13
8
13
8
45
9
45
9
29‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 46¡’π—°°’Ó 8 §π ¡“·¢àߢ—π°—π‚¥¬„™â°μ‘°“·∫∫·¢àߢ—π·∫∫·æ⧗¥ÕÕ°
∂“¡«à“ ®–μâÕß·¢àß°’˧√—Èß ®÷ß®–‰¥âºŸâ™π–
À√◊ÕÕ“®À“§”μÕ∫‚¥¬°“√‡¢’¬π·ºπº—ß
‚®∑¬å≈—°…≥–π’È„™â Ÿμ√®”π«π°“√·¢àߢ—π = a › 1
‡¡◊ËÕ a ·∑π®”π«π§π∑’Ë„Àâ∑”°“√·¢àߢ—πa = 8
®”π«π°“√·¢àߢ—π = 8 › 1= 7 §√—Èß
ºŸâ™π–
1
5
2
7
3
4
6
«‘∏’∑”
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé30
‚®∑¬å 47∂â“¡’°“√®—¥°“√·¢àߢ—πøÿμ∫Õ≈ 7 ∑’¡ ‚¥¬°”Àπ¥„Àâ¡’°“√®—¥°“√·¢àߢ—π
·∫∫æ∫°—πÀ¡¥ ∂“¡«à“ √Õ∫·√°®–μâÕß¡’°“√®—¥°“√·¢àߢ—π∑—ÈßÀ¡¥°’˧√—Èß
‚®∑¬å≈—°…≥–·¢àߢ—π·∫∫æ∫°—πÀ¡¥À√◊Õ¿“…“Õ—ß°ƒ…‡√’¬°°“√·¢àߢ—π·∫∫ League „™â Ÿμ√
«‘∏’∑”
‡¡◊ËÕ a ·∑π®”π«π∑’¡ a = 7
À√◊ÕÕ“®„™â°“√‡¢’¬π®—∫§Ÿà
6 + 5 + 4 + 3 + 2 + 1 = 21 §√—Èß
A B C D E F GAB BC CD DE EF FGAC BD CE DF EGAD BE CF DGAE BF CGAF BGAG
®”π«π°“√·¢àߢ—π = a Ó (a ›1)
2
®”π«π°“√·¢àߢ—π = 7 Ó 6
= 21 §√—Èß 2∴∴∴∴∴
31‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 48A, B ·≈– C ‡ªìπ®ÿ¥ “¡®ÿ¥ ∂â“μâÕß°“√‡¢’¬π à«π¢Õ߇ âπμ√߇™◊ËÕ¡√–À«à“ß
®ÿ¥∑—Èß “¡ ®–‡¢’¬π à«π¢Õ߇ âπμ√߉¥â°’ˇ âπ
«‘∏’∑”
‡¡◊ËÕ a ·∑π®”π«π®ÿ¥∑—ÈßÀ¡¥ a = 3
≈“°‡ âπ§√∫ ®–‰¥â 3 ‡ âπÀ√◊Õ„™â Ÿμ√
®”π«π à«π¢Õ߇ âπμ√ß = a Ó (a ›1)
2
®”π«π à«π¢Õ߇ âπμ√ß = 3 Ó 2
= 3 ‡ âπ 2
BC
A
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé32
«‘∏’∑”
‚®∑¬å 49¡’®ÿ¥ 5 ®ÿ¥ ∂â“μâÕß°“√‡¢’¬π à«π¢Õ߇ âπμ√߇™◊ËÕ¡√–À«à“ß®ÿ¥ ®–‰¥â à«π¢Õß
‡ âπμ√ß∑—ÈßÀ¡¥°’ˇ âπ
a = 5
„™â Ÿμ√
®”π«π à«π¢Õ߇ âπμ√ß = a Ó (a › 1)
2
®”π«π à«π¢Õ߇ âπμ√ß = 5 Ó 4
= 10 ‡ âπ 2∴∴∴∴∴
33‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 50¡’®ÿ¥ 15 ®ÿ¥ ∂â“μâÕß°“√‡¢’¬π à«π¢Õ߇ âπμ√ß ‡™◊ËÕ¡μàÕ®ÿ¥∑—Èß 15 ®ÿ¥
®–‰¥â à«π¢Õ߇ âπμ√ß°’ˇ âπ
«‘∏’∑”‡¡◊ËÕ¡’®ÿ¥∂÷ß 15 ®ÿ¥ °“√‡¢’¬π à«π¢Õ߇ âπμ√ß®√‘ß Ê Õ“® √â“ߧ«“¡¬ÿà߬“°
∑”„Àâ°“√„™â Ÿμ√¥Ÿ«à“®–ßà“¬°«à“
Ÿμ√
®”π«π à«π¢Õ߇ âπμ√ß = a Ó (a › 1)
2‡¡◊ËÕ a = 15
= 105 ‡ âπ
®”π«π à«π¢Õ߇ âπμ√ß = = 15 Ó 7 2
15 Ó 14∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé34
«‘∏’∑”
‚®∑¬å 51„ÀâÀ“®”π«π∑’ËÀ“¬‰ª„π ( ) ¢Õß
= +16
1( )
1( )
„™â°“√·¬°μ—«ª√–°Õ∫ 6 = 2 Ó 3
®–‰¥â«à“ =16
12 Ó 3
= Ó1
2 Ó 3(2 + 3)(2 + 3)
=2 + 3
2 Ó 3 Ó 5
= +2
2 Ó 3 Ó 53
2 Ó 3 Ó 5
= +115
110
35‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 52„ÀâÀ“§à“ A, B, C „π ( ) ¢Õß
«‘∏’∑”
18 = 2 Ó 3 Ó 3 ( —߇°μ«à“À“°μâÕß°“√ “¡«ß‡≈Á∫°Á·¬°‡ªìπ 3 ®”π«π§Ÿ≥°—π)
= + +1( A )
1( B )
118
1( C )
= + +172
148
148
= + +2
2 Ó 3 Ó 3 Ó 83
2 Ó 3 Ó 3 Ó 83
2 Ó 3 Ó 3 Ó 8
= Ó1
2 Ó 3 Ó 3(2 + 3 + 3)(2 + 3 + 3)
=1
2 Ó 3 Ó 31
18
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé36
«‘∏’∑”
‚®∑¬å 53„ÀâÀ“§à“¢Õß
= 199
101
= 2 Ó (1 › )1101
= 2 Ó ( + › + › + ... + › )12
12
13
13
14
1100
1101
= 2 Ó + 2 Ó ( › ) + 2 Ó ( › ) + ... + 2 Ó ( › )12
12
13
13
14
1100
1101
®“°‚®∑¬å = 1 + 2 Ó ( › ) + 2 ( › ) + ... + 2 Ó ( › )12
13
13
14
1100
1101
= 2 Ó ( › )11 + 2 + 3 + ... + 100
1100
1101
= = = 2 Ó ( › )11 + 2 + 3
1(1 + 3) Ó 3
2
23 Ó (1 + 3)
13
14
= = = 2 Ó ( › )11 + 2
1(1 + 2) Ó 2
2
22 Ó (1 + 2)
12
13
1 + + + ... +1
1 + 21
1 + 2 + 31
1 + 2 + 3 + ... + 100
= 2 › = =2
101202 › 2
101200101
37‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
«‘∏’∑’Ë 1
‚®∑¬å 54ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— √ŸªÀπ÷Ëß ¡’§«“¡¬“«¥â“π 23 ‡´π쑇¡μ√
∂ⓧ«“¡¬“«¥â“π·μà≈–¥â“π‡æ‘Ë¡¢÷Èπ 20% ∂“¡«à“ æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— √Ÿªπ’È®–‡æ‘Ë¡¢÷Èπ®“°‡¥‘¡°’ˇªÕ√凴Áπμå
23 ´¡.
A B
D C
120100
§«“¡¬“«¥â“π„À¡à Ó 23 = 27.6 ´¡.
§‘¥‡ªìπ Ó 100 = 44%232.76529
æ◊Èπ∑’Ë√Ÿª ABCD „À¡à ‡∑à“°—∫ 27.6 Ó 27.6 = 761.76 ´¡.2
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ®“°‡¥‘¡ 761.76 › 529 = 232.76 ´¡.2®
æ◊Èπ∑’Ë√Ÿª ABCD ‡∑à“°—∫ 23 Ó 23 = 529 ´¡.2
§«“¡¬“«¥â“π‡æ‘Ë¡¢÷Èπ 20%
®
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé38
«‘∏’∑’Ë 2
23 ´¡.
A B
D C10 ´¡.
A B
D C
§«“¡¬“«¥â“π‡æ‘Ë¡¢÷Èπ 20%
§«“¡¬“«¥â“π„À¡à Ó 10 = 12 ´¡.120100
∴∴∴∴∴
æ◊Èπ∑’Ë√Ÿª ABCD „À¡à ‡∑à“°—∫ 12 Ó 12 = 144 ´¡.2
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 144 › 100 = 44%
®
„À⧑¥«à“√Ÿª ¡’§«“¡¬“«¥â“π
10 ´¡. æ◊Èπ∑’Ë√Ÿª ABCD‡∑à“°—∫ 10 Ó 10 ´¡.2
®
®
39‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
«‘∏’∑’Ë 1
79 π‘È«
M N
P O
§«“¡¬“«„À¡à Ó 79 = 86.9 π‘È«110100
§‘¥‡ªìπ Ó 100 = 21%1,310.616,241∴∴∴∴∴
æ◊Èπ∑’Ë√Ÿª MNOP „À¡à ‡∑à“°—∫ 86.9 Ó 86.9 = 7,551.61 π‘È«2
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ®“°‡¥‘¡ 7,551.61 › 6,241 = 1,310.61 π‘È«2
®
æ◊Èπ∑’Ë√Ÿª MNOP ‡∑à“°—∫ 79 Ó 79 = 6,241 π‘È«2
§«“¡¬“«‡æ‘Ë¡¢÷Èπ 10%
®
‚®∑¬å 55√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— MNOP ¡’§«“¡¬“«¥â“π 79 π‘È« ∂ⓧ«“¡¬“«·μà≈–¥â“π
‡æ‘Ë¡¢÷Èπ 10% æ◊Èπ∑’Ë√Ÿª MNOP ‡æ‘Ë¡¢÷Èπ°’ˇªÕ√凴Áπμå®
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé40
«‘∏’∑’Ë 2
79 π‘È«
M N
P O10 π‘È«
M N
P O
§«“¡¬“«¥â“π‡æ‘Ë¡¢÷Èπ 10%
§«“¡¬“«¥â“π„À¡à Ó 10 = 11 π‘È«110100
æ◊Èπ∑’Ë√Ÿª MNOP „À¡à ‡∑à“°—∫ 11 Ó 11 = 121 π‘È«2
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 121›100 = 21%
®
„À⧑¥«à“√Ÿª ¡’§«“¡¬“«¥â“π
10 π‘È« æ◊Èπ∑’Ë√Ÿª MNOP‡∑à“°—∫ 10 Ó 10 = 100 π‘È«2
®
®
41‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
«‘∏’∑’Ë 1
‚®∑¬å 56EFGH ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“ ¡’§«“¡¬“«¥â“π¥—ß√Ÿª
«‘∏’∑’Ë 2
æ◊Èπ∑’Ë„À¡à 64.8 Ó 33.6 = 2,177.28 ´¡.2
§‘¥‡ªìπ Ó 100 = 44%665.281,512
EH = Ó 28 = 33.6 ´¡.120100
EF = Ó 54 = 64.8 ´¡.120100
§«“¡¬“«„À¡à Ó 10 = 12 ´¡.120100
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 144›100 = 44%
54 ´¡.
