Difference between revisions of "2011 AMC 8 Problems/Problem 14"
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+ | ==Problem== | ||
There are <math>270</math> students at Colfax Middle School, where the ratio of boys to girls is <math>5 : 4</math>. There are <math>180</math> students at Winthrop Middle School, where the ratio of boys to girls is <math>4 : 5</math>. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls? | There are <math>270</math> students at Colfax Middle School, where the ratio of boys to girls is <math>5 : 4</math>. There are <math>180</math> students at Winthrop Middle School, where the ratio of boys to girls is <math>4 : 5</math>. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls? | ||
− | <math> \ | + | <math> \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} </math> |
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/rQUwNC0gqdg?t=697 | ||
+ | |||
+ | ==Solution== | ||
+ | |||
+ | 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}=\boxed{\textbf{(C)} \frac{22}{45}}</math> | ||
+ | |||
+ | ==Video Solution 1 by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=mYn6tNxrWBU | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2011|num-b=13|num-a=15}} | ||
+ | {{MAA Notice}} |
Latest revision as of 15:10, 17 December 2023
Contents
Problem
There are students at Colfax Middle School, where the ratio of boys to girls is . There are students at Winthrop Middle School, where the ratio of boys to girls is . The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?
Video Solution
https://youtu.be/rQUwNC0gqdg?t=697
Solution
At Colfax Middle School, there are girls. At Winthrop Middle School, there are girls. The ratio of girls to the total number of students is
Solution 2 (Guess and Check)
Since we know there are kids at Colfax Middle School and we know the ratio of boys to girls is , we can use guess and check to find the number of girls at the dance (and vice versa). and . Now that the ratios sum to and , we know that in Colfax there are girls and in Winthrop, there are girls. total girls and total people.
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=mYn6tNxrWBU
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 |
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