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 "2011 AMC 8 Problems/Problem 14"

(→‎Solution)
(→‎Solution)
Line 10: Line 10:
  
 
At Colfax Middle School, there are <math>\frac49 \times 270 = 120</math> girls. At Winthrop Middle School, there are <math>\frac59 \times 180 = 100</math> girls. The ratio of girls to the total number of students is <math>\frac{120+100}{270+180} = \frac{220}{450} = \boxed{\textbf{(C)}\ \frac{22}{45}}</math>
 
At Colfax Middle School, there are <math>\frac49 \times 270 = 120</math> girls. At Winthrop Middle School, there are <math>\frac59 \times 180 = 100</math> girls. The ratio of girls to the total number of students is <math>\frac{120+100}{270+180} = \frac{220}{450} = \boxed{\textbf{(C)}\ \frac{22}{45}}</math>
 +
 +
==Solution 2 (Guess and Check)==
 +
Since we know there are <math>270</math> kids at Colfax Middle School and we know the ratio of boys to girls is <math>5:4</math>, we can use guess and check to find the number of girls at the dance (and vice versa). <math>5:4=150:120</math> and <math>4:5=80:100</math>. Now that the ratios sum to <math>270</math> and <math>180</math>, we know that in Colfax there are <math>120</math> girls and in Winthrop, there are <math>100</math> girls. <math>120+100=220</math> total girls and <math>270+180=450</math> total people. <math>\frac{220}{450}=\frac{22}{45}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2011|num-b=13|num-a=15}}
 
{{AMC8 box|year=2011|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 11:42, 14 January 2023

Problem

There are $270$ students at Colfax Middle School, where the ratio of boys to girls is $5 : 4$. There are $180$ students at Winthrop Middle School, where the ratio of boys to girls is $4 : 5$. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?

$\textbf{(A) } \dfrac7{18} \qquad\textbf{(B) } \dfrac7{15} \qquad\textbf{(C) } \dfrac{22}{45} \qquad\textbf{(D) } \dfrac12 \qquad\textbf{(E) } \dfrac{23}{45}$

Video Solution

https://youtu.be/rQUwNC0gqdg?t=697

Solution

At Colfax Middle School, there are $\frac49 \times 270 = 120$ girls. At Winthrop Middle School, there are $\frac59 \times 180 = 100$ girls. The ratio of girls to the total number of students is $\frac{120+100}{270+180} = \frac{220}{450} = \boxed{\textbf{(C)}\ \frac{22}{45}}$

Solution 2 (Guess and Check)

Since we know there are $270$ kids at Colfax Middle School and we know the ratio of boys to girls is $5:4$, we can use guess and check to find the number of girls at the dance (and vice versa). $5:4=150:120$ and $4:5=80:100$. Now that the ratios sum to $270$ and $180$, we know that in Colfax there are $120$ girls and in Winthrop, there are $100$ girls. $120+100=220$ total girls and $270+180=450$ total people. $\frac{220}{450}=\frac{22}{45}$

See Also

2011 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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=2011_AMC_8_Problems/Problem_14&oldid=186594"