E F
H G
28 ´¡.
æ◊Èπ∑’Ë√Ÿª EFGH „À¡à‡æ‘Ë¡¢÷Èπ 2,177.28›1,512 = 665.28 ´¡.2º
æ◊Èπ∑’Ë√Ÿª EFGH ‡¥‘¡ ‡∑à“°—∫ 54 Ó 28 = 1,512 ´¡.2
§«“¡¬“«¥â“π„À¡à
º
∂ⓧ«“¡¬“«·μà≈–¥â“π‡æ‘Ë¡¢÷Èπ 20% æ◊Èπ∑’Ë√Ÿª EFGH ®–‡æ‘Ë¡¢÷Èπ®“°‡¥‘¡°’ˇªÕ√凴Áπμå
º
„À⧑¥«à“√Ÿª ¡’§«“¡¬“«¥â“π ¥â“π≈– 10 ´¡.º
æ◊Èπ∑’Ë√Ÿª EFGH ‡∑à“°—∫ 10 Ó 10 = 100 ´¡.2º
æ◊Èπ∑’Ë√Ÿª EFGH „À¡à ‡∑à“°—∫ 12 Ó 12 = 144 ´¡.2º
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé42
«‘∏’∑”
«‘∏’∑’Ë 1
«‘∏’∑’Ë 2
§‘¥‡ªìπ Ó 100 = 32%627.21,960
¥â“π°«â“ß„À¡à Ó 35 = 38.5 ´¡.110100
¥â“𬓫„À¡à Ó 56 = 67.2 ´¡.120100
§«“¡¬“«¥â“𬓫‡æ‘Ë¡‡ªìπ Ó 10 = 12 ´¡.120100
§«“¡¬“«¥â“π°«â“߇æ‘Ë¡‡ªìπ Ó 10 = 11 ´¡.110100
æ◊Èπ∑’Ë√Ÿª ABCD „À¡à ‡∑à“°—∫ 12 Ó 11 = 132 ´¡.2
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 132 › 100 = 32%
º
„À⧑¥«à“√Ÿª ¡’§«“¡¬“«¥â“π·μà≈–¥â“π‡∑à“°—∫ 10 ´¡.
æ◊Èπ∑’Ë√Ÿª ABCD ‡∑à“°—∫ 10 Ó 10 = 100 ´¡.2
º
º
æ◊Èπ∑’Ë√Ÿª ABCD „À¡à ‡∑à“°—∫ 67.2 Ó 38.5 = 2,587.2 ´¡.2
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 2,587.2 › 1,960 = 627.2 ´¡.2º
æ◊Èπ∑’Ë√Ÿª ABCD ‡¥‘¡ ‡∑à“°—∫ 56 Ó 35 = 1,960 ´¡.2º
‚®∑¬å 57ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“ ¡’¥â“𬓫 50 ´¡. ¥â“π°«â“ß 35 ´¡.
∂â“¥â“𬓫¡’§«“¡¬“«‡æ‘Ë¡¢÷Èπ 20% ¥â“π°«â“ß¡’§«“¡¬“«‡æ‘Ë¡¢÷Èπ 10%
æ◊Èπ∑’Ë√Ÿª ABCD ®–‡æ‘Ë¡¢÷Èπ®“°‡¥‘¡°’ˇªÕ√å‡ Áπμåº
43‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
«‘∏’∑’Ë 1
= 4%
¥â“π°«â“ß„À¡à Ó 8 = 6.4 ¡.80100
¥â“𬓫„À¡à Ó 17 = 22.1 ¡.130100
«‘∏’∑’Ë 2
¥â“π°«â“ß„À¡à Ó 10 = 8 ¡.80
100
¥â“𬓫„À¡à Ó 10 = 13 ¡.130100
æ◊Èπ∑’Ë√Ÿª ABCD ‡¥‘¡ ‡∑à“°—∫ 17 Ó 8 = 136 ¡.2º
æ◊Èπ∑’Ë√Ÿª ABCD „À¡à ‡∑à“°—∫ 6.4 Ó 22.1 = 141.44 ¡.2º
æ◊Èπ∑’ˇæ‘Ë¡ §‘¥‡ªìπ Ó 100 = Ó 100141.44›136136 136
5.44
„À⧑¥«à“√Ÿª ¡’§«“¡¬“«¥â“π·μà≈–¥â“π‡∑à“°—∫ 10 ¡.
æ◊Èπ∑’Ë√Ÿª ABCD ‡¥‘¡ 10 Ó 10 = 100 ¡.2º
º
æ◊Èπ∑’Ë√Ÿª ABCD „À¡à ‡∑à“°—∫ 13 Ó 8 = 104 ¡.2
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 104 › 100 = 4%
º
‚®∑¬å 58ABCD ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“ ¡’¥â“𬓫 17 ¡. ¥â“π°«â“ß 8 ¡. ∂â“¥â“𬓫
‡æ‘Ë¡¢÷Èπ 30% ¥â“π°«â“ß≈¥≈ß 20% æ◊Èπ∑’Ë√Ÿª ABCD ®–‡æ‘Ë¡¢÷ÈπÀ√◊Õ≈¥≈ß°’ˇªÕ√凴Áπμå
º
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé44
«‘∏’∑”
«‘∏’∑’Ë 1
‚®∑¬å 59∂ⓧ«“¡¬“«∞“π·≈–§«“¡ Ÿß¢Õß√Ÿª “¡‡À≈’ˬ¡ ABC ‡∑à“°—∫ 30 ·≈– 40 ´¡.
μ“¡≈”¥—∫ ∂â“∑—Èߧ«“¡¬“«∞“π·≈–§«“¡ Ÿß‡æ‘Ë¡¢÷Èπ 10% æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡π’ȇæ‘Ë¡¢÷Èπ®“°‡¥‘¡°’ˇªÕ√凴Áπμå
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 726 › 600 = 126 ´¡.2
æ◊Èπ∑’Ë√Ÿª „À¡à ‡∑à“°—∫ Ó 33 Ó 44 = 726 ´¡.212
æ◊Èπ∑’Ë√Ÿª ‡¥‘¡ ‡∑à“°—∫ Ó ∞ Ó = Ó 30 Ó 40 = 600 ´¡.212
12
§«“¡¬“«∞“π„À¡à Ó 30 = 33 ´¡.110100
§«“¡ Ÿß„À¡à Ó 40 = 44 ´¡.110100
§‘¥‡ªìπ Ó 100 = 21%126600
«‘∏’∑’Ë 2
„À⧑¥«à“§«“¡¬“«¥â“π√Ÿª ‡∑à“°—∫ 10 ´¡.
æ◊Èπ∑’ˇæ‘Ë¡¢÷Èπ 60.5 › 50 = 10.5 ´¡.2
æ◊Èπ∑’Ë√Ÿª ‡¥‘¡ ‡∑à“°—∫ Ó 10 Ó 10 = 50 ´¡.212
§«“¡¬“«∞“π„À¡à Ó 10 = 11 ´¡.110100
§«“¡ Ÿß„À¡à Ó 10 = 11 ´¡.110100
æ◊Èπ∑’Ë„À¡à Ó 11 Ó 11 = = 60.5 ´¡.212
1212
§‘¥‡ªìπ = Ó 100 = 21%10.550
10.550
45‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
«‘∏’∑’Ë 1
‚®∑¬å 60§«“¡¬“« §«“¡°«â“ß ·≈–§«“¡ Ÿß¢Õß°≈àÕß„∫Àπ÷Ë߇∑à“°—∫ 50 ´¡. 30 ´¡.
·≈– 20 ´¡. μ“¡≈”¥—∫∂ⓧ«“¡¬“«·≈–§«“¡°«â“߇æ‘Ë¡¢÷Èπ 20% ·≈– 10% μ“¡≈”¥—∫ ·≈–§«“¡ Ÿß
≈¥≈ß 30%ª√‘¡“μ√®–‡æ‘Ë¡¢÷ÈπÀ√◊Õ≈¥≈ß®“°‡¥‘¡°’ˇªÕ√å‡ Áπμå
ª√‘¡“μ√„À¡à 60 Ó 33 Ó 14 = 27,720 ´¡.3
ª√‘¡“μ√≈¥≈ß 30,000 › 27,720 = 2,280 ´¡.3
ª√‘¡“μ√‡¥‘¡ 50 Ó 30 Ó 20 = 30,000 ´¡.3
§‘¥‡ªìπ Ó 100 = 7.6%2,28030,000
§«“¡ Ÿß„À¡à Ó 20 = 14 ´¡.70
100
§«“¡°«â“ß„À¡à Ó 30 = 33 ´¡.110100
§«“¡¬“«„À¡à Ó 50 = 60 ´¡.120100
«‘∏’∑’Ë 2„À⧑¥«à“§«“¡¬“«¥â“π·μà≈–¥â“π‡∑à“°—∫ 10 ´¡.ª√‘¡“μ√‡¥‘¡‡∑à“°—∫ 10 Ó 10 Ó 10 = 1,000 ´¡.3
ª√‘¡“μ√„À¡à ‡∑à“°—∫ 12 Ó 11 Ó 7 = 924 ´¡.3
ª√‘¡“μ√≈¥≈ß 1,000 › 924 = 76 ´¡.3 = 7.6%
§«“¡¬“«„À¡à Ó 10 = 12 ´¡.120100
§«“¡°«â“ß„À¡à Ó 10 = 11 ´¡.110100
§«“¡ Ÿß„À¡à Ó 10 = 7 ´¡.70
100
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé46
«‘∏’∑”
«‘∏’∑’Ë 1
‚®∑¬å 61«√πÿ™∑”¢âÕ Õ∫§≥‘μ»“ μ√å ®”π«π 100 ¢âÕ ∂â“∑”∂Ÿ°·μà≈–¢âÕ®–‰¥â¢âÕ≈–
1 §–·ππ ∂â“∑”º‘¥ 1 ¢âÕ ®–∂Ÿ°À—°¢âÕ≈– 2 §–·ππ ∂â“«√πÿ™‰¥â§–·ππ„π°“√ Õ∫§√—Èßπ’È 73 §–·ππ «√πÿ™∑”¢âÕ Õ∫™ÿ¥π’Ⱥ‘¥°’Ë¢âÕ
„Àâ«√πÿ™∑”¢âÕ Õ∫∂Ÿ° a ¢âÕ∑”º‘¥ 100 › a ¢âÕ‡¢’¬π ¡°“√‰¥â«à“
1a › 2 (100 › a) = 73a › 200 + 2a = 733a = 273a = 91
«√πÿ™∑”¢âÕ Õ∫º‘¥ 100 › 91 = 9 ¢âÕ
«‘∏’∑’Ë 2
· ¥ß«à“ À“°∑”º‘¥ 1 ¢âÕ §–·ππ®–À“¬‰ª 100 › 97 = 3 §–·ππ
√â“ßμ“√“ß
100 0 100 0 10099 1 99 ›2 97
¢âÕ Õ∫∑’Ë ¢âÕ Õ∫∑’Ë §–·ππ §–·ππ §–·ππ√«¡∑”∂Ÿ° ∑”º‘¥ ∑”∂Ÿ° ∑”º‘¥
«√πÿ™‰¥â 73 §–·ππ À“¬‰ª 27 §–·ππ ∑”º‘¥ = 9 ¢âÕ273
∴∴∴∴∴
47‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
‚®∑¬å 62°√–¥“…√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“·ºàπÀπ÷Ëß °«â“ß 20 ´¡. ¬“« 25 ´¡. μâÕß°“√μ—¥
‡ªìπ°√–¥“…¢π“¥ 2 Ó 2 ´¡.2 ®–‰¥â∑—ÈßÀ¡¥°’Ë·ºàπ
10 Ó 12 = 120®–μâÕ߉¥â∑—ÈßÀ¡¥ 120 ·ºàπ
= 10202
= 12.5 12 („Àâ„™â§à“®”π«π‡μÁ¡∑’Ë¡’§à“πâÕ¬‰¡àμâÕßªí¥‡ªìπ 13)252
‚®∑¬å 63°√–¥“…√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“ §«“¡¬“« 35 ´¡. §«“¡°«â“ß 18 ´¡. μâÕß°“√
μ—¥‡ªìπ°√–¥“…¢π“¥ 3 Ó 3 ´¡.2 ®–‰¥â∑—ÈßÀ¡¥°’Ë·ºàπ
«‘∏’∑”
6 Ó 11 = 66®–μ—¥‰¥â∑—ÈßÀ¡¥ 66 ·ºàπ
= 6183
≈≈≈≈≈ 11.67 11 („Àâ„™â§à“®”π«π‡μÁ¡∑’Ë¡’§à“πâÕ¬‰¡àμâÕßªí¥‡ªìπ 12)353
∴∴∴∴∴
