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 "2022 AMC 8 Problems/Problem 11"

m (→‎Solution)
m (→‎Solution 1)
Line 3: Line 3:
  
 
<math>\textbf{(A) } 34\qquad\textbf{(B) } 38\qquad\textbf{(C) } 41\qquad\textbf{(D) } 44\qquad\textbf{(E) } 47</math>
 
<math>\textbf{(A) } 34\qquad\textbf{(B) } 38\qquad\textbf{(C) } 41\qquad\textbf{(D) } 44\qquad\textbf{(E) } 47</math>
−
 
−
 
−
==Solution 1==
 
−
 
−
Let's say that the first strand of pasta he had is x. The first time he takes a bite of this strand will make it 2 pieces. This is x - 3. The second time he takes the bites of EACH strand will become (x - 3) * 3(2). The 2 is for the two bites he took, 1 for each strand. This equation is showing that for every bite that he is taking, he is taking off 3 inches. This is being showed in the 3(2). He took 2 bites, each taking off 3 inches. Now, the problem is stating that their are 10 pieces of pasta and the TOTAL length is 17 inches. We have to plug in the total number of bites to the equation (x - 3) * 3( ). To find this out, we can see that for the number of strands there are, he took 1 less bite than that number. For example, if there are 2 pieces, he took 1 bite to get those. Since there are 10 pieces, he took 9 bites. The equation then becomes, (x - 3) * 3(9) = 17. To solve this, x = 44. Therefore, the answer is <math>\boxed{\textbf{(D) } 44}</math>.
 
−
 
−
 
−
 
−
 
  
  
Line 17: Line 8:
  
  
 +
==Solution 1==
  
 +
Let's say that the first strand of pasta he had is x. The first time he takes a bite of this strand will make it 2 pieces. This is x - 3. The second time he takes the bites of EACH strand will become (x - 3) * 3(2). The 2 is for the two bites he took, 1 for each strand. This equation is showing that for every bite that he is taking, he is taking off 3 inches. This is being showed in the 3(2). He took 2 bites, each taking off 3 inches. Now, the problem is stating that their are 10 pieces of pasta and the TOTAL length is 17 inches. We have to plug in the total number of bites to the equation (x - 3) * 3( ). To find this out, we can see that for the number of strands there are, he took 1 less bite than that number. For example, if there are 2 pieces, he took 1 bite to get those. Since there are 10 pieces, he took 9 bites. The equation then becomes, (x - 3) * 3(9) = 17. To solve this, x = 44. Therefore, the answer is <math>\boxed{\textbf{(D) } 44}</math>.
  
 
==Solution 2==
 
==Solution 2==

Revision as of 13:11, 27 December 2024

Problem

Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is $17$ inches. How long, in inches, was the piece of pasta he started with?

$\textbf{(A) } 34\qquad\textbf{(B) } 38\qquad\textbf{(C) } 41\qquad\textbf{(D) } 44\qquad\textbf{(E) } 47$



Solution 1

Let's say that the first strand of pasta he had is x. The first time he takes a bite of this strand will make it 2 pieces. This is x - 3. The second time he takes the bites of EACH strand will become (x - 3) * 3(2). The 2 is for the two bites he took, 1 for each strand. This equation is showing that for every bite that he is taking, he is taking off 3 inches. This is being showed in the 3(2). He took 2 bites, each taking off 3 inches. Now, the problem is stating that their are 10 pieces of pasta and the TOTAL length is 17 inches. We have to plug in the total number of bites to the equation (x - 3) * 3( ). To find this out, we can see that for the number of strands there are, he took 1 less bite than that number. For example, if there are 2 pieces, he took 1 bite to get those. Since there are 10 pieces, he took 9 bites. The equation then becomes, (x - 3) * 3(9) = 17. To solve this, x = 44. Therefore, the answer is $\boxed{\textbf{(D) } 44}$.

Solution 2

If there are $10$ pieces of pasta, Henry took $10-1=9$ bites. Each of these $9$ bites took $3$ inches of pasta out, and thus his bites in total took away $9\cdot 3 = 27$ inches of pasta. Thus, the original piece of pasta was $27+17=\boxed{\textbf{(D) } 44}$ inches long.

~wamofan

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/oUEa7AjMF2A?si=1Ku2y-sqPm_i3J01&t=1518

~Math-X

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/q7bicopSq14

~Education, the Study of Everything

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=UmuvflX99ig

Video Solution

https://www.youtube.com/watch?v=H1K4WJ93isA

~Mathematical Dexterity

Video Solution

https://youtu.be/Ij9pAy6tQSg?t=887

~Interstigation

Video Solution

https://youtu.be/p29Fe2dLGs8

~STEMbreezy

Video Solution

https://youtu.be/S6kqa8VUj70

~savannahsolver

Video Solution by Dr. David

https://youtu.be/C4n4S-FbLl4

See Also

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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=2022_AMC_8_Problems/Problem_11&oldid=238008"