Răspuns :
[tex]\displaystyle\bf\\a,b~si~c,~sunt~direct~proportionale~cu~4,5~si~10 \implies \\\frac{a}{4} = \frac{b}{5} = \frac{c}{10} = k\implies a=4k,~b=5k~si~c=10k.\\5a+b+3c=5\cdot4k+5k+3\cdot 10k=20k+5k+30k=55k=11 \implies k = \frac{1}{5}.\\a=\frac{4}{5},~b=1~si~c=2.\\Diferenta~dintre~cel~mai~mare~numar~si~cel~mai~mic~este~:~\\c-a=2-\frac{4}{5} =\frac{6}{5}~.[/tex]
Răspuns: [tex]\bf c-a =\dfrac{6}{5}[/tex]
Explicație pas cu pas:
Salutare!
[tex]\bf \{a,b,c\}d.p.\{4,5,10\}[/tex]
[tex]\bf 5a+b+3c = 11[/tex]
[tex]\bf \dfrac{a}{4}= \dfrac{b}{5}= \dfrac{c}{10} = k \implies[/tex] [tex]\bf a = 4k[/tex]
[tex]\bf b = 5k[/tex]
[tex]\bf c = 10k[/tex]
[tex]\text{\it Inlocuim noile valori ale lui a, b, c in suma:}[/tex]
[tex]\bf 5\cdot 4k+5k+3\cdot 10k = 11[/tex]
[tex]\bf 20k + 5k + 30k = 11[/tex]
[tex]\bf 55k = 11\:\:\:\Big|:11[/tex]
[tex]\bf 5k = 1[/tex]
[tex]\boxed{\bf k =\dfrac{1}{5}}[/tex]
[tex]\bf a = 4\cdot \dfrac{1}{5} \implies \boxed{\bf a = \dfrac{1}{5}}[/tex]
[tex]\bf b = 5\cdot\dfrac{1}{5}\implies b = \dfrac{5}{5}\implies \boxed{\bf b =1}[/tex]
[tex]\bf c = 10\cdot\dfrac{1}{5}\implies c = \dfrac{10}{5}\implies \boxed{\bf c =2}[/tex]
[tex]\bf c- a = 2-\dfrac{4}{5}\implies c-a = \dfrac{10-4}{5}\implies \boxed{\bf c-a =\dfrac{6}{5}}[/tex]
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