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé48
«‘∏’∑”
‚®∑¬å 64∑àÕπ‰¡â∑√ß ’ˇÀ≈’ˬ¡¡ÿ¡©“°¢π“¥ 15 Ó 16 Ó 17 ´¡.3 π”¡“μ—¥‡ªìπ∑àÕπ‰¡â
¢π“¥ 2 Ó 2 Ó 2 ´¡.3 ®–‰¥â¡“°∑’Ë ÿ¥°’Ë∑àÕπ
7 Ó 8 Ó 8 = 448®–‰¥â¡“°∑’Ë ÿ¥ 448 ∑àÕπ
= 7.5 7152
= 8162
= 8.5 8172
‚®∑¬å 65º≈√«¡¢Õß®”π«π 2 ®”π«π ‡∑à“°—∫ 68 ·≈–º≈μà“ߢÕß Õß®”π«ππ’È
‡∑à“°—∫ 32 „ÀâÀ“®”π«π Õß®”π«ππ’È
«‘∏’∑”
= 50
= 18
= 362
= ( )68 › 322
®”π«π∑’Ë¡’§à“πâÕ¬ = (º≈√«¡ › º≈μà“ß)2
= 1002
®”π«π∑’Ë¡’§à“¡“° =(º≈√«¡ + º≈μà“ß)
2
∴∴∴∴∴
= ( )68 + 322
49‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
°”Àπ¥μ—«‡≈¢≈ß„π™àÕß√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°
‚®∑¬å 66®“°√Ÿª
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°°’Ë√Ÿª
«‘∏’∑”
«‘∏’∑’Ë 1°”Àπ¥Õ—°…√·∑π√Ÿª
®–‰¥â«à“√Ÿª a = 1 √Ÿªb = 1 √Ÿªa + b = 1 √Ÿª
®–¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“° 3 √Ÿª
a b
1 2
«‘∏’∑’Ë 2
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥ 1 + 2 = 3 √Ÿª∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé50
‚®∑¬å 67®“°√Ÿª
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥°’Ë√Ÿª
‡μ‘¡μ—«‡≈¢≈ß„π™àÕß«à“ß
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥ 1 + 2 + 3 = 6 √Ÿª
«‘∏’∑”
1 2 3
‚®∑¬å 68®“°√Ÿª
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥°’Ë√Ÿª
‡μ‘¡μ—«‡≈¢≈ß„π™àÕß«à“ß
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥ 1 + 2 + 3 + 4 + 5 = 15 √Ÿª
«‘∏’∑”
1 2 3 4 5
∴∴∴∴∴
∴∴∴∴∴
51‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 69®“°√Ÿª
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥°’Ë√Ÿª
‡μ‘¡μ—«‡≈¢≈ß„π™àÕß«à“ß·∂«·√°¥â“π≈à“ß
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥ 1 + 2 + 2 + 4 = 9 √Ÿª
‡μ‘¡μ—«‡≈¢≈ß„π™àÕß«à“ß·∂«∑’Ë Õß
«‘∏’∑”
1 2
1 2
2 4Ó 2
∴∴∴∴∴
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé52
‚®∑¬å 70®“°√Ÿª
«‘∏’∑”
‡μ‘¡μ—«‡≈¢≈ß„π™àÕß«à“ß·∂«∑’Ë Õß
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥°’Ë√Ÿª
‡μ‘¡μ—«‡≈¢≈ß„π™àÕß«à“ß·∂«·√°¥â“π≈à“ß
¡’√Ÿª ’ˇÀ≈’ˬ¡¡ÿ¡©“°∑—ÈßÀ¡¥ 1 + 2 + 3 + 4 + 2 + 4 + 6 + 8 = 30 √Ÿª
1 2 3 4
1 2 3 4
2 4 6 8
∴∴∴∴∴
53‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 71°”Àπ¥ A ·≈– B ‡ªìπ®”π«π‡μÁ¡∫«° (®”π«ππ—∫) ∑’Ë¡’§à“‰¡à‡∑à“°—π ·≈–
·μà≈–®”π«π¡’§à“√–À«à“ß 10 ∂÷ß 150 „ÀâÀ“§à“∑’Ë¡“°∑’Ë ÿ¥∑’ˇªìπ‰ª‰¥â¢Õß
«‘∏’∑”
A + BA › B
°“√À“§”μÕ∫¢Õß‚®∑¬å¢âÕπ’È ∂â“μâÕß°“√º≈≈—æ∏å∑’Ë¡’§à“¡“°∑’Ë ÿ¥ μ—«‡»…μâÕß¡’§à“¡“°∑’Ë ÿ¥ ·≈–μ—« à«πμâÕß¡’§à“πâÕ¬∑’Ë ÿ¥
μ—«‡»…∑’Ë¡’§à“¡“°∑’Ë ÿ¥ A + B μâÕß¡’§à“¡“°∑’Ë ÿ¥·≈–μ—« à«π∑’Ë¡’§à“πâÕ¬∑’Ë ÿ¥ A › B μâÕß¡’§à“πâÕ¬∑’Ë ÿ¥¥—ßπ—Èπ A = 149 ·≈– B = 148
= 297149 + 148149 › 148
μÕ∫ 297
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé54
‚®∑¬å 72°”Àπ¥ °° + ¢¢ + §§ = ¢°§‡¡◊ËÕ °, ¢ ·≈– § ·∑π¥â«¬μ—«‡≈¢‚¥¥∑’Ë¡’§à“·μ°μà“ß°—π„ÀâÀ“§à“ °, ¢ ·≈– §
«‘∏’∑”
‡¡◊ËÕº≈√«¡„πÀ≈—°Àπ૬‡∑à“°—∫ §· ¥ß«à“ ° + ¢ = 10 ¡’°“√∑¥ 1 „πÀ≈—° ‘∫‡¡◊ËÕº≈√«¡„πÀ≈—° ‘∫‡∑à“°—∫ °· ¥ß«à“ ¢ + § + 1 = 10À√◊Õ ¢ + § = 9‡¡◊ËÕ ¢ §◊Õº≈∫«°„πÀ≈—°√âÕ¬ ‡ªìπ‰ª‰¥â«à“ ¢ ‡∑à“°—∫ 1 À√◊Õ 2∂â“ ¢ = 2, ° = 8 ·≈– § = 7·μàº≈√«¡§◊Õ 88 + 22 + 77 = 187·≈–‡¡◊ËÕ ¢ = 1, ° = 9 ·≈– § = 8º≈√«¡®–‡∑à“°—∫ 99 + 11 + 88 = 198
°°, ¢¢, §§ À¡“¬∂÷ß ®”π«π∑’Ë¡’ 2 À≈—°¢°§ À¡“¬∂÷ß ®”π«π∑’Ë¡’ 3 À≈—°
μÕ∫ ° = 9, ¢ = 1 ·≈– § = 8
55‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 73
«‘∏’∑”
MNOP ‡ªìπ√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ¡’§«“¡¬“«¥â“π 2 ´¡. ∂â“≈“°‡ âπ®“° M ‰ª 0‡ âπ∑’Ë —Èπ∑’Ë ÿ¥®–¬“« 4 ´¡. ∂“¡«à“∂Ⓡ√“®–≈“°‡ âπ®“° M ‰ª 0 „Àâ¡’√–¬–∑“ß —Èπ∑’Ë ÿ¥ ‰¥â°’ˇ âπ∑“ß∑’ˉ¡à´È”°—π
1 ´¡.P
M
O
N
1 ´¡.
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé56
⨷Œ 74
«‘∏’∑”
∂“¡«à“ μ“√“ß 8 Ó 8 ®–¡’√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ∑—ÈßÀ¡¥°’Ë√Ÿª
®“°μ“√“ß 8 Ó 8
(¢π“¥¢Õߥâ“π)2 ®”π«π√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— 1 Ó 1 32 = 92 Ó 2 22 = 43 Ó 3 12 = 1
√«¡ 14
(¢π“¥¢Õߥâ“π)2 ®”π«π√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— 1 Ó 1 42 = 162 Ó 2 32 = 93 Ó 3 22 = 44 Ó 4 12 = 1
√«¡ 30
(¢π“¥¢Õߥâ“π)2 ®”π«π√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— 1 Ó 1 82 = 642 Ó 2 72 = 493 Ó 3 62 = 364 Ó 4 52 = 255 Ó 5 42 = 166 Ó 6 32 = 97 Ó 7 22 = 48 Ó 8 12 = 1
√«¡ 204
μ“√“ß 3 Ó 3
μ“√“ß 4 Ó 4
μÕ∫
57‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
π” §Ÿ≥μ≈Õ¥15
(1) › (2)
A = 145
A › A = 115
‚®∑¬å 75„ÀâÀ“§à“¢Õß 1 + + + + + ...1
51
251
1251
625
„Àâ A = 1 + + + + + ... (1)15
125
1125
1625
A = + + + + + ... (2)15
15
125
1125
1625
13,125
A = 1 Ó = = 154
54
14
μÕ∫ 14
1
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé58
«‘∏’∑”
‚®∑¬å 76„ÀâÀ“§à“¢Õß + + + + + ...1
214
18
116
132
π” §Ÿ≥μ≈Õ¥12
A = + + + + ... (2)14
18
116
132
12
„Àâ A = + + + + + ... (1)12
14
18
116
132
(1) › (2)
A = 1
A =12
12
A › A =12
12
μÕ∫ 1
59‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
‚®∑¬å 77∂â“ M = 3 + + + + + ...3
643
2563
1634
„ÀâÀ“§à“ M + 2
M = 3 + + + + + ... (1)364
3256
316
34
π” §Ÿ≥μ≈Õ¥14
M = + + + + ... (2)14
364
3256
316
34
(1) › (2)
M = 4
M = 334
M › M = 314
M + 2 = 4 + 2 = 6∴∴∴∴∴
μÕ∫ 6
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé60
«‘∏’∑”
‚®∑¬å 78∂â“ P = 0.5 + 0.05 + 0.005 + 0.0005 + ...
