Difference between revisions of "2022 AMC 8 Problems/Problem 16"
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==Problem== | ==Problem== | ||
| − | Four numbers are written in a row. The average of the first two is <math>21</math> | + | Four numbers are written in a row. The average of the first two is <math>21,</math> the average of the middle two is <math>26,</math> and the average of the last two is <math>30.</math> What is the average of the first and last of the numbers? |
<math>\textbf{(A) } 24 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 26 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 28</math> | <math>\textbf{(A) } 24 \qquad \textbf{(B) } 25 \qquad \textbf{(C) } 26 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 28</math> | ||
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~savannahsolver | ~savannahsolver | ||
| + | |||
| + | ==Video Solution by Dr. David== | ||
| + | https://youtu.be/Hh6DrF178lc | ||
==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 23:31, 21 December 2024
Contents
- 1 Problem
- 2 Solution 1 (Arithmetic)
- 3 Solution 2 (Algebra)
- 4 Solution 3 (Assumption)
- 5 Solution 4
- 6 Video Solution by Math-X (First understand the problem!!!)
- 7 Video Solution (🚀1 min solution 🚀)
- 8 Video Solution
- 9 Video Solution by Ismail.maths
- 10 Video Solution
- 11 Video Solution
- 12 Video Solution by Dr. David
- 13 See Also
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: ![\[\frac{a+d}{2}=21+30-26=\boxed{\textbf{(B) } 25}.\]](http://latex.artofproblemsolving.com/f/b/f/fbfec2d8a8f3b506dc010c0fae39d1777f6b929d.png)
~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
Solution 4
Let and be the four numbers in that order. We are given that
and we wish to find
If we subtract from 
, we get ![\[\frac{c+d}{2}-\frac{b+c}{2}=\frac{d-b}{2}=4.\]](http://latex.artofproblemsolving.com/0/d/d/0ddfc1e9680c4ce9e0d83871b20a14a831cce52a.png)
So if we multiply by 
then we get 
Looking at our question: ![\[\frac{a+d}{2}=\frac{a+b+8}{2}=\frac{a+b}{2}+\frac{8}{2}=21+4=\boxed{\textbf{(B) } 25}.\]](http://latex.artofproblemsolving.com/9/1/9/91976e9fe7a4dd1778db55f3f962c1b953ab3301.png)
~funnyarmadillo58_aops
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
Video Solution by Dr. David
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. 
