Problem 14 · 2011 AMC 8
Medium
Fractions, Decimals & Percents
ratio-totals
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?
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Answer: C — 22/45.
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Hint 1 of 2
You can't just average the two fractions of girls — the schools have different sizes. You need actual head counts of girls, then pool everyone together.
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Hint 2 of 2
Both ratios split a school into 9 equal parts (5+4 and 4+5). Conveniently 270 and 180 are both multiples of 9, so each part is a whole number of students.
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Approach: convert each ratio to real counts, then pool
- Colfax has 9 parts: girls are 4 of them, so girls = (4/9)(270) = 120.
- Winthrop has 9 parts: girls are 5 of them, so girls = (5/9)(180) = 100.
- Pool everyone: 120 + 100 = 220 girls out of 270 + 180 = 450 students ⇒ 220/450 = 22/45.
- Common trap: averaging 4/9 and 5/9 gives 1/2 (choice D) — wrong, because the bigger school (Colfax, mostly boys) pulls the combined fraction below 1/2. Always weight by size.
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