(1) › (2)
0.9 P = 0.5
P › 0.1 P = 0.5
P = 59
P = 0.5 + 0.05 + 0.005 + 0.0005 + ... (1)π” 0.1 §Ÿ≥μ≈Õ¥0.1 P = + 0.05 + 0.005 + 0.0005 + ... (2)
= P Ó = Ó =∴∴∴∴∴P9
19
59
19
581
„ÀâÀ“§à“ P9
μÕ∫ 581
61‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
∂â“ 5 ‰ªÀ“√®”π«π· ¥ßÀπⓉ¡à≈ßμ—«‡≈¢‚¥¥„πÀ≈—°Àπ૬μâÕ߉¡à„™à 5·≈–‡¡◊ËÕμ“¡‡ß◊ËÕπ‰¢‚®∑¬å 5 μâÕßÕ¬ŸàÀ≈—° ‘∫‡∑à“π—Èπ‡≈¢Àπâ“∑’ˇªìπ‰ªμ“¡‚®∑¬å°Á§◊Õ51, 52, 53, 54, 56, 57, 58, 59151, 152, 153, 154, 156, 157, 158, 159251, 252, 253, 254, 256, 257, 258, 259351, 352, 353, 354, 356, 357, 358, 359451, 452, 453, 454, 456, 457, 458, 459¡’∑—ÈßÀ¡¥ 40 Àπâ“
‚®∑¬å 79Àπ—ß ◊Õ§≥‘μ»“ μ√å‡≈à¡Àπ÷Ëß¡’ 500 Àπâ“ ¡’°’ËÀπâ“∑’Ë¡’‡≈¢Àπâ“¡’ 5 Õ¬Ÿà¥â«¬
·μà®”π«π∑’Ë· ¥ßÀπâ“π’È À“√¥â«¬ 5 ‰¡à≈ßμ—«
«‘∏’∑”
μÕ∫ 40 Àπâ“
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé62
«‘∏’∑”
‚®∑¬å 80„™â∑àÕπ‰¡â‡≈Á° Ê ∑’Ë¡’§«“¡¬“«‡∑à“°—π π”¡“μàÕ°—π¥—ß√Ÿª
= 630 ∑àÕπ= 60 + 3 Ó 190 = 60 + 570
√Ÿª∑’Ë n ®–¡’∑àÕπ‰¡â‡∑à“°—∫ n Ó 3 + 3 Ón (n › 1)
2
√Ÿª∑’Ë 20 ®–¡’∑àÕπ‰¡â‡∑à“°—∫ (20 Ó 3) + 3 Ó∴∴∴∴∴20 (20 › 1)
2
∂“¡«à“ √Ÿª∑’Ë 20 ¡’∑àÕπ‰¡â°’Ë∑àÕπ
π”¢âÕ¡Ÿ≈®“°°“√μàÕ¡“‡¢’¬π≈ß„πμ“√“ß
√Ÿª∑’Ë 1√Ÿª∑’Ë 2
√Ÿª∑’Ë 3
√Ÿª∑’Ë ®”π«π∑àÕπ‰¡â∑’Ë„™âμàÕ1 32 93 184 305 45. .. .. .
= 60 + 3 Ó400 › 20
2
μÕ∫ 630 ∑àÕπ
63‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
„ÀâÀ“º≈√«¡„π·∂«∑’Ë 2,008
«‘∏’∑”
= 2,017,036
·∂«∑’Ë n =n (n + 1)
2
·∂«∑’Ë 2,008 =2,008 (2,008 + 1)
2
‚®∑¬å 81°”Àπ¥„Àâ º≈√«¡·∂«∑’Ë 1... 1 = 1·∂«∑’Ë 2... 1 + 2 = 3·∂«∑’Ë 3... 1 + 2 + 3 = 6·∂«∑’Ë 4... 1 + 2 + 3 + 4 = 10
. . .. . .. . .
μÕ∫ 2,017,036
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé64
«‘∏’∑”
‚®∑¬å 82‡≈¢‚¥¥„πÀ≈—°Àπ૬¢Õß®”π«π∑’Ë¡’ ÕßÀ≈—° ¡’§à“‡æ‘Ë¡¢÷Èπ®“°‡¥‘¡ 50%
·≈–‡≈¢‚¥¥„πÀ≈—° ‘∫¢Õß®”π«π∑’Ë¡’ ÕßÀ≈—°π’È ¡’§à“‡æ‘Ë¡¢÷Èπ®“°‡¥‘¡ 100% æ∫«à“®”π«π„À¡àπ’È¡’§à“¡“°°«à“®”π«π‡¥‘¡Õ¬Ÿà 33
„ÀâÀ“®”π«π‡¥‘¡
„Àâ®”π«π‡¥‘¡ ‡∑à“°—∫ xy ¡’§à“ 10x + y
®”π«π‡¥‘¡ 36∴∴∴∴∴
®“°‚®∑¬å(20x + 1.5y) › (10x + y) = 3320x + 1.5y › 10x › y = 3310x + 0.5y = 33
®–‰¥â«à“ (10 Ó 3) + (0.5 Ó 6) = 33x = 3y = 6
®”π«π„À¡à 10 x + y = 20x + 1.5y[ ]200100
150100[ ]( ) ( )
μÕ∫ 36
65‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
‚®∑¬å 83 à«πº ¡π¡°—∫πÈ” ®”π«π 40 ≈‘μ√ ¡’Õ—μ√“ à«ππ¡μàÕπÈ” ‡∑à“°—∫ 3 : 1
®–μâÕß‡μ‘¡πÈ”°’Ë≈‘μ√ ‡æ◊ËÕ„ÀâÕ—μ√“ à«ππ¡μàÕπÈ”°≈“¬‡ªìπ 2 : 1
= 5 ≈‘μ√
πÈ”∑’Ë‡μ‘¡ ‡∑à“°—∫40 (3 Ó 1 › 1 Ó 2)
2 (3 + 1)
∂â“ à«πº ¡π¡°—∫πÈ” ®”π«π x ≈‘μ√ ¡’Õ—μ√“ à«ππ¡°—∫πÈ”‡∑à“°—∫ a : b ·≈â«μâÕß°“√À“ª√‘¡“≥πÈ”∑’Ë®–‡μ‘¡‡¢â“‰ª„π à«πº ¡π’ȇæ◊ËÕ„ÀâÕ—μ√“ à«ππ¡μàÕπÈ”‡ª≈’ˬπ‡ªìπ c : d
ª√‘¡“≥πÈ”∑’Ë‡μ‘¡ ‡∑à“°—∫x (ad › bc)c (a + b)
μÕ∫ 5 ≈‘μ√
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé66
«‘∏’∑”
‚®∑¬å 84 à«πº ¡ ¡’Õ—μ√“ à«ππ¡μàÕπÈ” ‡∑à“°—∫ 3 : 2 ∂â“‡μ‘¡πÈ”‡¢â“‰ª 4 ≈‘μ√
∑”„ÀâÕ—μ√“ à«ππ¡μàÕπÈ”‡∑à“°—π „ÀâÀ“ª√‘¡“≥π¡°—∫πÈ”„πμÕπ·√°
à«πº ¡ ¡’Õ—μ√“ à«ππ¡μàÕπÈ” ‡∑à“°—∫ a : b ∂â“‡μ‘¡πÈ”≈߉ªx ≈‘μ√ ∑”„ÀâÕ—μ√“ à«ππ¡μàÕπÈ”°≈“¬‡ªìπ a : c
·≈–¡’πÈ”bx
c › b
‡¥‘¡ à«πº ¡π’È®–¡’π¡Õ¬Ÿàax
c › b
ª√‘¡“≥πÈ” ‡∑à“°—∫ = 8 ≈‘μ√2 Ó 43 › 2
ª√‘¡“≥π¡ ‡∑à“°—∫ = 12 ≈‘μ√3 Ó 43 › 2
μÕ∫ π¡ 12 ≈‘μ√πÈ” 8 ≈‘μ√
67‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 85ªí®®ÿ∫—πæàÕ¡’Õ“¬ÿ‡ªìπ 4 ‡∑à“¢Õß≈Ÿ° à«π‡¡◊ËÕ 5 ªï°àÕπ æàÕÕ“¬ÿ‡ªìπ 9 ‡∑à“
¢Õß≈Ÿ° ªí®®ÿ∫—πæàÕÕ“¬ÿ°’˪ï
«‘∏’∑”
®“° Ÿμ√ A
ªí®®ÿ∫—πæàÕÕ“¬ÿ 4 Ó 8 = 32 ªï
= 8 ªï
=5 (9 › 1)
9 › 4
Õ“¬ÿ≈Ÿ° =t1 (X › 1)X › Y
t1 ªï°àÕπ ªí®®ÿ∫—π t
2 ªï¢â“ßÀπâ“
Õ“¬ÿæàÕμàÕÕ“¬ÿ≈Ÿ°À√◊Õ
Õ“¬ÿæàÕ‡ªìπ°’ˇ∑à“x ‡∑à“ y ‡∑à“ z ‡∑à“
¢ÕßÕ“¬ÿ≈Ÿ°
A Õ“¬ÿ≈Ÿ° = B Õ“¬ÿ≈Ÿ° =t1 (x › 1)
x › y
(z › 1) t2
y › z
C Õ“¬ÿ≈Ÿ° =t2 (z › 1) + t
1 (x › 1)
x › z
μÕ∫ 32 ªï
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé68
‚®∑¬å 865 ªï¢â“ßÀπâ“ æàÕÕ“¬ÿ‡ªìπ 3 ‡∑à“¢Õß≈Ÿ° ·≈– 5 ªï∑’Ë·≈â« æàÕÕ“¬ÿ‡ªìπ 7 ‡∑à“
¢Õß≈Ÿ°ªí®®ÿ∫—π·μà≈–§πÕ“¬ÿ°’˪ï
«‘∏’∑”®“° Ÿμ√ c
∂â“ F §◊ÕÕ“¬ÿæàÕ„πªí®®ÿ∫—πF + 5 = 3 (10 + 5)
F = 40 ªï
Õ“¬ÿ≈Ÿ° ‡∑à“°—∫ = 10 ªï5 (7 › 1) + 5 (3 ›1)
7 › 3
Õ¥’μ ªí®®ÿ∫—π Õπ“§μ
5 ªï 5 ªï
≈Ÿ° = X › 10 ªï ≈Ÿ° = X › 5 ªï ≈Ÿ° = X ªïæàÕ = 3X › 10 ªï æàÕ = 3X › 5 ªï æàÕ = 3X ªï
∴∴∴∴∴ 3X › 10 = 7 (X › 10)3X › 10 = 7X › 10
60 = 4X15 = X
μÕ∫ æàÕ 40 ªï≈Ÿ° 10 ªï
69‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 8710 ªï∑’Ë·≈â« A ¡’Õ“¬ÿ‡ªìπ§√÷ËßÀπ÷ËߢÕß B ∂â“ªí®®ÿ∫—πÕ—μ√“ à«π¢ÕßÕ“¬ÿ
¢Õß∑—Èß Õߧπ‡∑à“°—∫ 3 : 4 ªí®®ÿ∫—π∑—Èß ÕߧπÕ“¬ÿ§π≈–‡∑à“‰√
«‘∏’∑”
10 ªï∑’Ë·≈â« A Õ“¬ÿ ¢Õß B12
ªí®®ÿ∫—π A Õ“¬ÿ ¢Õß B34
®“° Ÿμ√ A Õ“¬ÿ¢Õß B ‡∑à“°—∫ = 20 ªï10 ( ›1)1
2
›12
34
A Õ“¬ÿ Ó 20 = 15 ªï34∴∴∴∴∴
μÕ∫ A Õ“¬ÿ 15 ªïB Õ“¬ÿ 20 ªï
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé70
«‘∏’∑”
‚®∑¬å 8888 + 898 + 8,998 + 89,998 + 899,998 ¡’§à“‡∑à“°—∫‡∑à“‰√
88 + 898 + 8,998 + 89,998 + 899,998= (90 › 2) + (900 › 2) + (9,000 › 2) + (90,000 › 2) + (900,000 › 2)= 90 + 900 + 9,000 + 90,000 + 900,000 › 10= 999,980
μÕ∫ 999,980
71‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
‚®∑¬å 894 › 54 › 1,234 › 1,766 + 37,531 + 6 + 2,469 › 146 ¡’§à“‡∑à“°—∫
‡∑à“‰√
4 › 54 › 1,234 › 1,766 + 37,531 + 6 + 2,469 › 146= (4 + 6) › (54 + 146) › (1,234 + 1,766) + (37,531 + 2,469)= 10 › 200 › 3,000 + 40,000= 36,810
μÕ∫ 36,810
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé72
«‘∏’∑”
‚®∑¬å 90„ÀâÀ“§à“¢Õß 15 Ó 64 Ó 75 Ó 375
15 Ó 64 Ó 75 Ó 375 = (3 Ó 5) Ó (2 Ó 4 Ó 8) Ó (3 Ó 25) Ó (3 Ó 125)= 3 Ó 3 Ó 3 Ó (5 Ó 2) Ó (4 Ó 25) Ó (8 Ó 125)= 27 Ó 10 Ó 100 Ó 1,000= 27,000,000
μÕ∫ 27,000,000
73‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
⨷Œ 91
„ÀâÀ“§à“ P
= 2,004
= 2,003 +2637
1137
2,003 + = P1137
234,234,234333,333,333
P = 2,003 + 1137
234,234,234333,333,333
= 2,003 +234 Ó 1,001,001333 Ó 1,001,001
1137
26
78
234
333
111
37
μÕ∫ 2,004
