Question

Determinati numerele naturale a,b,c stiind ca:
a/0,1(6) = b/0,125 =
c/0,08(3) si
12(ab + ac + bc) = 13abc

Answer 1

Răspuns:

4; 3; 2.

Explicație pas cu pas:

$0,1(6)=\frac{16-1}{90}=\frac{15}{90}=\frac{1}{6}\\ 0,125=\frac{125}{1000}=\frac{1}{8} \\0,08(3)=\frac{83-8}{900} =\frac{75}{900}=\frac{1}{12}\\a/0,1(6)=a:(1/6)=6a,~b/0,125=b/(1/8)=8b,~c/0,08(3)=c:(1/12)=12c\\Deci~6a=8b=12c,~a=\frac{4}{3}b,~c=\frac{2}{3}b,~inlocuim~in~12(ab+ac+bc)=13abc\\12*(\frac{4}{3}b^{2}+\frac{4}{3}b*\frac{2}{3}b+\frac{2}{3}b^{2})=13*\frac{4}{3}b*b*\frac{2}{3}b\\12*(\frac{12}{9}b^{2}+\frac{8}{9}b^{2}+\frac{6}{9}b^{2})=13*\frac{8}{9}b^{3},~12*\frac{26}{9}b^{2}=\frac{13*8}{9}b^{3}$

$b=\frac{12*26}{13*8}=3.~deci~a=\frac{4}{3}*b=4,~c=\frac{2}{3}*b=2$