Problem 12 · 2016 AMC 8
Medium
Fractions, Decimals & Percents
fraction-to-decimalratio
Jefferson Middle School has the same number of boys and girls. 34 of the girls and 23 of the boys went on a field trip. What fraction of the students on the field trip were girls?
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Answer: B — 9/17.
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Hint 1 of 2
No actual count of students is given — so pick a convenient one. Since the fractions are fourths and thirds, imagine 12 girls and 12 boys (12 makes both fractions come out as whole people), then literally count who goes.
Still stuck? Show hint 2 →
Hint 2 of 2
Out of 12 girls, 3/4 go = 9 girls; out of 12 boys, 2/3 go = 8 boys. Now the answer is just (girls on trip) over (everyone on trip).
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Approach: pick a friendly common size and count actual heads
- Equal numbers of boys and girls, so let there be 12 of each (12 is divisible by both 4 and 3, so no fractional people).
- Girls on the trip: 3/4 of 12 = 9. Boys on the trip: 2/3 of 12 = 8. Trip total = 9 + 8 = 17.
- Fraction that are girls = 9/17 = 9/17.
- Sanity check: more than half the trip is girls (9 of 17), which fits — the girls' fraction 3/4 beats the boys' 2/3, so girls should be the majority. That alone rules out the 1/2 and below choices.
- Why this transfers: when a problem gives only fractions/percents and no totals, invent a total that clears every denominator (here, 12) — counting real objects beats juggling abstract fractions.
Another way — compare via a common denominator (no chosen number):
- With equal group sizes, line up the trip fractions over a common denominator: 3/4 = 9/12 and 2/3 = 8/12.
- So girls : boys on the trip = 9 : 8, and the fraction of girls is 9 / (9 + 8) = 9/17.
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