2019 AMC 8 Problems/Problem 3: Difference between revisions

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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

Each one is in the form $\frac{x+4}{x}$ so we are really comparing $\frac{4}{11}, \frac{4}{15},$ and $\frac{4}{13}$ where you can see $\frac{4}{11}>\frac{4}{13}>\frac{4}{15}$ so the answer is $\boxed{\textbf{(E)}\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}$ .

Solution 2

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.

Solution 3

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 AMC 2012.

Solution 4(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 5

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 6(similar to Solution 1)

We convert them into mixed numbers, so we have 1 4/15, 1 4/11, and 1 4/13. Since all of them have a 1 as the mixed number, so we can just compare the fractions. Obviously, if you have a bigger denominator, than you have a smaller number, and the numerators are the same. Hence, our solution is $1 4/11> 1 4/13> 1 4/15$ .

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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.

@@ Line 29: / Line 29: @@
 ==Solution 5==
 Suppose each fraction is expressed with denominator <math>2145</math>: <math>\frac{2925}{2145}, \frac{2717}{2145}, \frac{2805}{2145}</math>. Clearly <math>2717<2805<2925</math> so the answer is <math>\boxed{\textbf{(E)}}</math>.
+==Solution 6(similar to Solution 1)==
+We convert them into mixed numbers, so we have 1 4/15, 1 4/11, and 1 4/13. Since all of them have a 1 as the mixed number, so we can just compare the fractions. Obviously, if you have a bigger denominator, than you have a smaller number, and the numerators are the same. Hence, our solution is <math>1 4/11> 1 4/13> 1 4/15</math>.
 ==See Also==

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