Question

Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M - m.$

Answer

Answer: 298

Concepts:inclusion-exclusion percentage bounding to limit cases

Difficulty rating: 2110

Solution:

Let $s$ and $f$ be the numbers of students studying Spanish and French. Since every student studies at least one language, the number studying both is $s + f - 2001.$ The bounds $1600.8 \lt s \lt 1700.85$ force $1601 \le s \le 1700,$ and $600.3 \lt f \lt 800.4$ force $601 \le f \le 800.$

The overlap is smallest when $s + f$ is smallest, giving $m = 1601 + 601 - 2001 = 201,$ and largest when $s + f$ is largest, giving $M = 1700 + 800 - 2001 = 499.$ Both extremes are achievable, so $M - m = 499 - 201 = 298.$

2001 AIME II Problem 2

Answer: 298

Difficulty rating: 2110

Solution:

Problem 2 in Other Years