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

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There are <math>\frac{45}{100}(4)=\frac{9}{5}</math> L of acid in Jar A. There are <math>\frac{48}{100}(5)=\frac{12}{5}</math> L of acid in Jar B. And there are <math>\frac{k}{100}</math> L of acid in Jar C. After transfering the solutions from jar C, there will be | There are <math>\frac{45}{100}(4)=\frac{9}{5}</math> L of acid in Jar A. There are <math>\frac{48}{100}(5)=\frac{12}{5}</math> L of acid in Jar B. And there are <math>\frac{k}{100}</math> L of acid in Jar C. After transfering the solutions from jar C, there will be | ||

<br> <math>4+\frac{m}{n}</math> L of solution in Jar A and <math>\frac{9}{5}+\frac{k}{100}\cdot\frac{m}{n}</math> L of acid in Jar A.<br> | <br> <math>4+\frac{m}{n}</math> L of solution in Jar A and <math>\frac{9}{5}+\frac{k}{100}\cdot\frac{m}{n}</math> L of acid in Jar A.<br> | ||

− | <br> <math>6-\frac{m}{n}</math> L of solution in Jar B and <math>\frac{12}{5}+\frac{k}{100}\cdot \left(1-\frac{m}{n}\right)</math> of acid in Jar B. | + | <br> <math>6-\frac{m}{n}</math> L of solution in Jar B and <math>\frac{12}{5}+\frac{k}{100}\cdot \left(1-\frac{m}{n}\right)=\frac{12}{5}+\frac{k}{100}-\frac{mk}{100n}\right</math> 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. | + | <br>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.<br> |

+ | <cmath>\frac{18}{5}+\frac{km}{50n}=4+\frac{m}{n}</cmath> | ||

+ | <cmath>\frac{24}{5}-\frac{km}{50n}+\frac{k}{50}=6-\frac{m}{n}</cmath> | ||

+ | Add the equations to get | ||

+ | <cmath>\frac{42}{5}+\frac{k}{50}=10</cmath> Solving gives <math>k=80</math>. | ||

+ | <br>If we substitute back in the original equation we get <math>\frac{m}{n}=\frac{2}{3}</math> so <math>3m=2n</math>. Since <math>m</math> and <math>n</math> are relatively prime, <math>m=2</math> and <math>n=3</math>. Thus <math>k+m+n=80+2+3=\boxed{085}</math>. |

## Revision as of 13:15, 19 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 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

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 $\frac{12}{5}+\frac{k}{100}\cdot \left(1-\frac{m}{n}\right)=\frac{12}{5}+\frac{k}{100}-\frac{mk}{100n}\right$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) 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 .