2019 AMC 8 Problems/Problem 3
Contents
Problem 3
Which of the following is the correct order of the fractions and from least to greatest?
Solution 1 (Bashing)
We take a common denominator:
Since it follows that the answer is .
-xMidnightFirex
~ dolphin7 - I took your idea and made it an explanation.
- Clearness by doulai1
Solution 2
When and , . Hence, the answer is . ~ 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 , , and . Thus, our answer is .
~~ by an insane math guy.
Solution 4
Suppose each fraction is expressed with denominator : . Clearly so the answer is .
Solution 5 -SweetMango77
We notice that each of these fraction's numerator denominator . If we take each of the fractions, and subtract from each, we get , , and . These are easy to order because the numerators are the same, we get . Because it is a subtraction by a constant, in order to order them, we keep the inequality signs to get .
Solution 6
Adding on to Solution 5, we can turn each of the fractions , , and into , , and , respectively. We now subtract from each to get , , and . Since their numerators are all 4, this is easy because we know that and therefore . Reverting them back to their original fractions, we can now see that the answer is .
~by 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
Video Solution 3
~savannahsolver
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.
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.