Question

A box contains chips, each of which is red, white, or blue. The number of blue chips is at least half the number of white chips and at most one-third the number of red chips. The number which are white or blue is at least $55.$ The minimum number of red chips is:

$24$

$33$

$45$

$54$

$57$

Answer: E

Concepts:inequality optimization bounding to limit cases

Difficulty rating: 1880

Small Hint:

Let the counts be $r,w,b$ and translate every condition into an inequality

Big Hint:

From $w\le2b$ and $w+b\ge55$ , find the least possible integer $b$

Solution:

Let the counts be $r,w,b.$ The conditions give $\begin{gathered} w\le2b,\\ r\ge3b,\\ w+b\ge55. \end{gathered}$ Hence $3b\ge w+b\ge55,$ so $b\ge19$ and $r\ge57.$ Equality is possible with $(w,b,r)=(36,19,57),$ so the minimum is $57.$

Therefore, the correct answer is E.

1971 AMC 12 Problem 27

Answer: E

Difficulty rating: 1880

Small Hint:

Big Hint:

Solution:

Problem 27 in Other Years