Value of$$\frac{0.97 \times 0.97 \times 0.97 + 0.03 \times 0.03 \times 0.03}{0.97 \times 1.94 - 1.94 \times 0.03 + 0.03 \times 0.06}$$​ is equal to:

Correct option is B

​$$\textbf{Given:}\\[6pt]\frac{0.97^3 + 0.03^3}{0.97 \times 1.94 - 1.94 \times 0.03 + 0.03 \times 0.06}\\[12pt]\textbf{Formula used:}\\[6pt]a^3 + b^3 = (a + b)(a^2 - ab + b^2)\\[12pt]\textbf{Solution:}\\[6pt]\text{Let } a = 0.97,\; b = 0.03,\; a + b = 1\\[10pt]\Rightarrow \text{Numerator} = a^3 + b^3= (a + b)(a^2 - ab + b^2)\\[6pt]= a^2 - ab + b^2\\[12pt]\text{Denominator:}\\[6pt]1.94 = 2a \quad \text{since } 2 \times 0.97 = 1.94\\[6pt]0.97 \times 1.94 = a \times 2a = 2a^2\\[6pt]1.94 \times 0.03 = 2a \times b = 2ab\\[6pt]0.03 \times 0.06 = b \times 2b = 2b^2\\[10pt]\Rightarrow \text{Denominator}= 2a^2 - 2ab + 2b^2= 2(a^2 - ab + b^2)\\[12pt]\Rightarrow\frac{a^2 - ab + b^2}{2(a^2 - ab + b^2)} = \frac{1}{2}\\[12pt]\therefore \text{The value} = \frac{1}{2}$$​​

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