Question

media aritmetica a 4 numere este 166. determinati numerele stiind ca primele trei sunt direct proportionale cu 2,3 si 4, iar ultimele doua sunt invers proportionale cu 5 si 7

Answer 1

(a+b+c+d)/4=166, a+b+c+d=41,5 a/2=b/3/=c/4, 5c=7d

a/2=c/4, a=2c/4=c/2, b/3=c/4, b=3c/4, d=5c/7, inlocuim a b si d in media aritmetica

c/2+3c/4+c+5c/7=41,5, numitor comun 28

14c+21c+28c+20c=1162,, 83c=1162, c=1162/83=14

a=14/2=7, b=3*14/4=10,5, d=5*14/7=10

Answer 2

Fie numerele a, b, c, d

Ma = ( a + b + c + d ) / 4 = 166 → media aritmetica

a + b + c + d = 4 × 166

a + b + c + d = 664 → suma celor 4 numere

[ a, b, c ] d. p. [ 2, 3, 4 ] , iar [ c, d ] i. p. [ 5; 7 ]

a / 2 = b / 3 = c / 4

⇒ a = 2 b / 3; c = 4 b / 3

c / ( 1 / 5 ) = d / ( 1 / 7 ) ⇒ c : 1 / 5 = c × 5 / 1 = 5 c = 7 d

5 × 4 b / 3 = 7 d ⇒ 7 d = 20 b / 3 ⇒ d = 20 b / 21

a + b + c + d = 664

2 b / 3 + b + 4 b / 3 + 20 b/ 21 = 664

14 b + 21 b + 28 b + 20 b = 21 × 664

83 b = 13 944

b = 13 944 : 83

b = 168 → al doilea nr.

a = 2 × 168 / 3 = 2 × 56 = 112 → primul numar

c = 4 × 168 / 3 = 4 × 56 = 224 → al treilea numar

d = 20 × 168 / 21 = 20 × 8 = 160 → al IV-lea nr.

Verific: ( 112 + 168 + 224 + 160) / 4 = 664 / 4 = 166 → media aritmetica

112 / 2 = 168 / 3 = 224 / 4 = 56

224 : 1 / 5 = 224 × 5 = 7 × 160 = 1 120

Raspuns: 112; 168; 224 si 160