Hoofdstuk.0 INTRO De som is, of 0, of. Dat zijn de enige met vier mogelijkheden, zie eerste twee kolommen. Som Mogelijkheden Product Manieren om het product te schrijven + 8 + 7 + + 5 8 8 0 8 of 7 of 5 0 + + 8 + 7 + + + 8 + 7 5 + 8 8 0 8 7 of 8 of of 8 7 5 Als je naar het product van de mogelijkheden kijkt, komt 8 (licht blauw) en (paars) twee keer voor, zie kolom drie. Als je naar het product van de mogelijkheden van de som is en kijkt, kun je maar twee van de vier mogelijkheden wegstrepen. Dus moet het wel en zijn, want het product, en is maar op één manier te schrijven als product van twee ongelijke gehele getallen onder de tien. Zie kolom vier.. DE OPLOSSING ZOEKEN a breedte 8 lengte 7 8 8 b bij tegels. Dan is de omtrek 8. c bij tegels. 7 rijen met per rij zitplaatsen of rijen met per rij 7 zitplaatsen. a 7 = 87. Er liggen of 7 stukjes op een rij. b + 7 + + 7 = 5 stukjes 5 dagen maanden dagen maanden 0 0 0 0 5 5 5 5 5 5 0 0 0 0 a bij, bij, bij 8 of bij tegels. b bij 5 of 5 bij 5 tegels bij of bij tegels bij tegels 7,, 5, 7,,, 7,,,, 8 b x + (x + ) = 5 x = 5 c x(x + ) = 5 x = 7 of x = -8 d x = 8 x = cm e x = - géén oplossingen f x = - x = - g x = x + 5 x = a Alleen de getallen - en voldoen. b Alleen de getallen - en voldoen. c Alleen de getallen - en voldoen. 0 x = x = 0 géén oplossingen x = - x = of x = - géén oplossingen x = x = - x = of x = -5 x = - a Vinja, want = =. b Met haakjes: ( ) c = = 8 = d = 8 = = 8 = 8 a Vinja, want - = -( ) = -. b (-) = - - = (-x) = -x -x = x c -(-) + - = - 8 = - - (-) - = - + = -(-) (-) (-) = - + = - d -0x x -5x = -0x 0x x 5x = 00x 0x -x 5x = -50x x = of x = - x = 0 of x = -0 x = of x = - x = 5 of x = -5 x = 5 of x = -7 x = 7 of x = - x = of x = - x = of x = x = of x = - x = 5 of x = -5 x = - x = of x = - géén oplossingen x = - of x = - x = 0 of x = - x = 5 = of x = - 5 = - 5 a x(x + 0) = 00 b x = 0 meter x(x + 5) =, dus x = 7 x(x + ) = 7, dus x = 8 de Wageningse Methode Antwoorden H
8 a x(x + ) = 80 x = 8 b x(x + ) = 80 x = 5 a x = b x = 8 0 a en 7 b x(0 x) = c x(0 x) = d = 08 7 5 8 7 8 e De getallen zijn en 8. -0 (-0 + ) = -0-8 = 80 x = -0 ; x = -8 ; x = - ; x = -, x = -5 x = - of x = - Rechter kolom x = 0 of x = x = 0 of x = of x = - x = of x = - 0 x(x 7) (x )(x ) x (x 7) x (x )(x ) (x )(x 5) x(x 5) x(x )(x 5) x (x 5). SYSTEMATISCH OPLOSSEN x + 0x = - x + 0x + = 0 (x + )(x + 8) = 0 x = - of x = -8 0x = x 0x x = 0 x(0 x) = 0 x = 0 of x = 0 x + x = x + x = 0 (x + 8)(x ) = 0 x = -8 of x = PLUS MIN x MIN x = 5 of x = - x = - of x = -5 x = of x =. HET PRODUCT IS 0 5 a Een van de getallen moet 0 zijn. b Dit kan op allerlei manieren, zoals 0 of 0 of 0. c 0 ; 0 d 0 = 0 e x = -, want - 0 = 0 f x = of x = - of x = - Voor andere getallen