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé74
«‘∏’∑”
⨷Œ 92
= 1,111
15,678
16,789
„ÀâÀ“§à“¢Õß 6,789 Ó ( › ) Ó 5,678
6,789 Ó ( › ) Ó 5,67815,678
16,789
= 6,789 Ó ( ) Ó 5,6786,789 › 5,6785,678 Ó 6,789
= 6,789 Ó Ó 5,6781,1115,678 Ó 6,789
μÕ∫ 1,111
75‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
⨷Œ 93
= 2
= 105
= + + +15
25
35
45
«‘∏’∑”
+ + +15
2255
333555
4,4445,555
„ÀâÀ“§”μÕ∫¢Õß + + +15
2255
333555
4,4445,555
μÕ∫ 2
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé76
«‘∏’∑”
⨷Œ 9412
14
18
1128
„ÀâÀ“§”μÕ∫¢Õß 128 + 64 + 32 + ... + 2
128 + 64 + 32 + ... + 212
14
18
1128
= 254 + (1 › )1128
= 254 127128
= 128 Ó 2 › 2 + (1 › ) + ( › ) + ( › ) + ... + ( › )12
12
14
14
18
164
1128
= 128 + 64 + 32 + 16 + 8 + 4 + 2 + ( + + + ... + )12
14
18
1128
μÕ∫ 254127128
77‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
‚®∑¬å 95„ÀâÀ“º≈≈—æ∏å¢Õß202 + 313 + 424 + 535 + 646 + 757 + 868 + 979
202 + 313 + 424 + 535 + 646 + 757 + 868 + 979
= 4,724
= (202 + 979) Ó 4
= 1,181 Ó 4
202 + 313 + 424 + 535 + 646 + 757 + 868 + 979
1,1811,181
1,1811,181
μÕ∫ 4,724
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé78
«‘∏’∑”
2.1 + 2.91 + 2.991 + 2.9991 + 2.99991= 2 + 3 + 3 + 3 + 3 + 0.1 › 0.09 › 0.009 › 0.0009 › 0.00009= 14.1 › 0.09999= 14.00001
‚®∑¬å 962.1 + 2.91 + 2.991 + 2.9991 + 2.99991 ¡’§à“‡∑à“°—∫‡∑à“‰√
μÕ∫ 14.00001
79‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
‚®∑¬å 97∂â“¡’°“√°”Àπ¥«à“
·≈â«„ÀâÀ“§à“¢Õß
12 * 3
A * B = ›1A Ó B
1(A + 1) Ó (B + 1)
= 112
= 2 › 112
2 * 3 = › = ›12 Ó 3
1(2 + 1) Ó (3 + 1)
16
13 Ó 4
= 12
= 1 ˚ 112∴∴∴∴∴
12 * 3
μÕ∫ 12
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé80
«‘∏’∑”
⨷Œ 98
2,004 ˚ 2,004 › 2,003 ˚ 2,0032,0042,005
2,0032,004
2,004 ˚ 2,004 › 2,003 ˚ 2,0032,0042,005
2,0032,004
=2
2,005
= 1 + ›1
2,0052,0042,005
= (1 + ) › 2,003 Ó1
2,0052,004
2,003 (2,004 + 1)
= (2,004 + ) ˚ 2,004 › 2,003 ˚ ( )2,0042,005
2,003 Ó 2,004 + 2,0032,004
= ( + ) › 2,003 Ó2,0042,004
2,0042,003 Ó 2,004 + 2,003
2,0042,005 Ó 2,004
=2,005 + 1 › 2,004
2,005
= + ›2,0052,005
12,005
2,0042,005
μÕ∫ 22,005
81‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
„ÀâÀ“§à“ PQ › (P + Q)
7P › 13Q = 10
› =P13
Q7
1091
= 1091
7P › 13Q91
·≈–°”Àπ¥„Àâ › =P13
Q7
1091
®“° 7 Ó P › 13 Ó Q ‡∑à“°—∫ 10 π”¡“‡¢’¬πμ“√“߉¥â«à“
P = 7, Q = 3∴∴∴∴∴
PQ › (P + Q) = 21 › 10 = 11∴∴∴∴∴
μ—«§Ÿ≥1 2 3 4 5 6 7 8 9 10
º≈§Ÿ≥
P Ó 7 7 14 21 28 35 42 49 56 63 70Q Ó 13 13 26 39 42 65 78 91 104 117 130
⨷Œ 99
P °—∫ Q ‡ªìπ®”π«π∑’Ë¡’§à“‡∑à“°—π
= (7 Ó 3) › (7 + 3)
μÕ∫ 11
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé82
⨷Œ 100
«‘∏’∑”
®”π«ππ—∫®“° 1 ∂÷ß 700 ®–¡’μ—«‡≈¢‚¥¥ 7 ª√“°Ø∑—ÈßÀ¡¥°’Ëμ—«
¡’‡≈¢‚¥¥ 7 ª√“°Ø¢÷Èπ (20 Ó 7) + 1 = 141 μ—«∴∴∴∴∴
1 › 99 ¡’ 7 20 μ—«100 › 999 ¡’ 7 20 μ—«200 › 299 ¡’ 7 20 μ—«300 › 399 ¡’ 7 20 μ—«400 › 499 ¡’ 7 20 μ—«500 › 599 ¡’ 7 20 μ—«600 › 699 ¡’ 7 20 μ—«700 ¡’ 7 1 μ—«
μÕ∫ 141 μ—«
83‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
‚®∑¬å 101∂â“°”Àπ¥„Àâ x3 = 250,047
y4 = 50,625
§.√.π. ¢Õß x ·≈– y ‡ªìπ°’ˇ∑à“¢Õß À.√.¡. ¢Õß x ·≈– y
x3 = 250,047= 7 Ó 7 Ó 7 Ó 9 Ó 9 Ó 9
x = 7 Ó 9= 7 Ó 3 Ó 3
y4 = 50,625= 3 Ó 3 Ó 3 Ó 3 Ó 5 Ó 5 Ó 5 Ó 5
y = 3 Ó 5À.√.¡. ¢Õß x ·≈– y §◊Õ 3§.√.π. ¢Õß x ·≈– y §◊Õ 315§.√.π. ¢Õß x ·≈– y §‘¥‡ªìπ 105 ‡∑à“ ¢Õß À.√.¡. x ·≈– y
∴∴∴∴∴
∴∴∴∴∴
∴∴∴∴∴
μÕ∫ 105 ‡∑à“
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé84
‚®∑¬å 102„ÀâÀ“§”μÕ∫¢Õß 38 Ó 3
«‘∏’∑”
·μà°“√𔇠πÕ§√—Èßπ’È ‡ªìπ°“√§Ÿ≥®“° ⓬‰ª¢«“ (ºŸâ‡¢’¬π»÷°…“‡Õ° “√®“°kenneth R Williams. 2006. VEDIC MATHEMATICS Clntermediate level). Delhi ; JainendraPrakashjain Shrijainendra Press.) ∂◊Õ‡ªìπ∑“߇≈◊Õ°Àπ÷Ëß¡“𔇠πÕ
°“√§Ÿ≥·∫∫π’ÈÕ“®¡’§«“¡·μ°μà“ß®“° ‘Ëß∑’Ëπ—°‡√’¬π‰¥â√—∫√Ÿâ¡“ ‡æ√“–À“°‡®Õ‚®∑¬å°“√§Ÿ≥ 38 Ó 3 °Á®–‡√‘Ë¡§Ÿ≥®“°À≈—°Àπ૬°àÕπ ¥—ßπ’È
§”Õ∏‘∫“¬
38 Ó 3 = 114∴∴∴∴∴
§Ÿ≥À≈—°Àπ૬ §Ÿ≥À≈—° ‘∫ ‰¥â§”μÕ∫
8 Ó 3 = 24‡¢’¬π 4 ∑¥ 2
3 Ó 3 = 99 + 2 = 11
34
Ó328
3114
Ó328 38 Ó 3 = 114
39, 24
Ó38
114
μÕ∫ 114
85‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
®“° 38 Ó 3 =
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
3Ó
38
9, 2411
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 38 Ó 3‰¥â§”μÕ∫ 9, 24 (3 Ó 3 = 9 ·≈– 8 Ó 3 = 24)
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 38 Ó 3 = 114
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé86
«‘∏’∑”
78 Ó 8 = 624∴∴∴∴∴
8156, 64
Ó78
2624
§”Õ∏‘∫“¬
®“° 78 Ó 8 =
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
8Ó
78
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 78 Ó 8‰¥â§”μÕ∫ 56, 64 (7 Ó 8 = 56 ·≈– 8 Ó 8 = 64)
‚®∑¬å 103„ÀâÀ“§”μÕ∫ 78 Ó 8
μÕ∫ 624
87‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 78 Ó 8 = 624
156, 642
624
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé88
‚®∑¬å 104„ÀâÀ“§”μÕ∫ 237 Ó 3
«‘∏’∑”
237 Ó 3 = 711∴∴∴∴∴
316, 9, 21
Ó237
1711
§”Õ∏‘∫“¬
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
3Ó
237
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 237 Ó 3‰¥â§”μÕ∫ 6, 9, 21 (2 Ó 3 = 6, 3 Ó 3 = 9, 7 Ó 3 = 21)
®“° 237 Ó 3
μÕ∫ 711
89‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 237 Ó 3 = 711
16, 9, 211
711
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé90
‚®∑¬å 105„ÀâÀ“§”μÕ∫ 357 Ó 6
«‘∏’∑”
357 Ó 6 = 2,142∴∴∴∴∴
·∫∫Ωñ°À—¥
6118, 30, 42
Ó357
42,1421
‡π◊ËÕß®“°ºŸâ‡¢’¬π§‘¥‡Õß«à“ «‘∏’°“√§Ÿ≥®“°´â“¬‰ª¢«“π’È π—°‡√’¬πÕ“®‰¡à§ÿâπ‡§¬π—°®÷ß¡’·∫∫Ωñ°À—¥„Àâ≈ÕßΩñ°∑”¥Ÿ (‚¥¬‰¡à„™â«‘∏’‡¥‘¡ Ê ‡æ◊ËÕÀ“§”μÕ∫) ·≈â«Õ“®„™â«‘∏’°“√§ÿâπ‡§¬À√◊Õμ√«® Õ∫§«“¡∂Ÿ°μâÕßÀ√◊Õ‡∑’¬∫°—∫‡©≈¬∑⓬·∫∫Ωñ°À—¥°Á‰¥â
„ÀâÀ“§”μÕ∫¢Õß°“√§Ÿ≥·μà≈–¢âÕμàÕ‰ªπ’È
1. 2. 3.
