2022 AMC 8 Problems/Problem 16
Contents
Problem
Four numbers are written in a row. The average of the first two is the average of the middle two is and the average of the last two is What is the average of the first and last of the numbers?
Solution 1 (Arithmetic)
Note that the sum of the first two numbers is the sum of the middle two numbers is and the sum of the last two numbers is
It follows that the sum of the four numbers is so the sum of the first and last numbers is Therefore, the average of the first and last numbers is
~MRENTHUSIASM
Solution 2 (Algebra)
Let and be the four numbers in that order. We are given that and we wish to find
We add and then subtract from the result: ~MRENTHUSIASM
Solution 3 (Assumption)
We can just assume some of the numbers. For example, let the first two numbers both be It follows that the third number is and the fourth number is Therefore, the average of the first and last numbers is
We can check this with other sequences, such as where the average of the first and last numbers is still
~wuwang2002
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution
https://youtu.be/Ij9pAy6tQSg?t=1394
~Interstigation
Video Solution by Ismail.maths
https://www.youtube.com/watch?v=38JjGdGI5a0
~Ismail.maths93
Video Solution
~STEMbreezy
Video Solution
~savannahsolver
See Also
2022 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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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