Roger bought some pencils and erasers at the stationery store. If he
bought more pencils than erasers, and the total number of the pencils
and erasers he bought is between 12 and 20 (inclusive), which of the
following statements must be true?Select all that apply.
a. Roger bought no fewer than 7 pencils.
b. Roger bought no more than 12 pencils.
c. Roger bought no fewer than 6 erasers.
d. Roger bought no more than 9 erasers.
To me, all of them are correct. Here is why:
With p = pencil and e = eraser.
a. p ≥ 7 ⇒ (p,e): (7,5), (8,4), (9,3), (10,2), (11,1), ect.
b. p ≤ 12 ⇒ (p,e): (12,8), (11,9)
c. e ≥ 6 ⇒ (p,e): (7,6), (8,7), (9,8), (10,9)
d. e ≤ 9 ⇒ (p,e): (10,9), (9,8), (8,7), (7,6), (7,5), ect.
However the correct answers according to the book are (a) and (d). Please help me understand why, thank you!
Best Answer
For a, if Roger had only bought $6$ pencils, he can have bought at most $5$ erasers, and he can no way make $12$ items, so he must have bought at least $7$ pencils.
Similarly for d, if Roger bought more than $9$ erasers, he must have bought at least $10$ pencils, and now he has too many items.
So a and d are absolutely true (this is the Pigeonhole principle btw).
b and c both have exceptions ($(11,0)$ is an exception to b, and $(11,10)$ is an exception to c), although they can possibly be true, as your answer suggests.