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 — Solution

2020 AMC 12B Problem 2

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What is the value of the following expression?

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

$1$

$\dfrac{9951}{9950}$

$\dfrac{4780}{4779}$

$\dfrac{108}{107}$

$\dfrac{81}{80}$

Answer: A

Concepts:difference of squares

Difficulty rating: 1020

Solution:

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.

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