Difference between revisions of "2011 AIME I Problems/Problem 1"

Revision as of 20:44, 28 March 2011

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 $k\%$ acid. From jar C, $\frac{m}{n}$ 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 $m$ and $n$ are relatively prime positive integers, find $k + m + n$ .

Solution 1

There are $\frac{45}{100}(4)=\frac{9}{5}$ L of acid in Jar A. There are $\frac{48}{100}(5)=\frac{12}{5}$ L of acid in Jar B. And there are $\frac{k}{100}$ L of acid in Jar C. After transfering the solutions from jar C, there will be
$4+\frac{m}{n}$ L of solution in Jar A and $\frac{9}{5}+\frac{k}{100}\cdot\frac{m}{n}$ L of acid in Jar A.

$6-\frac{m}{n}$ L of solution in Jar B and $\frac{12}{5}+\frac{k}{100}\cdot \left(1-\frac{m}{n}\right)=\frac{12}{5}+\frac{k}{100}-\frac{mk}{100n}\right$ (Error compiling LaTeX. Unknown error_msg) 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.
$\frac{18}{5}+\frac{km}{50n}=4+\frac{m}{n}$ $\frac{24}{5}-\frac{km}{50n}+\frac{k}{50}=6-\frac{m}{n}$ Add the equations to get $\frac{42}{5}+\frac{k}{50}=10$ Solving gives $k=80$ .
If we substitute back in the original equation we get $\frac{m}{n}=\frac{2}{3}$ so $3m=2n$ . Since $m$ and $n$ are relatively prime, $m=2$ and $n=3$ . Thus $k+m+n=80+2+3=\boxed{085}$ .

Solution 2

One might cleverly change the content of both Jars.

Since the end result of both Jars are $50\%$ acid, we can turn Jar A into a 1 gallon liquid with $50\%-4(5\%) = 30\%$ acid

and Jar B into 1 gallon liquid with $50\%-5(2\%) =40\%$ 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 $\dfrac{2}{3}$ of Jar C will be pour into Jar A.

Thus, $m=2$ and $n=3$ .

$\dfrac{30\% + \frac{2}{3} \cdot k\%}{\frac{5}{3}} = 50\%$

Solving for $k$ yields $k=80$

So the answer is $80+2+3 = \boxed{85}$

Solution 3

One may first combine all three jars in to a single container. That container will have $10$ liters of liquid, and it should be $50\%$ acidic. Thus there must be $5$ liters of acid.

Jug A contained $45\% \cdot 4L$ , or $1.8L$ of acid, and jug B $48\% \cdot 5L$ or $2.4L$ . Solving for the amount of acid in jug C, $k = (5 - 2.4 - 1.8) = .8$ , or $80\%$ .

Once one knows that the jug C is $80\%$ acid, use solution 1 to figure out m and n.

@@ Line 33: / Line 33: @@
 One may first combine all three jars in to a single container.  That container will have <math>10</math> liters of liquid, and it should be <math>50\%</math> acidic.  Thus there must be <math>5</math> liters of acid.
-Jug A contained <math>45\% \cdot 4L</math>, or <math>1.8L</math> of acid, and jug B <math>48\% \cdot 5L</math> or <math>2.4L</math>.  Solving for the amount of acid in jug Cm <math>k = (5 - 2.4 - 1.8) = .8</math>, or <math>80\%</math>.
+Jug A contained <math>45\% \cdot 4L</math>, or <math>1.8L</math> of acid, and jug B <math>48\% \cdot 5L</math> or <math>2.4L</math>.  Solving for the amount of acid in jug C, <math>k = (5 - 2.4 - 1.8) = .8</math>, or <math>80\%</math>.
 Once one knows that the jug C is <math>80\%</math> acid, use solution 1 to figure out m and n.

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