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 "2025 AMC 8 Problems/Problem 1"

m (Fixed link)
(→‎Solution 6)
 
(7 intermediate revisions by 4 users not shown)
Line 52: Line 52:
 
~derekwang2048
 
~derekwang2048
  
−
== Video Solution 1 by SpreadTheMathLove ==
+
== Solution 5 ==
 +
 
 +
The first thing that we can see is that the shaded region is a <math>2</math>x<math>2</math> square(in the middle of the figure) combined with <math>8</math> small triangles. Since each small triangle is <math>1 \cdot \frac{1}{2} = \frac{1}{2}</math> unit squares, we get <math>4 + 8 \cdot \frac{1}{2} = 8</math> unit squares. Now we can easily see the answer is <math>\boxed{\textbf{(B)}~50}</math>.
 +
 
 +
-vockey
 +
 
 +
== Solution 6 ==
 +
We can see that there is 16 squares in the box. 2 triangles make up 1 square. Counting all the triangles on the side, makes 4 squares, and all the squares in the middle make 8 SQUARES. 8/16 is 1/2, therefore 50/100 so,the answer is <math>\boxed{\textbf{(B)}~50}</math>.
 +
 
 +
-AADHYA2012
 +
 
 +
== Video Solution 1 (Detailed Explanation) 🚀⚡📊 ==
 +
Youtube Link ⬇️
 +
 
 +
https://youtu.be/rGEGn7U4uHk
 +
 
 +
~ ChillGuyDoesMath :)
 +
 
 +
== Video Solution by SpreadTheMathLove ==
  
 
https://www.youtube.com/watch?v=jTTcscvcQmI
 
https://www.youtube.com/watch?v=jTTcscvcQmI
 +
 +
==Video Solution (A Clever Explanation You’ll Get Instantly)==
 +
https://youtu.be/VP7g-s8akMY?si=rhpz6FZayHNW--j8&t=7
 +
~hsnacademy
  
 
== Video Solution by Daily Dose of Math ==
 
== Video Solution by Daily Dose of Math ==

Latest revision as of 12:34, 1 February 2025

Problem

The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire \(4\times4\) grid is covered by the star?

[asy] path x = (0,1)--(1,2)--(2,2)--(1,1)--cycle; path y = reflect((0,0),(4,4)) * x; fill(x, gray(0.6)); fill(rotate(90, (2,2)) * x, gray(0.6)); fill(rotate(180, (2,2)) * x, gray(0.6)); fill(rotate(270, (2,2)) * x, gray(0.6)); fill(y, gray(0.8)); fill(rotate(90, (2,2)) * y, gray(0.8)); fill(rotate(180, (2,2)) * y, gray(0.8)); fill(rotate(270, (2,2)) * y, gray(0.8)); draw((1,1)--(3,3)); draw((3,1)--(1,3)); add(grid(4,4)); path w = (1,0)--(2,1)--(3,0); draw(w); draw(rotate(90, (2,2)) * w); draw(rotate(180, (2,2)) * w); draw(rotate(270, (2,2)) * w); [/asy]

$\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 80$

Solution 1

Each of the side triangles has base length $2$ and height $1$, so they all have area $\frac{1}{2} \cdot 2 \cdot 1 = 1$. Each of the four corner squares has side length $1$ and hence area $1$. Then the area of the shaded region is $4^2 - 4 \cdot 1 - 4 \cdot 1 = 8$. Since $\frac{8}{16}=\frac{1}{2}$, our answer is $\frac{1}{2} \cdot 100 = \boxed{\textbf{(B)}~50}$. ~cxsmi

Solution 2

There are $16$ total squares in the diagram and each square can have $2$ triangles. Thus, the total number of triangles in the diagram is $16 \cdot 2 = 32$ triangles. There are $16$ shaded triangles in the diagram (you can just count this up), so the percentage of the shaded triangles is $\dfrac{16}{32} \cdot 100 =$ $\boxed{\textbf{(B)}~50}$. ~Pi_in_da_box

Solution 3

Quite simply, you can "rearrange" the triangles to make a 2 column section in the middle, which equates to $\boxed{\textbf{(B)}~50}$. Probably the most accurate in-test solution. ~shreyan.chethan

Solution 4

Note that we can move one triangle from each of the four cells in the middle to each of the four corners. This will leave every cell in the grid with one triangle each, and each triangle has an area half of the area of each cell. Thus, our final answer is $\frac12=\boxed{\textbf{(B)}~50}.$ ~derekwang2048

Solution 5

The first thing that we can see is that the shaded region is a $2$x$2$ square(in the middle of the figure) combined with $8$ small triangles. Since each small triangle is $1 \cdot \frac{1}{2} = \frac{1}{2}$ unit squares, we get $4 + 8 \cdot \frac{1}{2} = 8$ unit squares. Now we can easily see the answer is $\boxed{\textbf{(B)}~50}$.

-vockey

Solution 6

We can see that there is 16 squares in the box. 2 triangles make up 1 square. Counting all the triangles on the side, makes 4 squares, and all the squares in the middle make 8 SQUARES. 8/16 is 1/2, therefore 50/100 so,the answer is $\boxed{\textbf{(B)}~50}$.

-AADHYA2012

Video Solution 1 (Detailed Explanation) 🚀⚡📊

Youtube Link ⬇️

https://youtu.be/rGEGn7U4uHk

~ ChillGuyDoesMath :)

Video Solution by SpreadTheMathLove

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

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/VP7g-s8akMY?si=rhpz6FZayHNW--j8&t=7 ~hsnacademy

Video Solution by Daily Dose of Math

https://youtu.be/rjd0gigUsd0

~Thesmartgreekmathdude

Video Solution by Thinking Feet

https://youtu.be/PKMpTS6b988

See Also

2025 AMC 8 (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
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=2025_AMC_8_Problems/Problem_1&oldid=241617"