4. 5.
4Ó
83
8Ó
79
3Ó
256
8Ó
334
6Ó
5,432
μÕ∫ 2,142
91‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‡©≈¬·∫∫Ωñ°À—¥
(μ“¡«‘∏’°“√§Ÿ≥®“° ⓬‰ª¢«“)
1. «‘∏’∑”4
Ó83
83 Ó 4 = 332∴∴∴∴∴
432, 12
Ó83
3332
§”Õ∏‘∫“¬
4Ó
83
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 83 Ó 4‰¥â§”μÕ∫ 32, 12 (8 Ó 4 = 32 ·≈– 3 Ó 4 = 12)
μÕ∫ 332
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé92
32, 123
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 83 Ó 4 = 332
2. «‘∏’∑”8
Ó79
79 Ó 8 = 632∴∴∴∴∴
8156, 72
Ó79
3632
§”Õ∏‘∫“¬
8Ó
79
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
μÕ∫ 632
93‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 79 Ó 8‰¥â§”μÕ∫ 56, 72 (7 Ó 8 = 56 ·≈– 9 Ó 8 = 72)
156, 723
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 79 Ó 8 = 632
3. «‘∏’∑”3
Ó256
256 Ó 3 = 768∴∴∴∴∴
36, 15, 18
Ó256
6768
7
§”Õ∏‘∫“¬
3Ó
256
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
μÕ∫ 768
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé94
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 256 Ó 3‰¥â§”μÕ∫ 6, 15, 18 (2 Ó 3 = 6, 5 Ó 3 = 15 ·≈– 6 Ó 3 = 18)
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 256 Ó 3 = 768
6, 15, 1867
4. «‘∏’∑”8
Ó334
334 Ó 8 = 2,672∴∴∴∴∴
824, 24, 32
Ó334
72,6726
μÕ∫ 2,672
95‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
§”Õ∏‘∫“¬
8Ó
334
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 334 Ó 8‰¥â§”μÕ∫ 24, 24, 32 (3 Ó 8 = 24, 3 Ó 8 = 24 ·≈– 4 Ó 8 = 32)
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 334 Ó 8 = 2,672
24, 24, 3276
5. «‘∏’∑”
6Ó
5,432
5,432 Ó 6 = 32,592∴∴∴∴∴
630, 24, 18, 12
Ó5,432
932,592
52
μÕ∫ 32,592
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé96
§”Õ∏‘∫“¬
6Ó
5,432
¢—Èπ∑’Ë 1‡¢’¬π®”π«πμ“¡·π«μ—Èß
¢—Èπ∑’Ë 2‡√‘Ë¡§Ÿ≥®“°®”π«π∑“ߴ⓬¡◊Õ 5,432 Ó 6‰¥â§”μÕ∫ 30, 24, 18, 12 (5 Ó 6 = 30, 4 Ó 6 = 24, 3 Ó 6 = 18·≈– 2 Ó 6 = 12)
¢—Èπ∑’Ë 3∫«°®”π«πμ“¡‡ âπ‚§âß
¢—Èπ∑’Ë 4‰¥â§”μÕ∫ 5,432 Ó 6 = 32,592
30, 24, 18, 12952
97‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 106„ÀâÀ“§”μÕ∫¢Õß 88 Ó 97
«‘∏’∑”
88 Ó 97 = 8,536∴∴∴∴∴
§”Õ∏‘∫“¬
¢—Èπ∑’Ë 1„Àâæ‘®“√≥“«à“ 88 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 12·≈– 97 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 3
®“°‚®∑¬å 106 ‡ªìπ°“√§Ÿ≥®”π«π Õß®”π«π∑’Ë¡’§à“„°≈⇧’¬ß 100 ·≈–¡’«‘∏’°“√∑’Ëμà“ß®“°°“√§Ÿ≥®”π«π∑’Ë¡’ ÕßÀ≈—°¥â«¬®”π«π∑’Ë¡’ ÕßÀ≈—°·∫∫∑’˧ÿâπ‡§¬°—π®÷ß¡’§”Õ∏‘∫“¬ ¥—ßπ’È
¢—Èπ∑’Ë 2π”®”π«π Õß®”π«π∑’˧Ÿ≥°—π¡“‡¢’¬π„π√Ÿª∑’ˇ πÕπ’È
88 88 › 12
97 97 › 3Ó
85 / 36
88 88 › 12
97 97 › 3Ó
μÕ∫ 8,536
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé98
¢—Èπ∑’Ë 3§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°·√°∑“ߴ⓬¡◊Õ ‡°‘¥®“°°“√≈∫„π·π«∑·¬ß88 › 3 = 85 À√◊Õ 97 › 12 = 85
¢—Èπ∑’Ë 4§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°∂—¥‰ª∑“ߢ«“¡◊Õ ‡°‘¥®“°°“√§Ÿ≥ 12 Ó 3 = 36
®–‰¥â§”μÕ∫¢Õß 88 Ó 97 = 8,536
85 /
88 › 12
97 › 3Ó
85 / 36
88 › 12
97 › 3Ó
99‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 107„ÀâÀ“§”μÕ∫¢Õß 93 Ó 96
«‘∏’∑”
93 Ó 96 = 8,928∴∴∴∴∴
89 / 28 89 / 28
93 93 › 7 93 › 07
96 96 › 4 96 › 04Ó À√◊Õ Ó
§”Õ∏‘∫“¬«‘∏’°“√À“§”μÕ∫§≈⓬ Ê °—∫‚®∑¬å 109
¢—Èπ∑’Ë 1„Àâæ‘®“√≥“«à“ 93 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 7·≈– 96 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 4
¢—Èπ∑’Ë 2π”®”π«π Õß®”π«π∑’˧Ÿ≥°—π¡“‡¢’¬π„π√Ÿª∑’ˇ πÕπ’È
93 93 › 07
96 96 › 04Ó
μÕ∫ 8,928
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé100
¢—Èπ∑’Ë 3§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°·∂«∑“ߴ⓬¡◊Õ ‡°‘¥®“°°“√≈∫„π·π«∑·¬ß96 › 7 = 89 À√◊Õ 93 › 4 = 89
¢—Èπ∑’Ë 4§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°∂—¥‰ª∑“ߢ«“¡◊Õ ‡°‘¥®“°°“√§Ÿ≥ 7 Ó 4 = 28
®–‰¥â§”μÕ∫ 93 Ó 96 = 8,928
89 /
93 › 07
96 › 04Ó
89 / 28
93 › 07
96 › 04Ó
·∫∫Ωñ°À—¥
„ÀâÀ“§”μÕ∫
1. 2.
3. 4.
