AMC12 2020 B

AMC12 2020 B · Q2

AMC12 2020 B · Q2. It mainly tests Factoring.

What is the value of the following expression? $\frac{100^2 - 7^2}{70^2 - 11^2} \cdot \frac{(70 - 11)(70 + 11)}{(100 - 7)(100 + 7)}$

下列表达式的值是多少？ $\frac{100^2 - 7^2}{70^2 - 11^2} \cdot \frac{(70 - 11)(70 + 11)}{(100 - 7)(100 + 7)}$

(A) 1 1

(B) $\frac{9951}{9950}$ $\frac{9951}{9950}$

(D) $\frac{108}{107}$ $\frac{108}{107}$

(E) $\frac{81}{80}$ $\frac{81}{80}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Because $a^2 - b^2$ factors as $(a-b)(a+b)$, the given expression simplifies to 1: $\frac{100^2 - 7^2}{70^2 - 11^2} \cdot \frac{(70 - 11)(70 + 11)}{(100 - 7)(100 + 7)} = \frac{100^2 - 7^2}{(100 - 7)(100 + 7)} \cdot \frac{(70 - 11)(70 + 11)}{70^2 - 11^2} = 1$.

因为$a^2 - b^2$可因式分解为$(a-b)(a+b)$，给定的表达式简化为1： $\frac{100^2 - 7^2}{70^2 - 11^2} \cdot \frac{(70 - 11)(70 + 11)}{(100 - 7)(100 + 7)} = \frac{100^2 - 7^2}{(100 - 7)(100 + 7)} \cdot \frac{(70 - 11)(70 + 11)}{70^2 - 11^2} = 1$。

Topics

AL.006 Factoring