1

Part of 2011 AIME Problems

Problems(2)

Large beverage [2011.II.1]

Source:

Gary purchased a large beverage, but drank only

m/n

of this beverage, where

m

n

are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only

\frac{2}{9}

as much beverage. Find

m+n

number theoryrelatively primeAMC

Jars of Acid Solution

Source:

Jar A contains four liters of a solution that is

45\%

acid. Jar B contains five liters of a solution that is

48\%

acid. Jar C contains one liter of a solution that is

k\%

acid. From jar C,

\tfrac{m}{n}

liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end, both jar A and jar B contain solutions that are

50\%

acid. Given that

m

n

are relatively prime positive integers, find

k+m+n

rationumber theoryrelatively primeAMC