Question : The value of $\frac{(157 \times 157)+(157 \times 133)+(133 \times 133)}{(157 \times 157 \times 157)-(133 \times 133 \times 133)}$ is:
Option 1: $\frac{1}{24}$
Option 2: $24$
Option 3: $290$
Option 4: $\frac{1}{290}$
Correct Answer: $\frac{1}{24}$
Solution :
Given: $\frac{(157 \times 157)+(157 \times 133)+(133 \times 133)}{(157 \times 157 \times 157)-(133 \times 133 \times 133)}$
Let $a=157$ and $b=133$
= $\frac{a^2+ab+b^2}{a^3-b^3}$
= $\frac{a^2+ab+b^2}{(a-b)(a^2+ab+b^2)}$
= $\frac{1}{(a-b)}$
Putting the values, we get:
= $\frac{1}{157-133}$
= $\frac{1}{24}$
Hence, the correct answer is $\frac{1}{24}$.
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