Difference between revisions of "2019 AMC 8 Problems/Problem 3"

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==Problem 3==
 
Which of the following is the correct order of the fractions <math>\frac{15}{11},\frac{19}{15},</math> and <math>\frac{17}{13},</math> from least to greatest?
 
  
<math>\textbf{(A) }\frac{15}{11}< \frac{17}{13}< \frac{19}{15}  \qquad\textbf{(B) }\frac{15}{11}< \frac{19}{15}<\frac{17}{13}    \qquad\textbf{(C) }\frac{17}{13}<\frac{19}{15}<\frac{15}{11}    \qquad\textbf{(D) } \frac{19}{15}<\frac{15}{11}<\frac{17}{13}  \qquad\textbf{(E) }  \frac{19}{15}<\frac{17}{13}<\frac{15}{11}</math>
 
==Solution 1==
 
Each one is in the form <math>\frac{x+4}{x}</math> so we are really comparing <math>\frac{4}{11}, \frac{4}{15},</math> and <math>\frac{4}{13}</math> where you can see <math>\frac{4}{11}>\frac{4}{13}>\frac{4}{15}</math> so the answer is  <math>\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}</math>.
 
 
==Solution 2==
 
We take a common denominator:
 
<cmath>\frac{15}{11},\frac{19}{15}, \frac{17}{13} = \frac{15\cdot 15 \cdot 13}{11\cdot 15 \cdot 13},\frac{19 \cdot 11 \cdot 13}{15\cdot 11 \cdot 13}, \frac{17 \cdot 11 \cdot 15}{13\cdot 11 \cdot 15} = \frac{2925}{2145},\frac{2717}{2145},\frac{2805}{2145}.</cmath>
 
 
Since <math>2717<2805<2925</math> it follows that the answer is <math>\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}</math>.
 
 
-xMidnightFirex
 
 
~ dolphin7 - I took your idea and made it an explanation.
 
 
==Solution 3==
 
When <math>\frac{x}{y}>1</math> and <math>z>0</math>, <math>\frac{x+z}{y+z}<\frac{x}{y}</math>. Hence, the answer is <math>\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}</math>.
 
~ ryjs
 
 
This is also similar to Problem 20 on the AMC 2012.
 
 
==Solution 4(probably won't use this solution)==
 
We use our insane mental calculator to find out that <math>\frac{15}{11} \approx 1.36</math>, <math>\frac{19}{15} \approx 1.27</math>, and <math>\frac{17}{13} \approx 1.31</math>. Thus, our answer is <math>\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}</math>.
 
 
~~ by an insane math guy
 
 
==See Also==
 
{{AMC8 box|year=2019|num-b=2|num-a=4}}
 
 
{{MAA Notice}} The butterfly method is a method when you multiply the denominator of the second fraction and multiply it by the numerator from the first fraction.
 

Revision as of 17:32, 15 October 2020