Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2000 AMC 8 Problems/Problem 23"

(solution and see also box)
(→‎Video Solution)
 
(8 intermediate revisions by 7 users not shown)
Line 4: Line 4:
  
 
<math> \text{(A)}\ 5\frac{3}{7}\qquad\text{(B)}\ 6\qquad\text{(C)}\ 6\frac{4}{7}\qquad\text{(D)}\ 7\qquad\text{(E)}\ 7\frac{3}{7} </math>
 
<math> \text{(A)}\ 5\frac{3}{7}\qquad\text{(B)}\ 6\qquad\text{(C)}\ 6\frac{4}{7}\qquad\text{(D)}\ 7\qquad\text{(E)}\ 7\frac{3}{7} </math>
−
 
  
 
==Solution==
 
==Solution==
  
−
Remember that if a list of <math>n</math> numbers has an average of <math>k</math>, then then sum <math>S</math> of all the numbers on the list is <math>S = nk</math>.
+
Remember that if a list of <math>n</math> numbers has an average of <math>k</math>, then the sum <math>S</math> of all the numbers on the list is <math>S = nk</math>.
  
 
So if the average of the first <math>4</math> numbers is <math>5</math>, then the first four numbers total <math>4 \cdot 5 = 20</math>.
 
So if the average of the first <math>4</math> numbers is <math>5</math>, then the first four numbers total <math>4 \cdot 5 = 20</math>.
Line 16: Line 15:
 
If the average of all <math>7</math> numbers is <math>6\frac{4}{7}</math>, then the total of all seven numbers is <math>7 \cdot 6\frac{4}{7} = 7\cdot 6 + 4 =  46</math>.
 
If the average of all <math>7</math> numbers is <math>6\frac{4}{7}</math>, then the total of all seven numbers is <math>7 \cdot 6\frac{4}{7} = 7\cdot 6 + 4 =  46</math>.
  
−
If the first four numbers are <math>20</math>, and the last four numbers are <math>32</math>, then all "eight" numbers are <math>20 + 32 = 52</math>.  But that's counting one number twice.  Since the sum of all seven numbers is <math>46</math>, then the number that was counted twice is <math>52 - 46 = 6</math>, and the answer is <math>\boxed{B}</math>
+
If the first four numbers are <math>20</math>, and the last four numbers are <math>32</math>, then all "eight" numbers are <math>20 + 32 = 52</math>.  But that's counting one number twice.  Since the sum of all seven numbers is <math>46</math>, then the number that was counted twice is <math>52 - 46 = 6</math>, and the answer is <math>\boxed{(B) 6}</math>
 +
 
 +
Algebraically, if <math>a + b + c + d = 20</math>, and <math>d + e + f + g = 32</math>, you can add both equations to get <math>a + b + c + 2d + e + f + g = 52</math>.  You know that <math>a + b + c + d + e + f + g = 46</math>, so you can subtract that from the last equation to get <math>d = \boxed{(B) 6}</math>, and <math>d</math> is the number that appeared twice.
 +
 
 +
Yay!  :D
 +
 
 +
 
 +
== Video Solution ==
 +
https://youtu.be/xdNSN7hYK6k  Soo, DRMS, NM
 +
 
 +
https://www.youtube.com/watch?v=DZ_AzlW3qqM      ~David
  
−
Algebraically, if <math>a + b + c + d = 20</math>, and <math>d + e + f + g = 32</math>, you can add both equations to get <math>a + b + c + 2d + e + f + g = 52</math>.  You know that <math>a + b + c + d + e + f + g = 46</math>, so you can subtract that from the last equation to get <math>d = 6</math>, and <math>d</math> is the number that appeared twice.
 
−
 
 
==See Also==
 
==See Also==
  
 
{{AMC8 box|year=2000|num-b=22|num-a=24}}
 
{{AMC8 box|year=2000|num-b=22|num-a=24}}
 +
{{MAA Notice}}

Latest revision as of 19:07, 15 April 2023

Problem

There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is

$\text{(A)}\ 5\frac{3}{7}\qquad\text{(B)}\ 6\qquad\text{(C)}\ 6\frac{4}{7}\qquad\text{(D)}\ 7\qquad\text{(E)}\ 7\frac{3}{7}$

Solution

Remember that if a list of $n$ numbers has an average of $k$, then the sum $S$ of all the numbers on the list is $S = nk$.

So if the average of the first $4$ numbers is $5$, then the first four numbers total $4 \cdot 5 = 20$.

If the average of the last $4$ numbers is $8$, then the last four numbers total $4 \cdot 8 = 32$.

If the average of all $7$ numbers is $6\frac{4}{7}$, then the total of all seven numbers is $7 \cdot 6\frac{4}{7} = 7\cdot 6 + 4 = 46$.

If the first four numbers are $20$, and the last four numbers are $32$, then all "eight" numbers are $20 + 32 = 52$. But that's counting one number twice. Since the sum of all seven numbers is $46$, then the number that was counted twice is $52 - 46 = 6$, and the answer is $\boxed{(B) 6}$

Algebraically, if $a + b + c + d = 20$, and $d + e + f + g = 32$, you can add both equations to get $a + b + c + 2d + e + f + g = 52$. You know that $a + b + c + d + e + f + g = 46$, so you can subtract that from the last equation to get $d = \boxed{(B) 6}$, and $d$ is the number that appeared twice.

Yay!  :D


Video Solution

https://youtu.be/xdNSN7hYK6k Soo, DRMS, NM

https://www.youtube.com/watch?v=DZ_AzlW3qqM ~David

See Also

2000 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2000_AMC_8_Problems/Problem_23&oldid=192215"