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Mathematics

If (3x - 4y) : (2x - 3y) = (5x - 6y) : (4x - 5y), find : x : y.

Ratio Proportion

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Answer

Given,

(3x - 4y) : (2x - 3y) = (5x - 6y) : (4x - 5y)

⇒3x−4y2x−3y=5x−6y4x−5y⇒(3x−4y)(4x−5y)=(5x−6y)(2x−3y)⇒12x2−15xy−16xy+20y2=10x2−15xy−12xy+18y2⇒12x2−10x2−15xy+15xy−16xy+12xy+20y2−18y2=0⇒2x2−4xy+2y2=0⇒2(x2−2xy+y2)=0⇒x2−2xy+y2=0⇒(x−y)2=0⇒x−y=0⇒x=y⇒xy=1\Rightarrow \dfrac{3x - 4y}{2x - 3y} = \dfrac{5x - 6y}{4x - 5y} \\[1em] \Rightarrow (3x - 4y)(4x - 5y) = (5x - 6y)(2x - 3y) \\[1em] \Rightarrow 12x^2 - 15xy - 16xy + 20y^2 = 10x^2 - 15xy - 12xy + 18y^2 \\[1em] \Rightarrow 12x^2 - 10x^2 - 15xy + 15xy - 16xy + 12xy + 20y^2 - 18y^2 = 0 \\[1em] \Rightarrow 2x^2 - 4xy + 2y^2 = 0 \\[1em] \Rightarrow 2(x^2 - 2xy + y^2) = 0 \\[1em] \Rightarrow x^2 - 2xy + y^2 = 0 \\[1em] \Rightarrow (x - y)^2 = 0 \\[1em] \Rightarrow x - y = 0 \\[1em] \Rightarrow x = y \\[1em] \Rightarrow \dfrac{x}{y} = 1 \\[1em]

Hence, x : y = 1 : 1.

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