2022 AMC 8 Problems/Problem 6
Contents
Problem
Three positive integers are equally spaced on a number line. The middle number is and the largest number is times the smallest number. What is the smallest of these three numbers?
Solution 1
Let the smallest number be It follows that the largest number is
Since and are equally spaced on a number line, we have ~MRENTHUSIASM
Solution 2
Let the common difference of the arithmetic sequence be . Consequently, the smallest number is and the largest number is . As the largest number is times the smallest number, . Finally, we find that the smallest number is .
~MathFun1000
Solution 3
Let the smallest number be . Since the integers are equally spaced, and there are three of them, the middle number () is the arithmetic mean of the other two numbers ( and ). Thus, we set up the equation , and, solving for , get . Since is the smallest number out of the list ( because it equals ), the answer is .
~scthecool
Solution 4
Let the smallest number be . Because and are equally spaced from , must be the average. By adding and and dividing by , we get that the mean is also . We get that , and solving gets .
~DrDominic
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=bwDG0eKuI9uNqoOW&t=677
~Math-X
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution
~STEMbreezy
Video Solution
~savannahsolver
Video Solution
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=409
~Interstigation
Video Solution
~harungurcan
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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.