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

(→‎Solution 5 -SweetMango77)
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== Solution 6 ==
 
== Solution 6 ==
 
Adding on to Solution 5, we can turn each of the fractions <math>\frac{15}{11}</math>, <math>\frac{17}{13}</math>, and <math>\frac{19}{15}</math> into <math>1</math><math>\frac{4}{11}</math>, <math>1</math><math>\frac{4}{13}</math>, and <math>1</math><math>\frac{4}{15}</math>, respectively. We now subtract <math>1</math> from each to get <math>\frac{4}{11}</math>, <math>\frac{4}{15}</math>, and <math>\frac{4}{13}</math>. Since their numerators are all 4, this is easy because we know that <math>\frac{1}{15}<\frac{1}{13}<\frac{1}{11}</math> and therefore <math>\frac{4}{15}<\frac{4}{13}<\frac{4}{11}</math>. Reverting them back to their original fractions, we can now see that the answer is <math>\boxed{\textbf{(E)}\;\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}</math>.
 
Adding on to Solution 5, we can turn each of the fractions <math>\frac{15}{11}</math>, <math>\frac{17}{13}</math>, and <math>\frac{19}{15}</math> into <math>1</math><math>\frac{4}{11}</math>, <math>1</math><math>\frac{4}{13}</math>, and <math>1</math><math>\frac{4}{15}</math>, respectively. We now subtract <math>1</math> from each to get <math>\frac{4}{11}</math>, <math>\frac{4}{15}</math>, and <math>\frac{4}{13}</math>. Since their numerators are all 4, this is easy because we know that <math>\frac{1}{15}<\frac{1}{13}<\frac{1}{11}</math> and therefore <math>\frac{4}{15}<\frac{4}{13}<\frac{4}{11}</math>. Reverting them back to their original fractions, we can now see that the answer is <math>\boxed{\textbf{(E)}\;\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}</math>.
 +
 +
~ ChipmunkT
  
 
== Video Solution ==
 
== Video Solution ==

Revision as of 16:41, 1 January 2022

Problem 3

Which of the following is the correct order of the fractions $\frac{15}{11},\frac{19}{15},$ and $\frac{17}{13},$ from least to greatest?

$\textbf{(A) }\frac{15}{11}< \frac{17}{13}< \frac{19}{15} \qquad\textbf{(B) }\frac{15}{11}< \frac{19}{15}<\frac{17}{13} \qquad\textbf{(C) }\frac{17}{13}<\frac{19}{15}<\frac{15}{11} \qquad\textbf{(D) } \frac{19}{15}<\frac{15}{11}<\frac{17}{13} \qquad\textbf{(E) } \frac{19}{15}<\frac{17}{13}<\frac{15}{11}$

Solution 1 (Bashing)

We take a common denominator: \[\frac{15}{11},\frac{19}{15}, \frac{17}{13} = \frac{15\cdot 15 \cdot 13}{11\cdot 15 \cdot 13},\frac{19 \cdot 11 \cdot 13}{15\cdot 11 \cdot 13}, \frac{17 \cdot 11 \cdot 15}{13\cdot 11 \cdot 15} = \frac{2925}{2145},\frac{2717}{2145},\frac{2805}{2145}.\]

Since $2717<2805<2925$ it follows that the answer is $\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$.

-xMidnightFirex

~ dolphin7 - I took your idea and made it an explanation.

- Clearness by doulai1

Solution 2

When $\frac{x}{y}>1$ and $z>0$, $\frac{x+z}{y+z}<\frac{x}{y}$. Hence, the answer is $\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$. ~ ryjs

This is also similar to Problem 20 on the 2012 AMC 8.

Solution 3 (probably won't use this solution)

We use our insane mental calculator to find out that $\frac{15}{11} \approx 1.36$, $\frac{19}{15} \approx 1.27$, and $\frac{17}{13} \approx 1.31$. Thus, our answer is $\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$.

~~ by an insane math guy.

Solution 4

Suppose each fraction is expressed with denominator $2145$: $\frac{2925}{2145}, \frac{2717}{2145}, \frac{2805}{2145}$. Clearly $2717<2805<2925$ so the answer is $\boxed{\textbf{(E)}}$.

Solution 5 -SweetMango77

We notice that each of these fraction's numerator $-$ denominator $=4$. If we take each of the fractions, and subtract $1$ from each, we get $\frac{4}{11}$, $\frac{4}{15}$, and $\frac{4}{13}$. These are easy to order because the numerators are the same, we get $\frac{4}{15}<\frac{4}{13}<\frac{4}{11}$. Because it is a subtraction by a constant, in order to order them, we keep the inequality signs to get $\boxed{\textbf{(E)}\;\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$.

Solution 6

Adding on to Solution 5, we can turn each of the fractions $\frac{15}{11}$, $\frac{17}{13}$, and $\frac{19}{15}$ into $1$$\frac{4}{11}$, $1$$\frac{4}{13}$, and $1$$\frac{4}{15}$, respectively. We now subtract $1$ from each to get $\frac{4}{11}$, $\frac{4}{15}$, and $\frac{4}{13}$. Since their numerators are all 4, this is easy because we know that $\frac{1}{15}<\frac{1}{13}<\frac{1}{11}$ and therefore $\frac{4}{15}<\frac{4}{13}<\frac{4}{11}$. Reverting them back to their original fractions, we can now see that the answer is $\boxed{\textbf{(E)}\;\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$.

~ ChipmunkT

Video Solution

The Learning Royal: https://youtu.be/IiFFDDITE6Q

Video Solution 2

Solution detailing how to solve the problem: https://www.youtube.com/watch?v=q27qEcr7TbQ&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=4

See also

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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.

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