Difference between revisions of "2012 AMC 8 Problems/Problem 20"

Problem

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

$\textbf{(A)}\hspace{.05in}\frac{9}{23}<\frac{7}{21}<\frac{5}{19}\quad\textbf{(B)}\hspace{.05in}\frac{5}{19}<\frac{7}{21}<\frac{9}{23}\quad\textbf{(C)}\hspace{.05in}\frac{9}{23}<\frac{5}{19}<\frac{7}{21}$

$\textbf{(D)}\hspace{.05in}\frac{5}{19}<\frac{9}{23}<\frac{7}{21}\quad\textbf{(E)}\hspace{.05in}\frac{7}{21}<\frac{5}{19}<\frac{9}{23}$

Solution 1

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

Solution 2

Instead of finding the LCD, we can subtract each fraction from $1$ to get a common numerator. Thus,

$1-\dfrac{5}{19}=\dfrac{14}{19}$

$1-\dfrac{7}{21}=\dfrac{14}{21}$

$1-\dfrac{9}{23}=\dfrac{14}{23}$

All three fractions have common numerator $14$. Now it is obvious the order of the fractions. $\dfrac{14}{19}>\dfrac{14}{21}>\dfrac{14}{23}\implies\dfrac{5}{19}<\dfrac{7}{21}<\dfrac{9}{23}$. Therefore, our answer is $\boxed{\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$.

Solution 3

Change $7/21$ into $1/3$; $$\frac{1}{3}\cdot\frac{5}{5}=\frac{5}{15}$$ $$\frac{5}{15}>\frac{5}{19}$$ $$\frac{7}{21}>\frac{5}{19}$$ And $$\frac{1}{3}\cdot\frac{9}{9}=\frac{9}{27}$$ $$\frac{9}{27}<\frac{9}{23}$$ $$\frac{7}{21}<\frac{9}{23}$$ Therefore, our answer is $\boxed{\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$.

Solution 4

When $\frac{x}{y}<1$ and $z>0$, $\frac{x+z}{y+z}>\frac{x}{y}$. Hence, the answer is ${\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$. ~ ryjs

This is also similar to Problem 3 on the AMC 8 2019, but with the rule switched.

Solution 5

By dividing, we see that 5/19 ≈ 0.26, 7/21 ≈ 0.33, and 9/23 ≈ 0.39. When we put this in order, $0.26$ < $0.33$ < $0.39$. So our answer is $\boxed{\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$ ~ math_genius_11

Solution 6

$\frac{5}{19}$ is very close to $\frac{1}{4}$, so you can round it to that. Similarly, $\frac{7}{21} = \frac{1}{3}$ and $\frac{9}{23}$ can be rounded to $\frac{1}{2}$, so our ordering is 1/4, 1/3, and 1/2, or $\boxed{\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$.

Solution 7

The numbers are in form $\frac{x}{x+14}$. Using quotient rule on $\frac{d}{dx}(\frac{x}{x+14})$ gives $\frac{14}{(x+14)^2}$ and this is always positive. Because the derivative is always positive, a greater $x$ implies a greater $\frac{x}{x+14}$, thus giving us the answer of $\boxed{\textbf{(B)}\ \frac{5}{19}<\frac{7}{21}<\frac{9}{23}}$. ~lopkiloinm