Question

determinati numerele x si y ştiind că sunt invers proporţionale cu 1 pe 2 şi 1 pe 3 iar suma lor este 360

Answer 1

x*1/2=k     x=2k
y*1/3=k     y=3k
x+y=360
2k+3k=360
5k=360
k=72
x=2k=2*72=144
y=3k=3*72=216

Answer 2

{x,y} i.p -> {$\frac{1}{2}, \frac{1}{3}$)
⇒x/1/1/2 = y/1/1/3 =k.
(1) x/1/1/2 = $\frac{x}{1}* \frac{2}{1} = \frac{2x}{1}= 2x$
2x= k ⇒ x=$\frac{k}{2}$
(2) y/1/1/3 = $\frac{y}{1} * \frac{3}{1}= \frac{3y}{1}= 3y$
3y= k ⇒ y= $\frac{k}{3}$
(1),(2)⇒$\frac{ ^{3)} k}{2} + \frac{^{2)} k}{3} = 360~ ( ^{x)} = amplificare) \\ \frac{3k}{6} + \frac{2k}{6}=360 \\ \frac{5k}{6}= 360. \\ 5k= 2160 \\ \boxed {k=432.} \\ x= \frac{k}{2} = \frac{432}{2} \\ \boxed { x=216} \\ y= \frac{k}{3} = \frac{432}{3} \\ \boxed{ y=144 }$