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A glass contains vinegar and water in the ratio 1:3. Another glass twice the capacity of the first has vinegar and water in the ratio 1:4. If the contents of both glasses were mixed together in another container what is then the ratio of vinegar to water?

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    $\begingroup$ To delay using fractions, let the small glass have capacity $20$. Then the big glass has capacity $40$. $\endgroup$ – André Nicolas Feb 24 '15 at 19:58
  • $\begingroup$ $[\color{red}{1}\cdot\frac13+\color{red}{1}\cdot\frac23]:[\color{blue}{3}\cdot \frac13+\color{blue}{4}\cdot\frac23]$, where the vinegar is red and the water is blue. $\endgroup$ – barak manos Feb 24 '15 at 19:59
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For vinegar and water respectively:

Let first glass have volume 5 = 5/4 + 15/4

Let second glass have volume 10 = 10/5 + 40/5

( 5/4 + 10/5) : (15/4 + 40/5) = 65 : 235 = 13 : 47

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Some hints.

  1. Call the capacity of the first glass $x$. What is the capacity of the second one, that's twice as big?

For the first glass, the ratio of vinegar to water is $1:3$. This means the amount of vinegar is $x/4$ and the amount of water is $3x/4$.

  1. What would be the amount of vinegar in the second container? (Remember, it's twice as big.)
  2. What is the amount of water in the second container?
  3. What is the total amount of vinegar after the containers are mixed?
  4. What is the total amount of water after the containers are mixed?
  5. What, then, is the ratio after the mixing?
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