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Mathematics

Find the compounded ratio of

(a + b)2 : (a - b)2, (a2 - b2) : (a2 + b2) and (a4 - b4) : (a + b)4.

Ratio Proportion

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Answer

The compounded ratio is :

(a+b)2(a−b)2×(a2−b2)(a2+b2)×(a4−b4)(a+b)4=(a+b)2(a−b)2×(a−b)(a+b)(a2+b2)×(a2−b2)(a2+b2)(a+b)4=(a+b)2(a−b)2×(a−b)(a+b)(a2+b2)×(a−b)(a+b)(a2+b2)(a+b)4=(a+b)4(a−b)2(a2+b2)(a−b)2(a2+b2)(a+b)4=11=1:1.\dfrac{(a + b)^2}{(a - b)^2} \times \dfrac{(a^2 - b^2)}{(a^2 + b^2)} \times \dfrac{(a^4 - b^4)}{(a + b)^4} \\[1em] = \dfrac{(a + b)^2}{(a - b)^2} \times \dfrac{(a - b)(a + b)}{(a^2 + b^2)} \times \dfrac{(a^2 - b^2)(a^2 + b^2)}{(a + b)^4} \\[1em] = \dfrac{(a + b)^2}{(a - b)^2} \times \dfrac{(a - b)(a + b)}{(a^2 + b^2)} \times \dfrac{(a - b)(a + b)(a^2 + b^2)}{(a + b)^4} \\[1em] = \dfrac{(a + b)^4(a - b)^2(a^2 + b^2)}{(a - b)^2(a^2 + b^2)(a + b)^4} \\[1em] = \dfrac{1}{1} = 1 : 1 .

Hence, the compounded ratio is 1 : 1.

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