98Ó
87
87Ó
99
96Ó
98
99Ó
99
≈ÕßΩñ°°—∫·∫∫Ωñ°À—¥ ·≈⫇∑’¬∫°—∫‡©≈¬ (¢Õ„Àâ≈ÕßΩñ°À“§”μÕ∫¥â«¬«‘∏’μ“¡μ—«Õ¬à“ßπ–§√—∫)
101‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
1. «‘∏’∑”98
Ó87
87 Ó 98 = 8,526∴∴∴∴∴
‡©≈¬
98 › 02
85 / 26
Ó87 › 13
§”Õ∏‘∫“¬
¢—Èπ∑’Ë 1„Àâæ‘®“√≥“«à“ 87 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 13·≈– 98 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 2
¢—Èπ∑’Ë 2π”®”π«π Õß®”π«π∑’˧Ÿ≥°—π¡“‡¢’¬π„π√Ÿª∑’ˇ πÕπ’È
87 87 › 13
98 98 › 02Ó
μÕ∫ 8,526
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé102
¢—Èπ∑’Ë 3§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°·√°∑“ߴ⓬¡◊Õ ‡°‘¥®“°°“√≈∫„π·π«∑·¬ß98 › 13 = 85 À√◊Õ 87 › 2 = 85
¢—Èπ∑’Ë 4§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°∂—¥‰ª∑“ߢ«“¡◊Õ ‡°‘¥®“°°“√§Ÿ≥ 13 Ó 2 = 26
®–‰¥â§”μÕ∫ 87 Ó 98 = 8,526
85 /
87 › 13
98 › 02Ó
85 / 26
87 › 13
98 › 02Ó
2. «‘∏’∑”87
Ó99
99 Ó 87 = 8,613∴∴∴∴∴
87 › 13
86 / 13
Ó99 › 01
μÕ∫ 8,613
103‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
§”Õ∏‘∫“¬
¢—Èπ∑’Ë 1„Àâæ‘®“√≥“«à“ 99 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 1
87 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 13
¢—Èπ∑’Ë 2π”®”π«π Õß®”π«π∑’˧Ÿ≥°—π¡“‡¢’¬π„π√Ÿª∑’ˇ πÕπ’È
99 99 › 01
87 87 › 13Ó
¢—Èπ∑’Ë 3§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°·√°∑“ß â“¬¡◊Õ ‡°‘¥®“°°“√≈∫„π·π«∑·¬ß99 › 13 = 85 À√◊Õ 87 › 01 = 86
®–‰¥â§”μÕ∫ 99 Ó 87 = 8,613
86 /
99 › 01
87 › 13Ó
86 / 13
99 › 01
87 › 13Ó
¢—Èπ∑’Ë 4§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°∂—¥‰ª∑“ߢ«“¡◊Õ ‡°‘¥®“°°“√§Ÿ≥ 1 Ó 13 = 13
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé104
3. «‘∏’∑”96
Ó98
98 Ó 96 = 9,408∴∴∴∴∴
96 › 04
94 / 08
Ó98 › 02
§”Õ∏‘∫“¬
¢—Èπ∑’Ë 1„Àâæ‘®“√≥“«à“ 98 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 2
96 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 4
¢—Èπ∑’Ë 2π”®”π«π Õß®”π«π∑’˧Ÿ≥°—π¡“‡¢’¬π„π√Ÿª∑’ˇ πÕπ’È
98 98 › 02
96 96 › 04Ó
¢—Èπ∑’Ë 3§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°·√°∑“ߴ⓬¡◊Õ ‡°‘¥®“°°“√≈∫„π·π«∑·¬ß96 › 02 = 94 À√◊Õ 98 › 04 = 94
94 /
98 › 02
96 › 04Ó
μÕ∫ 9,408
105‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
®–‰¥â§”μÕ∫ 98 Ó 96 = 9,408
94 / 08
98 › 02
96 › 04Ó
¢—Èπ∑’Ë 4§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°∂—¥‰ª∑“ߢ«“¡◊Õ‡°‘¥®“°°“√§Ÿ≥ 2 Ó 4 = 8 „π°√≥’π’ȉ¥â§”μÕ∫‡ªìπÀπ÷ËßÀ≈—°®÷ßμâÕ߇¢’¬πº≈§Ÿ≥∑’ˇªìπ ÕßÀ≈—° ‚¥¬‡μ‘¡ 0 „πÀ≈—° ‘∫‡ªìπ 08
4. «‘∏’∑”99
Ó99
99 Ó 99 = 9,801∴∴∴∴∴
99 › 01
98 / 01
Ó99 › 01
§”Õ∏‘∫“¬
¢—Èπ∑’Ë 1„Àâæ‘®“√≥“«à“ 99 ¡’§à“πâÕ¬°«à“ 100 Õ¬Ÿà 1
μÕ∫ 9,801
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé106
¢—Èπ∑’Ë 2π”®”π«π Õß®”π«π∑’˧Ÿ≥°—π¡“‡¢’¬π„π√Ÿª∑’ˇ πÕπ’È
99 99 › 01
99 99 › 01Ó
¢—Èπ∑’Ë 3§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°·√°∑“ß â“¬¡◊Õ ‡°‘¥®“°°“√≈∫„π·π«∑·¬ß99 › 01 = 98
98 /
99 › 01
99 › 01Ó
®–‰¥â§”μÕ∫ 99 Ó 99 = 9,801
98 / 01
99 › 01
99 › 01Ó
¢—Èπ∑’Ë 4§”μÕ∫¢Õߺ≈§Ÿ≥ ÕßÀ≈—°∂—¥‰ª∑“ߢ«“¡◊Õ‡°‘¥®“°°“√§Ÿ≥ 1 Ó 1 = 1 „π°√≥’π’ȉ¥â§”μÕ∫‡ªìπÀπ÷ËßÀ≈—°®÷ßμâÕ߇¢’¬πº≈§Ÿ≥π’ȇªìπ ÕßÀ≈—° ‚¥¬‡μ‘¡ 0 „πÀ≈—° ‘∫‡ªìπ 01
107‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 108„ÀâÀ“º≈∫«°¢Õß 76 + 89
«‘∏’∑”
§”Õ∏‘∫“¬«‘∏’°“√À“§”μÕ∫≈—°…≥–π’È ‡À¡◊Õπ°—∫«à“‡ªìπ°“√À“º≈∫«°®”π«π‡≈Á° Ê
°“√∫«°‡√“®–∫«°®”π«π∑’≈–À≈—° ‚¥¬‡√“∫«°®”π«π„πÀ≈—°„¥ °Á‡¢’¬πº≈∫«°„ÀâÀ≈—°Àπ૬¢Õߺ≈∫«°μ√ß°—∫®”π«π∑’Ëπ”¡“∫«°°—𠇙àπ 76 + 89 °Á¥”‡π‘π°“√¥—ßπ’È
¢—Èπ∑’Ë 1∫«°À≈—°Àπ૬
15 15
76 76
89 89+
À√◊Õ
+
15 15
165 165
15
76
89+
μÕ∫ 165
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé108
¢—Èπ∑’Ë 2∫«°À≈—° ‘∫
15
76
89+
15
¢—Èπ∑’Ë 3π”º≈∫«°„π¢—Èπ∑’Ë 1 ·≈–º≈∫«°„π¢—Èπ∑’Ë 2 ¡“∫«°°—π
®–‰¥â§”μÕ∫¢Õß 76 + 89 = 165
15
76
89+
15
165
109‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 109„ÀâÀ“º≈∫«°¢Õß 712 + 399
«‘∏’∑”
11
712
399+
10
1,111
10
§”Õ∏‘∫“¬
¢—Èπ∑’Ë 1∫«°À≈—°Àπ૬
11
712
399+
¢—Èπ∑’Ë 2∫«°À≈—° ‘∫
11
712
399+
11
μÕ∫ 1,111
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé110
¢—Èπ∑’Ë 3∫«°À≈—°√âÕ¬
11
712
399+
11
11
¢—Èπ∑’Ë 4π”º≈∫«°„π¢—Èπ∑’Ë 1-3 ¡“∫«°°—π
®–‰¥â§”μÕ∫¢Õß 712 + 399 = 1,111
11
712
399+
10
10
1,111
111‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
°‘®°√√¡°“√§‘¥°‘®°√√¡°“√§‘¥ (Thinking Activities)
„π‡Õ° “√©∫—∫π’ȇªìπ à«πÀπ÷ËߢÕß·π«∑“ß„Àâ§ÿ≥§√Ÿ‰¥âΩñ°·≈–‡°‘¥§«“¡¡—Ëπ„®
„π°“√®—¥ √â“ß°‘®°√√¡°“√§‘¥§≥‘μ»“ μ√å„Àâ°—∫‡¥Á° Ê μàÕ‰ª ‚¥¬§“¥«à“®–π”√Ÿª·∫∫«‘∏’°“√π’È
‰ªª√—∫„™â À√◊Õ§‘¥‡æ‘Ë¡‰¥â
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé112
„Àâπ—°‡√’¬π‡μ‘¡®”π«π∑’ËÀ“¬‰ª ·≈–∫Õ°‡Àμÿº≈¥â«¬«à“∑”‰¡‡ªìπ®”π«ππ—Èπ
59 64 3 2 81 70
2 19 80 37 6 45
40 5 12 69 37 8
18 56 40 2 39
67 38 49 1 50 2
3 20 7 58 49 16
‡©≈¬7 ‡æ√“–·μà≈–·∂«μâÕß¡’‡≈¢‚¥¥§√∫ 10 μ—« §◊Õ 0 ∂÷ß 9
113‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
®“°·∫∫√Ÿªπ’È ®”π«π∑’ËÀ“¬‰ª§◊Õ®”π«π„¥ ‡æ√“–‡Àμÿ„¥
6 10 18 34
‡©≈¬66 ‡æ√“– ®”π«π∑’Ë Õß¡“°°«à“®”π«π·√° 4
®”π«π∑’Ë “¡¡“°°«à“®”π«π∑’Ë Õß 8®”π«π∑’Ë ’Ë¡“°°«à“®”π«π∑’Ë “¡ 16¥—ßπ—Èπ ®”π«π∑’ËÀâ“¡“°°«à“®”π«π∑’Ë ’Ë 32
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé114
8 515
815
16
0123456789
„Àâ‡≈◊Õ°‡μ‘¡ 0 ∂÷ß 9 ≈ß„π™àÕß«à“ß ‚¥¬®”π«π∑’Ë‡μ‘¡μâÕ߉¡à´È”°—π ·≈â«∑”„Àâº≈∫«°‡∑à“°—∫º≈∫«°∑’Ë°”Àπ¥„π·μà≈–¢âÕ
‡©≈¬μ—«Õ¬à“߇™àπ
3
4
8 0
6
2 5
1 5 9
6 2
8 3 4 1 4
0
5
8
6
3
7
115‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
„Àâ‡μ‘¡®”π«π∑’ËÀ“¬‰ª
3 7 5
97
11
911
15
‡©≈¬13
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé116
„À⇢’¬π®”π«π„ à≈ß„π™àÕß«à“ß„Àâ∂Ÿ°μâÕß
12 3
42
15
21 57
39
913
19 7
5
41
3
‡©≈¬14
117‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
„Àâ‡≈◊Õ°®”π«π 3 ®”π«π∑’Ë°”Àπ¥„Àâ „ à≈ß„π ‡æ◊ËÕ∑”„Àâª√–‚¬§ —≠≈—°…≥凪ìπ®√‘ß
1. 873
51
Ó + = 8
3.34
51
2
Ó › = 3
2.132
94
5
Ó + = 13
‡©≈¬μ—«Õ¬à“߇™àπ1) 1 Ó 3 + 5 = 82) 2 Ó 4 + 5 = 133) 2 Ó 4 › 5 = 3
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé118
5 27 6
9 1013 16
13 1819 26
7 610 11
1116 21
„Àâ‡μ‘¡®”π«π∑’ËÀ“¬‰ª
‡©≈¬14
119‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
13 15 17 19 21 23 25 27
0
3 7 1 0 6 2
2 9 16 1 4
3 9 22 2 8
4 7 3
5 7 2
0
„Àâ‡≈◊Õ°®”π«π∑’ËÕ¬Ÿà·∂«≈à“ß ÿ¥¡“ 1 ®”π«π ‡μ‘¡„π™àÕß«à“ß æ√âÕ¡Õ∏‘∫“¬‡Àμÿº≈
‡©≈¬19
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé120
„Àâ‡μ‘¡®”π«π∑’ËÀ“¬‰ª
8
13
1824
39
‡©≈¬6 À√◊Õ 54
121‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
? §◊Õ®”π«π„¥
6 4
34
11 3
47
8 5
53
9 2
?
‡©≈¬29
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé122
? §◊Õ®”π«π„¥
@ 33
?
33
27
24 63 24 21@ @ !
! Σ ! Ω
Ω Σ Ω Ω
! ! ! @
‡©≈¬39
9 9 9 6
6 24 6 3
3 24 3 3
6 6 6 9
39
123‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
3.
ONEMORETIME
+
2.
ADDMEUPSUM+
+
1.
WHATNOS
WORK+ +
„Àâ‡≈◊Õ°‡≈¢‚¥¥·∑πμ—«Õ—°…√„π·μà≈–¢âÕ ‚¥¬μ—«Õ—°…√‡¥’¬«°—π·∑π¥â«¬‡≈¢‚¥¥∑’ˇÀ¡◊Õπ°—π μ—«Õ—°…√μà“ß°—π·∑π¥â«¬‡≈¢‚¥¥∑’Ëμà“ß°—π
‡©≈¬μ—«Õ¬à“߇™àπ (1) (2) (3)384
7512
78966590
570
7160
+
5692
459
311
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé124
∑√ß≈Ÿ°∫“»°å·μà≈–¢âÕ‡¡◊ËÕ§≈’ËÕÕ°¡“·≈⫧◊Õ¢âÕ a, b, c À√◊Õ d
3.a. b. c. d.
2.a. b. c. d.
1.a. b. c. d.