x is x niet 0, is x + niet 0 en is x + niet 0. x = - of x = -5 x = of x = x = 0 of x = 7 x = of x = - of x = of x = -5 x = 0 of x = of x = - x = 0 of x = - of x = of x = - 8 a (x )(x )(x 5)(x 7)(x ) = 0 b x(x )(x + )(x ) = 0 Linker kolom x = 0 of x = of x = - of x = of x = - x = 0 of x = of x = x = of x = of x = - x + = 8x x 8x + = 0 (x ) = 0 x = x = 0x x 0x = 0 x 5x = 0 x(x 5) = 0 x = 0 of x = 5 x 5x = x 5x = 0 (x )(x + ) = 0 x = of x = - x + x = 0 x (x + ) = 0 x = 0 of x = - x = x x + x = 0 (x + )(x ) = 0 x = - of x = x = x x x + = 0 x x + = 0 (x )(x ) = 0 x = MIN 8x MIN 0x DELEN DOOR MIN PLUS x, MIN MIN x, PLUS DELEN DOOR de Wageningse Methode Antwoorden H
x = x x x + = 0 x x + = 0 (x )(x ) = 0 x = (x + ) = x + 5 x + = x + 5 x x + = 0 (x )(x ) = 0 x = of x = PLUS x, MIN DELEN DOOR MIN x, MIN x 0x + = 0 ( + x) = (x )(x ) = 0 +x = 7 of +x = -7 x = of x = x = of x = -0 x (x 0x + ) = 0 x (x )(x ) = 0 x = 0 of x = of x = géén oplossingen (x )(x 0x + ) = 0 ( + x) = 0 (x )(x )(x ) = 0 + x = 0 x = of x = x = - (x + )(x + ) = x x + x + = x x + x + = 0 x + x + = 0 (x + )(x + ) = 0 x = - (x + ) = x + x + x + = x + x + x = 0 x + x = 0 x(x + ) = 0 x = 0 of x = - x(x ) = x (x + ) x x = x + x x + x = 0 x + x = 0 x(x + ) = 0 x = 0 of x = - (x ) = (x ) x = x x 8 = 0 x = 0 (x )(x + ) = 0 x = of x = - (x + ) = (x ) x + x + = x 8x + x = x = x = of x = - PLUS x, MIN DELEN DOOR MIN x, MIN DELEN DOOR MIN x, PLUS x DELEN DOOR MIN x, PLUS DELEN DOOR MIN x, PLUS 8x, MIN DELEN DOOR 5x = 80 x = x x = x + x = 0 x = of x = - (x + )(x ) = 0 x = - of x = 5x = 80x x = x 5x 80x = 0 x + x = 0 x x = 0 x + x = 0 x (x ) = 0 (x + )(x ) =0 x = 0 of x = of x = - x = - of x = 5x = 80x x = x 5x 80x = 0 x + x = 0 x x = 0 x + x = 0 x (x ) = 0 (x + )(x ) = 0 x = 0 of x = x = - of x = x + = (x + ) = x = x + = x = (x + ) = (x + ) = x+ = 8 of x+ = -8 x+ = of x+ = - x = of x = -0 x = 0 of x = -. VERGELIJKINGEN OPSTELLEN a x + (x + ) = (x + ) b x + (x + ) = (x + ) x + x + = x + 8x + x x = 0 (x )(x + ) = 0 x = of x = - 5 x(x + ) + = 8 x + x + = 8 x + x 5 = 0 (x + 5)(x ) = 0 x = -5 of x = Alleen het antwoord x = kan, omdat het om een lengte gaat. x(x + 7) = 0 x + 7x 0 = 0 (x + 0)(x ) = 0 x = -0 of x = Alleen x = kan. 5x = 05 x = 8 x = of x = - Alleen x = kan. x + (x ) = 8 x = 8 x = 0 x = 7 7 a h(h + ) = 0 h + h 0 = 0 (h + )(h 0) = 0 h = - of h = 0 Er zijn 0 honden en katten. de Wageningse Methode Antwoorden H
