Question : Simplify the given equation:
$\frac{0.09 \times 0.09+0.04 \times 0.04+0.16 \times 0.16+2 \times 0.09 \times 0.04+2 \times 0.04 \times 0.16+2 \times 0.09 \times 0.16}{0.3 \times 0.3+0.2 \times 0.2+0.4 \times 0.4}$
Option 1: 0.38
Option 2: 0.24
Option 3: 0.32
Option 4: 0.29
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Correct Answer: 0.29
Solution :
Use the algebraic identity,
\((a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca\)
Where \(a=0.09\), \(b=0.04\) and \(c=0.16\).
Substituting these values into the formula, we get:
$=\frac{0.09 \times 0.09+0.04 \times 0.04+0.16 \times 0.16+2 \times 0.09 \times 0.04+2 \times 0.04 \times 0.16+2 \times 0.09 \times 0.16}{0.3 \times 0.3+0.2 \times 0.2+0.4 \times 0.4}$
$=\frac{(0.09+0.04+0.16)^2}{(0.3^2+0.2^2+0.4^2)}$
$= \frac{(0.29)^2}{0.29}$
$= 0.29$
Hence, the correct answer is 0.29.
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