Question

if x/x+2y+z = y/y+2z+x = z/z+2x+y and x+y+2≠0 then show that each ratio is equal to 1/4​

Answer 1

Given :-

$\longrightarrow \dfrac{x}{x + 2y + z} = \dfrac{y}{y + 2z +x } = \dfrac{z}{z + 2y + x}$

$x + y + 2 \neq 0$

Aim :-

• To show that each ratio is equal to ¼

Let us take a look at the numerators alone.

$\implies x = y = z$

Therefore, all the variables are equal.

Let us represent all of these variables as 1 :- k

Substituting,

$\implies \dfrac{k}{k + 2k + k} = \dfrac{k}{ k+ 2k+k } = \dfrac{k}{ k+ 2k+ k}$

Adding all the terms in the denominators, we get :-

$\implies \dfrac{k}{4k} = \dfrac{k}{4k} = \dfrac{k}{4k}$

Since the numerators and denominators have a common factor k, by cancelling k out we get :-

$\implies \dfrac{1}{4} = \dfrac{1}{4} = \dfrac{1}{4}$

Hence we have shown that each fraction, when reduced to the lowest terms results in ¼.

Ratio :-

Ratio is the comparison of any two or more quantities usually expressed in it's simplest form.

Answer 2

Answer:

$Given :-</p><p></p><p>\longrightarrow \dfrac{x}{x + 2y + z} = \dfrac{y}{y + 2z +x } = \dfrac{z}{z + 2y + x}⟶x+2y+zx=y+2z+xy=z+2y+xz</p><p></p><p>x + y + 2 \neq 0x+y+2=0</p><p></p><p>Aim :-</p><p></p><p>To show that each ratio is equal to ¼</p><p></p><p>Let us take a look at the numerators alone.</p><p></p><p>\implies x = y = z⟹x=y=z</p><p></p><p>Therefore, all the variables are equal.</p><p></p><p>Let us represent all of these variables as 1 :- k</p><p></p><p>Substituting,</p><p></p><p>\implies \dfrac{k}{k + 2k + k} = \dfrac{k}{ k+ 2k+k } = \dfrac{k}{ k+ 2k+ k}⟹k+2k+kk=k+2k+kk=k+2k+kk</p><p></p><p>Adding all the terms in the denominators, we get :-</p><p></p><p>\implies \dfrac{k}{4k} = \dfrac{k}{4k} = \dfrac{k}{4k}⟹4kk=4kk=4kk</p><p></p><p>Since the numerators and denominators have a common factor k, by cancelling k out we get :-</p><p></p><p>\implies \dfrac{1}{4} = \dfrac{1}{4} = \dfrac{1}{4}⟹41=41=41</p><p></p><p>Hence we have shown that each fraction, when reduced to the lowest terms results in ¼.</p><p></p><p>Ratio :-</p><p></p><p>Ratio is the comparison of any two or more quantities usually expressed in it's simplest form</p><p></p><p>$