Difference between revisions of "2011 AIME I Problems/Problem 1"
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== Problem 1 == | == Problem 1 == | ||
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 <math>k\%</math> acid. From jar C, <math>\frac{m}{n}</math> 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 <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>k + m + n</math>. | 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 <math>k\%</math> acid. From jar C, <math>\frac{m}{n}</math> 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 <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>k + m + n</math>. |
Revision as of 15:44, 9 August 2018
Problem
Problem 1
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 acid. From jar C,
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
and
are relatively prime positive integers, find
.
Solution 1
There are L of acid in Jar A. There are
L of acid in Jar B. And there are
L of acid in Jar C. After transfering the solutions from jar C, there will be
L of solution in Jar A and
L of acid in Jar A.
L of solution in Jar B and
of acid in Jar B.
Since the solutions are 50% acid, we can multiply the amount of acid for each jar by 2, then equate them to the amount of solution.
Add the equations to get
Solving gives
.
If we substitute back in the original equation we get so
. Since
and
are relatively prime,
and
. Thus
.
Solution 2
One might cleverly change the content of both Jars.
Since the end result of both Jars are acid, we can turn Jar A into a 1 gallon liquid with
acid
and Jar B into 1 gallon liquid with acid.
Now, since Jar A and Jar B contain the same amount of liquid, twice as much liquid will be pour into Jar A than Jar B, so of Jar C will be pour into Jar A.
Thus, and
.
Solving for yields
So the answer is
Solution 3
One may first combine all three jars in to a single container. That container will have liters of liquid, and it should be
acidic. Thus there must be
liters of acid.
Jug A contained , or
of acid, and jug B
or
. Solving for the amount of acid in jug C,
, or
.
Once one knows that the jug C is acid, use solution 1 to figure out m and n for
.
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.