anonymous
  • anonymous
Liquid X is composed of 30% alcohol and 70% water, while Liquid Y is composed of 18% alcohol and 82% water. The two liquids are combined to form Mixture Z, which is composed of 21% alcohol and 79% water. What is the ratio in Mixture Z of Liquid X to Liquid Y?
Mathematics
katieb
  • katieb
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ybarrap
  • ybarrap
Let \(a\) be the amount of solution \(X\) in \(Z\) and \(b\) be the amount of solution \(Y\) in \(Z\). \(A\) is Alcohol and \(W\) is water. Then $$ X=0.3A+.7W\\ Y=.18A+.82W\\ Z=.2A+.79W=aX+bY\\ =a(0.3A+.7W)+b(.18A+.82W)\\ =A(.3a+.18b)+W(.7a+.82b) $$ This means that $$ Z=.2A+.79W=A(.3a+.18b)+W(.7a+.82b)\\ \implies \\ .2=(.3a+.18b)\\ .79=(.7a+.82b) $$ Two equations. Two unknowns. Solvable. When you get a and b you can then determine the ratio of X to Y in Z: $$ \frac{a}{b} $$ Does this make sense?
anonymous
  • anonymous
what is the Y?
anonymous
  • anonymous
oh nvm i see it

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ybarrap
  • ybarrap
*Above \(Z=.2\color{red}1A+.79W\) -- notice I missed the "1"
anonymous
  • anonymous
hmmmm I'm going to have to look at this several times ....
ybarrap
  • ybarrap
OK
anonymous
  • anonymous
Hey this is a hard problem right? It's not just me right?

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