Question

18. Media aritmetică a trei numere naturale este egală cu 72. Al doilea număr este cu 50%
mai mare decât primul, iar al treilea număr este cu o treime mai mare decât al doilea. Cars
sunt cele trei numere?​

Answer 1

Răspuns:

$x,y,z$ - cele trei numere

$\frac{x+y+z}{3}=72$

$x=100%$%$=> y = (\frac{100}{100}+\frac{50}{100})x= \frac{150x}{100} <=> \frac{3x}{2}$

$z=\frac{1}{3}*y+y=\frac{y}{3}+\frac{3y}{3} <=> \frac{4y}{3}$

si cum y = $\frac{3x}{2} => z = \frac{4*(\frac{3x}{2}) }{3} <=>\frac{6x}{3} => z = 2x$

$\frac{x+\frac{3x}{2}+z }{3} = \frac{\frac{2x}{2}+\frac{3x}{2}+\frac{2z}{2} }{3} <=> \frac{5x+2z}{6}=72 <=> 5x+2z=432$

$5x+2*(2x)=432 <=> 9x=432 => x = 48$

$y=\frac{3x}{2}=\frac{3*48}{2}=3*24 = 72$

$z=2x = 2*48 = 96$

Explicație pas cu pas:

Answer 2

Răspuns:

$\red{\Large \bf a = 48~primul ~ numar}$

$\pink{\Large\bf b = 72~al ~doilea ~numar}$

$\green{\Large\bf c = 96~~al ~treilea~numar}$

Explicație pas cu pas:

$\red{\large\boxed{\bf Media_{aritmetica}=\dfrac{a_{1}+a_{2}+a_{3}+a_{4}+....+a_{n}}{n}}}$

Notam cu:

a - primul număr

b - al doilea număr

c - al treilea număr

$\left.\begin{matrix}\large\bf M_{a}=\dfrac{a+b+c}{3}\\\\ \large\bf M_{a}=72\end{matrix}\right\}\implies \large\bf \dfrac{a+b+c}{3}= 72 \implies \boxed{\bf a+b+c = 216}</p><p>$$\large\bf a+\dfrac{3a}{2} +2a= 126~~~\bigg|\cdot 2$

$\it~~$

$\large \bf b = a + 50\% \cdot a \implies b = a + \dfrac{\not50a}{\not100} \implies \boxed{\bf b = \dfrac{3a}{2} }$

$\it~~~$

$\left.\begin{matrix} \large\bf c=\dfrac{1}{3} \cdot b +b\\\\\large \bf b = \dfrac{3a}{2} \end{matrix}\right\}\implies\large \bf c =\dfrac{1}{\not3}\cdot\dfrac{\not3a}{2}+\dfrac{3a}{2}\implies c = \dfrac{a}{2}+\dfrac{3a}{2}\implies$

$\large \bf \implies c =\dfrac{4a}{2}\implies c = \dfrac{\not4a}{\not2}\implies \boxed{\bf c= 2a }$

Înlocuim in suma si vom avea:

$\large\bf a+\dfrac{3a}{2} +2a= 216~~~\bigg|\cdot 2$

$\large\bf 2a+3a +4a= 432$

$\large\bf 9a= 432~~~\bigg|:9$

$\red{\Large\boxed{\bf a = 48}}$

$\large \bf b = \dfrac{3\cdot 48}{2} \implies b = 3\cdot 24 \implies \pink{\Large\boxed{\bf b = 72}}$

$\large \bf c = 2\cdot 48 \implies \green{\Large\boxed{\bf c = 96}}$

PS: Daca esti pe telefon te rog sa glisezi spre stânga pentru a vedea rezolvarea

#copaceibrainly