Find the compounded ratio of $(x^{2} - y^{2}) : (x^{2} + y^{2})$ and $(x^{4} - y^{4}) : (x + y)^{4}$

(x - y)^{3} : (x + y)^{3}

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(x + y)^{2} : (x^{2} - y^{2})

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1 : 1

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(x - y)^{2} : (x + y)^{2}

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Solution

The correct option is D $(x - y)^{2} : (x + y)^{2}$
By the definition of compounded ratio these ratio can be expressed as
$\frac{(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \frac{(x^{4} - y^{4})}{(x + y)^{4}}$
$\frac{(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \frac{(x^{2} - y^{2}) (x^{2} + y^{2})}{(x + y)^{4}}$
$= \frac{(x^{2} - y^{2})^{2}}{(x + y)^{4}}$
$= \frac{[(x - y) (x + y)]^{2}}{(x + y)^{4}}$
$= \frac{(x - y)^{2}}{(x + y)^{2}}$
Hence $(x - y)^{2} : (x + y)^{2}$

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