Question : Simplify the expression: $\frac{143 \times 143+143 \times 139+139 \times 139}{143 \times 143 \times 143-139 \times 139 \times 139}$
Option 1: $\frac{1}{2}$
Option 2: $282$
Option 3: $\frac{1}{4}$
Option 4: $4$
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Correct Answer: $\frac{1}{4}$
Solution :
Given: $\frac{143 \times 143+143 \times 139+139 \times 139}{143 \times 143 \times 143-139 \times 139 \times 139}$
$=\frac{(143)^2+143\times139+(139)^2}{(143)^3-(139)^3}$
We know, $a^3-b^3=(a-b)(a^2+ab+b^2)$
$=\frac{(143)^2+143\times139+(139)^2}{(143-139)((143)^2+143\times139+(139)^2)}$
$=\frac{1}{143-139}$
$=\frac{1}{4}$
Hence, the correct answer is $\frac{1}{4}$.
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