What is the value of the following expression? (100^2 - 7^2)/(70^2 - 11^2) × ((70 - 11)(70 + 11))/((100 - 7)(100 + 7))

2020 AMC 12B Problem 2

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What is the value of the following expression?

\frac{100^2 - 7^2}{70^2 - 11^2} \cdot \frac{(70 - 11)(70 + 11)}{(100 - 7)(100 + 7)}

1

\dfrac{9951}{9950}

\dfrac{4780}{4779}

\dfrac{108}{107}

\dfrac{81}{80}

a^2 - b^2 = (a - b)(a + b)

Factor

100^2 - 7^2

and

70^2 - 11^2

, then look for cancellation

Using the difference of squares,

100^2 - 7^2 = (100 - 7)(100 + 7)

and

70^2 - 11^2 = (70 - 11)(70 + 11).

The expression becomes

\begin{aligned} &\frac{(100 - 7)(100 + 7)}{(70 - 11)(70 + 11)} \\ &\quad {}\cdot \frac{(70 - 11)(70 + 11)}{(100 - 7)(100 + 7)} = 1. \end{aligned}

Thus, the correct answer is A.