2011 AMC 8 Problem 14

Attempt Problem 14 of the 2011 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

There are 270270 students at Colfax Middle School, where the ratio of boys to girls is 5:4.5 : 4. There are 180180 students at Winthrop Middle School, where the ratio of boys to girls is 4:5.4 : 5. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?
718\dfrac{7}{18}
715\dfrac{7}{15}
2245\dfrac{22}{45}
12\dfrac{1}{2}
2345\dfrac{23}{45}
Answer: C
Concepts:ratio and proportionfraction
Small hint:
Then divide the total number of girls by 270+180270+180.
Big hint:
Find the number of girls at each school from its ratio.
Solution:
The total number of girls is 49⋅270+59⋅180=\dfrac{4}{9} \cdot 270 + \dfrac{5}{9} \cdot 180 = 120+100=220. 120 + 100 = 220.
There are 270+180=450270 + 180 = 450 students total, so the fraction of girls is 220450=2245.\dfrac{220}{450} = \dfrac{22}{45}.
Thus, C is the correct answer.
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