Problem 14 · 1998 AJHSME
Medium
Fractions, Decimals & Percents
percent-overlap
At Annville Junior High School, 30% of the students in the Math Club are in the Science Club, and 80% of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?
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Answer: E — 40 students.
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Hint 1 of 2
The 'in both clubs' group is the bridge between the two clubs — and you only know a real number for the Science Club (15). Count the overlap from the side you can actually compute.
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Hint 2 of 2
Once you have the overlap as a head count, it equals 30% of the Math Club. Going from 'a part and its percent' back to the whole means divide.
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Approach: pin the shared overlap as a number, then divide back to the whole
- The overlap (students in both clubs) is described two ways, but only the Science side gives a number: 80% of the 15 Science Club students = 0.8 × 15 = 12 students are in both.
- That same 12 is 30% of the Math Club. To recover the whole from a known part and its percent, divide: 12 ÷ 0.30 = 40 students.
- Why this transfers: when two groups overlap, the shared piece links them — compute it from whichever side you have a concrete number for, then use 'part ÷ percent = whole' to unlock the other side. Sanity check: 30% of 40 is 12, matching.
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