b h(7 h) = 0 7h h 0 = 0 h 7h + 0 = 0 (h 0)(h 7) = 0 h = 0 of h = 7 Er zijn 0 honden en 7 katten of 7 honden en 0 katten. c honden 5 7 0 katten 0 05 70 5 0 5 honden 5 0 5 70 05 0 katten 0 7 5 Opgave a: Rood gekleurd (=donker gekleurd) Opgave b: Groen gekleurd (=licht gekleurd) 8 Opp. Berends = x Opp. Ermers = (x 0)(x 0) x = (x 0)(x 0) x = x 0x + 00 x 0x + 00 = 0 (x 0)(x 0) = 0 x = 0 of x = 0 x = 0 kan niet, want dan wordt de lengte en breedte van het land van Ermers negatief. Afmetingen Berends is 0 bij 0 meter. Afmetingen Ermers is 0 bij 80 meter. x(x + ) = x + x = 0 (x + )(x ) = 0 x = - of x = Afmetingen kamer is bij meter. 0 (x + ) = x + = of x + = - x = 5 of x = -7 Opp. kleine vierkant = 5 = 8 Zijde kleine vierkant = 8 = cm d = + = = cm.5 HOGEREGRAADS VERGELIJKINGEN Derdegraads; vierdegraads x = x x x x + x = 0 x(x x + ) = 0 x(x )(x ) = 0 x = 0 of x = x x = x x x x = 0 x(x x ) = 0 x(x )(x + ) = 0 x = 0 of x = of x = - x + x = x x + x x = 0 x (x + x ) = 0 x (x + )(x ) = 0 x = 0 of x = - of x = x = 8x x 8x + = 0 (x )(x ) = 0 x = x = 8x x 8x + = 0 (x )(x ) = 0 x = of x = - x x = x x = 0 (x )(x + ) = 0 x = of x = - x + 7x = 8 x + 7x 8 = 0 (x + 8)(x ) = 0 x = - of x = OKEROPGAVEN jongste middelste oudste som 8 8 0 Omdat Ofelia het nog steeds niet wist, moet de som zijn (komt twee keer voor). Omdat er één oudste is, blijft alleen de mogelijkheid over:, en jaar. x = x = 5 of x = x = of x = - x = géén oplossingen géén oplossingen x = - géén oplossingen x = of x = - x = - 8 a = 0 prijzen b x prijzen c 0 = 0,- d x( x) e x( x) = 0 x = of x = 8 Er zijn prijzen van 8,- of 8 prijzen van,-. x + 5x = x(x + 5) = x = of x = -8 de Wageningse Methode Antwoorden H
5x 5x = 50 x x = 70 x(x ) = 70 x = 0 of x = -7 -x + x = x x = - x(x ) = - x = of x = 5 a Eén van de getallen moet 0 zijn. b Dit kan op allerlei manieren, zoals 0 of 0 of 0. c x a moet 0 zijn of x b moet 0 zijn. Dus x = a of x = b. 0 a k = -5, x 5x = (x )(x + ) k = -, x x = (x 8)(x + ) k = 0, x = (x )(x + ) k = 5, x + 5x = (x )(x + ) b k = -7 of k = 0 of k = 5 a + + 0 = + + 0 = 0 b (x )(x + x 0) = x + x x 0x x + 0 = x + x x + 0 - -x -x 0 x x x -0x x x -0 c x + x 7x + 5 = 0 (x + 5)(x + x ) = 0 (x + 5)(x + )(x ) = 0 x = -5 of x = - of x = x x x = 0 (x )(x + x) = 0 (x )(x + )x = 0 x = of x = - of x = 0 (x )(x + ) = x + x x + x + 7x x 8 = 0 (x + x )(x + 5x + ) = 0 (x )(x + )(x + )(x + ) = 0 x = of x = - of x = - of x = - a x + (x + ) + (x + ) + (x + ) = (x + ) + (x + 5) + (x + ) x + x + = x + 0x + 77 x 8x = 0 (x )(x + ) = 0 x = of x = - De kleinste positieve oplossing van de zeven opeenvolgende getallen is. b x + (x + ) + (x + ) + (x + ) + (x + ) = (x + 5) + (x + ) + (x + 7) + (x + 8) 5x + 0x + 0 = x + 5x + 7 x x = 0 (x )(x + ) = 0 x = of x = - De kleinste positieve oplossing van de negen opeenvolgende getallen is. a prijs per persoon: 5 0,50 7 =,50 totale prijs: 7,50 = 5,50 b prijs per persoon: 5 0,5(x 0) = 0 0,5x totale prijs: x (0 0,5x) = -0,5x + 0x c -0,5x + 0x = x 0x + 8 = 0 (x )(x ) = 0 x = of x = x = kan niet, dus x =. 