2012 AMC 8 Problems/Problem 20
Contents
Problem
What is the correct ordering of the three numbers , , and , in increasing order?
Solution 1
The value of is . Now we give all the fractions a common denominator.
Ordering the fractions from least to greatest, we find that they are in the order listed. Therefore, our final answer is .
Solution 2
Change into ; And Therefore, our answer is .
Solution 3
When and , . Hence, the answer is . ~ ryjs
This is also similar to Problem 3 on the AMC 8 2019, but with the rule switched.
Solution 4
By dividing, we see that 5/19 ≈ 0.26, 7/21 ≈ 0.33, and 9/23 ≈ 0.39. When we put this in order, < < . So our answer is ~ math_genius_11
Solution 5
is very close to , so you can round it to that. Similarly, and can be rounded to , so our ordering is 1/4, 1/3, and 1/2, or .
Solution 6
The numbers are in form . Using quotient rule on gives and this is positive. Because the derivative is always positive and the values of given by this question can be put on an interval that does not contain the critical point , a greater implies a greater , thus giving us the answer of . ~lopkiloinm
Solution 7
We calculating each fraction subtracted from 1. Doing so, we find that , (we leave this unsimplified), and . Clearly, . Using the fact that if , then , we find that the correct ordering is .
Video Solution
https://youtu.be/pU1zjw--K8M ~savannahsolver
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=TpsuRedYOiM&t=250s
See Also
2012 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 |
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