Difference between revisions of "2022 AMC 8 Problems/Problem 16"
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We add <math>(1)</math> and <math>(3),</math> then subtract <math>(2)</math> from the result: <cmath>\frac{a+d}{2}=21+30-26=\boxed{\textbf{(B) } 25}.</cmath> | We add <math>(1)</math> and <math>(3),</math> then subtract <math>(2)</math> from the result: <cmath>\frac{a+d}{2}=21+30-26=\boxed{\textbf{(B) } 25}.</cmath> | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 3 (Assumption)== | ||
+ | |||
+ | We can just assume some of the numbers. For example, let the first two numbers both be <math>21.</math> It follows that the third number is <math>31,</math> and the fourth number is <math>29.</math> Therefore, the average of the first and last numbers is <math>\dfrac{21+29}2=\dfrac{50}2=\boxed{\textbf{(B) } 25}.</math> | ||
+ | |||
+ | We can check this with other sequences, such as <math>20,22,30,30,</math> where the average of the first and last numbers is still <math>25.</math> | ||
+ | |||
+ | ~wuwang2002 | ||
+ | |||
+ | ==Video Solution by Math-X (First understand the problem!!!)== | ||
+ | https://youtu.be/oUEa7AjMF2A?si=2goytlOr7qxq69I9&t=2801 | ||
+ | |||
+ | ~Math-X | ||
+ | |||
+ | ==Video Solution (🚀1 min solution 🚀)== | ||
+ | https://youtu.be/oVG6zqPVPfM | ||
+ | |||
+ | ~Education, the Study of Everything | ||
==Video Solution== | ==Video Solution== | ||
Line 30: | Line 48: | ||
~Interstigation | ~Interstigation | ||
+ | |||
+ | ==Video Solution by Ismail.maths== | ||
+ | |||
+ | https://www.youtube.com/watch?v=38JjGdGI5a0 | ||
+ | |||
+ | ~Ismail.maths93 | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/hs6y4PWnoWg | ||
+ | |||
+ | ~STEMbreezy | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/tpzsowaoQRc | ||
+ | |||
+ | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2022|num-b=15|num-a=17}} | {{AMC8 box|year=2022|num-b=15|num-a=17}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 20:48, 17 May 2024
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 by Math-X (First understand the problem!!!)
https://youtu.be/oUEa7AjMF2A?si=2goytlOr7qxq69I9&t=2801
~Math-X
Video Solution (🚀1 min solution 🚀)
~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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.