‡©≈¬1. c2. d3. a
125‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
„ÀâÀ“®”π«π∑—ÈßÀ¡¥¢Õß®ÿ¥ ( ) ∑’Ë¡Õ߉¡à‡ÀÁπ¢Õß∑√ß≈Ÿ°∫“»°å≈Ÿ°‡μã“®”≈Õß
‡©≈¬41
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé126
®“°¿“æ ∂â“√–¥—∫πÈ”≈¥≈߇¢Á¡≈Ÿ°»√®–™’È¢÷ÈπÀ√◊Õ≈ß ‡æ√“–‡Àμÿ„¥
¢÷Èπ
ŧ
∑ÿàπ
·¡àπÈ”
¢÷Èπ
ŧ
∑ÿàπ
·¡àπÈ”
127‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
§≥‘μ¬Õ¥§‘¥‡¬’ˬ¡
μÕππ’ȇªìπ°“√𔇠πÕ‚®∑¬å∑’Ëπà“ π„®(¡“° Ê „𧫓¡§‘¥‡ÀÁπ¢ÕߺŸâ‡¢’¬π)‡æ√“–‡ªìπ‚®∑¬åªí≠À“∑’Ë∂Ÿ°‡≈◊Õ°„™â
„π°“√·¢àߢ—π§≥‘μ»“ μ√å√–¥—∫π“π“™“μ‘Õ¬à“ßπâÕ¬ °Á§“¥«à“®–‰¥â√Ÿâ«à“ ‚®∑¬å‡À≈à“π’ȉ¡à¬“°Õ¬à“ß∑’˧‘¥
‡æ’¬ß·μà‡≈◊Õ°π”§«“¡√Ÿâæ◊Èπ∞“π¡“ª√–¬ÿ°μå„™â„À⇪ìπ‡∑à“π—Èπ‡Õß ≈ÕߥŸ§√—∫
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé128
23
∴∴∴∴∴
«‘∏’∑” (§”μÕ∫∑’ˇªìπ‰ª‰¥âª√–°“√Àπ÷Ëß)‡¡◊ËÕ≈Ÿ°∫“»°å¢π“¥„À≠à¡’πÈ”Àπ—°‡∑à“°—∫ 1 ‡∑à“¢Õß∑√ß≈Ÿ°∫“»°å¢π“¥‡≈Á°¥—ßπ—Èπ≈Ÿ°∫“»°å¢π“¥„À≠à 3 ≈Ÿ° = ≈Ÿ°∫“»°å¢π“¥‡≈Á° 5 ≈Ÿ°´÷Ëß∂â“≈Ÿ°∫“»°å‡≈Á° 12 ≈Ÿ° = ≈Ÿ°∫“»°å¢π“¥„À≠à 6 ≈Ÿ°°—∫≈Ÿ°∫“»°å¢π“¥‡≈Á°
2 ≈Ÿ°®÷ß‡μ‘¡≈Ÿ°∫“»°å¢π“¥„À≠à 2 ≈Ÿ° ¢π“¥‡≈Á° 2 ≈Ÿ°„πμ“™—Ëß∑“ߢ«“¡◊Õ ®–∑”„Àâ
μ“™—Ëß ¡¥ÿ≈
‚®∑¬å 1¡’∑√ß≈Ÿ°∫“»°å 2 ¢π“¥ ‚¥¬∑√ß≈Ÿ°∫“»°å¢π“¥„À≠à¡’πÈ”Àπ—°‡ªìπ 1 ‡∑à“
¢Õß∑√ß≈Ÿ°∫“»°å‡≈Á° μ“™—Ëß∑“ߴ⓬¡◊Õ¡’∑√ß≈Ÿ°∫“»°å¢π“¥‡≈Á° 12 ≈Ÿ° μ“™—Ëß∑“ߢ«“¡◊Õ¡’∑√ß≈Ÿ°∫“»°å¢π“¥„À≠à 4 ≈Ÿ° ®–‡μ‘¡≈Ÿ°∫“»°å¢π“¥„À≠àÀ√◊Õ¢π“¥‡≈Á°Õ¬à“߉√ ∑’Ë®–∑”„Àâμ“™—Ë߇°‘¥°“√ ¡¥ÿ≈
23
129‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
‚®∑¬å 2®”π«π‡μÁ¡∫«°∑’Ë¡’ 6 À≈—° 2 abcde ·≈– abcde 2 ‡¢’¬π ¡°“√‰¥â«à“2 abcde Ó 3 = abcde 2 „ÀâÀ“§à“ abcde
∴∴∴∴∴
∴∴∴∴∴
∴∴∴∴∴
∴∴∴∴∴
∴∴∴∴∴∴∴∴∴∴
«‘∏’∑”
2 abcde Ó 3 = abcde 2æ‘®“√≥“μ—«‡≈¢‚¥¥μ—« ÿ¥∑⓬ (∑“ߢ«“¡◊Õ À≈—°Àπ૬)
e Ó 3 §◊Õ 2e = 4
μ—«‡≈¢‚¥¥μ—«∑’Ë Õß (À≈—° ‘∫)d4 Ó 3 §◊Õ 4d = 1
μ—«‡≈¢‚¥¥μ—«∑’Ë “¡ (À≈—°√âÕ¬)c14 Ó 3 §◊Õ 1c = 7
μ—«‡≈¢‚¥¥μ—«∑’Ë ’Ë (À≈—°æ—π)b714 Ó 3 §◊Õ 7b = 5
μ—«‡≈¢‚¥¥μ—«∑’ËÀâ“ (À≈—°À¡◊Ëπ)a5714 Ó 3 §◊Õ 5a = 8
285,714 Ó 3 = 857,142 ¥—ßπ—Èπ abcde = 85,714
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé130
«‘∏’∑”
‚®∑¬å 3„ÀâÀ“§à“ x ®“°
=1 + 1 + 1 + x +
5788
11
11
15
=1 + 1 + 1 + x +
5788
11
1
115
1 + = = 1 + 1 + 1 + x +
8857
111
15
3157
= 1 + 1 + x +
3157
11
115
1 + = = 1+ 1 + x +
5731
1
115
2631
131‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
= 1 + x +
2631
1
115
1 + = = 1 + x +
3126
115
526
= x +
526
115
x + =265
15
x = › = = 5265
15
255
= 1 + 2 + m +
1825
11
114
1 + = = 1 + 2 + m +
2518
1
114
718
‚®∑¬å 4„ÀâÀ“§à“ m ®“° =1
1
114
1 + 2 + m +
1825
«‘∏’∑”
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé132
= 2 + m +
718
1
114
2 + = = 2 + m +
187
114
47
= m +
114
47
m + =14
74
m = › = = = 174
14
64
32
12
⨷Œ 5
°”Àπ¥æ◊Èπ∑’Ë√Ÿª ’ˇÀ≈’ˬ¡®—μÿ√— ∑—Èß 3 √Ÿª¡’§à“‡∑à“°—∫ 360, 40 ·≈– 40 ¡.2 μ“¡≈”¥—∫
„ÀâÀ“æ◊Èπ∑’Ë√Ÿª “¡‡À≈’ˬ¡ ABC ∑’Ë∂Ÿ°·√‡ß“«à“°’Ë ´¡.2
A
C
B
360
40
40
133‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
«‘∏’∑”
A
C
B
D
F
E
æ◊Èπ∑’Ë√Ÿª BDC ‡∑à“°—∫ 40 ´¡.2
æ◊Èπ∑’Ë√Ÿª FDC ≅≅≅≅≅ æ◊Èπ∑’Ë√Ÿª AEF¥—ßπ—Èπ
æ◊Èπ∑’Ë√Ÿª ABC = æ◊Èπ∑’Ë√Ÿª ABE + æ◊Èπ∑’Ë√Ÿª BDC= 160 + 40 ´¡.2
= 200 ´¡.2
⨷Œ 6 BAD MAD + DAM
º≈√«¡∑’Ë¡’§à“¡“°∑’Ë ÿ¥A, B, D ·≈– M ·∑π¥â«¬ 2, 5, 9 ·≈– 8 ‚¥¬μ—«Õ—°…√∑’Ë·μ°μà“ß°—π
·∑π¥â«¬μ—«‡≈¢∑’Ë·μ°μà“ß°—π μ—«Õ—°…√∑’ˇÀ¡◊Õπ°—π·∑π¥â«¬μ—«‡≈¢∑’ˇÀ¡◊Õπ°—π„ÀâÀ“§à“º≈√«¡∑’Ë¡’§à“¡“°∑’Ë ÿ¥π’È
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé134
«‘∏’∑”
BAD + MAD + DAM= 100 B + 10 A + D 100 M + 10 A + D + 100 D + 10 A + M= 102 D + 101 M + 100 B + 30 A
∂â“„Àâ¡’§à“º≈∫«°¡“°∑’Ë ÿ¥= 102 Ó 9 + 101 Ó 8 + 100 Ó 5 + 30 Ó 2= 2,286
«‘∏’∑”
⨷Œ 7
¥—ßπ—Èπ Ó Ó Ó ... Ó = 4,028,04941
94
169 2,0062
2,0072
‡¡◊ËÕ N = Ó Ó Ó ... Ó Ó2
0 + 12
3
1 + 13
4
2 + 14
2,006
2,004 + 12,006
2,007
2,005 + 12,007
„ÀâÀ“§à“ N
=4
2 + 14
169
=3
1 + 13
94
‡¡◊ËÕ =2
0 + 12
41
135‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
§‘¥‡º‘π ʇ¥Á°À≈“¬§π‡∫◊ËÕ§”«à“§≥‘μ»“ μ√å‡∫◊ËÕ‡√’¬π ‡∫◊ËÕ∑”°“√∫â“πÕ“®¡“®“°°‘®°√√¡∑’˧√Ÿπ”‡ πÕ¡“°àÕπ°Á‰¥âç°≈§≥‘μ §‘¥∑—π‚≈°é®–∑”„Àâ™à«ß‡«≈“Àπ÷ËߢÕß°‘®°√√¡°“√‡√’¬π√Ÿâ¡’§«“¡ πÿ° π“π¡’§«“¡À¡“¬¢÷Èπ‡μÁ¡‰ª¥â«¬∫√√¬“°“»·Ààß°“√∑â“∑“¬
§‘¥∑—π‚≈°é°≈§≥‘μç
‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé136
2 Ó 3 Ó 4 = 24
2 3
4 241 Ó 4 Ó 7 = 28
1 4
28 7
1 Ó 9 Ó 2 = 18
1 18
9 22 Ó 2 Ó 5 = 20
20 2
5 2
‡©≈¬
2 3 1 4
4 24 28 7
1 18 20 2
9 2 5 2
°‘®°√√¡ 1º‘«Àπâ“¢Õß∑√ß≈Ÿ°∫“»°å ∂Ÿ°‡¢’¬πμ—«‡≈¢≈ß„πμ“√“ߧ√∫∂â«π∂Ÿ°μâÕß ∂“¡«à“
°“√«“ßμ—«‡≈¢‡À≈à“π’È∂Ÿ°μâÕߥ⫬‡ß◊ËÕπ‰¢À√◊Õ°μ‘°“„¥„ÀâÕ∏‘∫“¬
137‡ √‘¡§«“¡√Ÿâ§≥‘μ»“ μ√å çSpeed Mathsé
°‘®°√√¡ 2®”π«π∑’ËÀ“¬‰ª„π C §◊Õ®”π«π„¥
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