7 a = kubusjes b ribbe 7 aantal kubusjes 58 c Aantal buitenkant = ((n ) 5) = n n Aantal binnenin = (n 5) = n 0 A = n n + (n 0) = n n d n n = 58 n n 5 = 0 n n 8 = 0 (n )(n + ) = 0 n = of n = - De ribbe van de kubus is. a 5 7 + = 8 = + = 55.05 = 505 b x(x + )(x + )(x + ) + c x + x + x + x + d x + x + x + x + = (x + x + ) x x x x x x x x x x x x.7 EXTRA OPGAVEN p(p +,5) p = 0 p + p 0 = 0 p + p 0 = 0 (p + 8)(p 5) = 0 p = -8 of p = 5 Antwoord p = -8 kan niet. Afmetingen tuin is,5 bij meter. de Wageningse Methode Antwoorden H 5
x = x x x = 0 x(x ) = 0 x = 0 of x = (x + )(x ) = x x = x x + x 0 = 0 (x + 5)(x ) = 0 x = -5 of x = (x + ) = x(x + ) x + 8x + = x + x x = - x = - (x ) = x x 8x + = x x 7x + = 0 (x )(x ) = 0 x = of x = (x + ) = x x + 8x + = x x 8x + = 0 (x ) = 0 x = (x + ) = x + = of x + = - x = 0 of x = -8 Noem de lengte van de dozen x. Inhoud grote doos = 0 x x = 5x Inhoud kleine doos = 0 x x = 0x 0x + 000 = 5x 000 = 5x 00 = x x = 0 of x = -0 Antwoord x = -0 kan niet. Afmetingen kleine doos is 0 bij 0 bij 0 cm. ( x)( x) = 0 58 0x + x = 0 x 0x + = 0 x 0x + = 0 (x 7)(x ) = 0 x = 7 of x = Antwoord x = kan niet, want dan wordt een zijde = - meter. Dus x = 7 meter. 5 Noem de breedte van het pad x (in meters). Opp. hele tuin = (x + 8) Opp. paden = 00 Opp. perken = Opp. hele tuin = Opp. paden + Opp. perken (x + 8) = 00 + (x + 8) = 70 x + 8 = 5 of x + 8 = -5 x = of x = -00 x = of x = -5 Antwoord x = -5 kan niet. Breedte van de paden zijn meter. a Henk: x (0 x) + x 5 = -x + 0x Erik: 5 x + x (0 x) = -x + 5x -x + 0x = -x + 5x x 5x = 0 x(x 5) = 0 x = 0 of x = 5 Breedte van de letter is 5 cm. b 7, gram geeft een inhoud van cm. -x + 0x + = -x + 5x x 5x + = 0 (x )(x ) = 0 x = of x = De breedte van de letter is cm of cm. 7 a bij bij cm ; bij bij cm b bij bij cm cd lengte breedte hoogte oppervlakte 8 8 7 70 8 5 5 e Bij de afmeting bij bij cm. 8 a De mensen lopen elkaar een beetje in de weg. b (0 ) = mp-spelers 5 (0 5) = 5 mp-spelers 8 (0 8) = 5 mp-spelers n (0 (n 0)) c n (0 (n 0)) = n (0 n + 0) = n(0 n) d 5 (0 5) = 5 5 = 5 mp-spelers e n(0 n) = 5 n = 5, want 5 5 = 5 mp-spelers f n(0 n) = g n(0 n) = n 0n + = 0 (n )(n 8) = 0 n = of n = 8 de Wageningse Methode Antwoorden H
x + (x 7) = (x + ) x + x x + = x + x + x x + 8 = 0 (x )(x ) = 0 x = of x = Het antwoord x = kan niet, omdat de hoogte van de driehoek dan negatief is. Antwoord x =. 0 x + ( x) = 0 x + 8x + x = 0 x x + = 0 (x )(x ) = 0 x = of x = Inhoud balk = x x (x ) = x x Opp. balk = (x + x(x ) + x(x )) = (x + x x + x x) = 0x x x x = 0x x x x + x = 0 x 7x + x = 0 x(x 7x + ) = 0 x(x )(x ) = 0 x = 0 of x = of x = De antwoorden x = 0 en x = kunnen niet. Afmetingen van de balk zijn bij bij. de Wageningse Methode Antwoorden H 7