AMC8 2012

AMC8 2012 · Q20

AMC8 2012 · Q20. It mainly tests Manipulating equations, Fractions.

What is the correct ordering of the three numbers $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, in increasing order?

下列三个分数 $\frac{5}{19}$、$\frac{7}{21}$、$\frac{9}{23}$ 由小到大的正确顺序是？

(A) $\frac{9}{23} < \frac{7}{21} < \frac{5}{19}$ $\frac{9}{23} < \frac{7}{21} < \frac{5}{19}$

(B) $\frac{5}{19} < \frac{7}{21} < \frac{9}{23}$ $\frac{5}{19} < \frac{7}{21} < \frac{9}{23}$

(D) $\frac{5}{19} < \frac{9}{23} < \frac{7}{21}$ $\frac{5}{19} < \frac{9}{23} < \frac{7}{21}$

(E) $\frac{7}{21} < \frac{5}{19} < \frac{9}{23}$ $\frac{7}{21} < \frac{5}{19} < \frac{9}{23}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

The value of $\frac{7}{21}$ is $\frac{1}{3}$. Now we give all the fractions a common denominator. $\frac{5}{19} \implies \frac{345}{1311}$ $\frac{1}{3} \implies \frac{437}{1311}$ $\frac{9}{23} \implies \frac{513}{1311}$ Ordering the fractions from least to greatest, we find that they are in the order listed. Therefore, our final answer is $\boxed{\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$.

$\frac{7}{21} = \frac{1}{3}$。现在给所有分数通分，公分母为 $1311$。 $\frac{5}{19} = \frac{345}{1311}$ $\frac{1}{3} = \frac{437}{1311}$ $\frac{9}{23} = \frac{513}{1311}$ 由小到大排序，正好是给出的顺序。因此最终答案是 $\boxed{\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$。

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