Multiple choice

Two identical jars are filled with alcohol solutions. The ratio of the volume of alcohol to the volume of water is $p : 1$ in one jar and $q : 1$ in the other jar. If the entire contents of the two jars are mixed/ together, then what is the ratio of the volume of alcohol to the volume of water in the mixture?

  1. $\displaystyle\frac{p+q+2pq}{p+q+2}$
  2. $\displaystyle\frac{p+q}{p+q+2}$
  3. $\displaystyle\frac{p+q+1}{p+q+2}$

  4. None of these
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A Correct answer
Explanation

Jar 1: Alcohol = p/(p+1), Water = 1/(p+1). Jar 2: Alcohol = q/(q+1), Water = 1/(q+1). Total Alcohol = p/(p+1) + q/(q+1) = (pq+p+pq+q)/((p+1)(q+1)). Total Water = 1/(p+1) + 1/(q+1) = (q+1+p+1)/((p+1)(q+1)). Ratio = (2pq+p+q)/(p+q+2).

AI explanation

Let the volume of each jar be V. In the first jar, the volume of alcohol is V multiplied by p/(p+1) and the volume of water is V multiplied by 1/(p+1). In the second jar, the volume of alcohol is V multiplied by q/(q+1) and the volume of water is V multiplied by 1/(q+1). When mixed, the total alcohol is V multiplied by the quantity p/(p+1) plus q/(q+1), which simplifies to V multiplied by (2pq plus p plus q) divided by (pq plus p plus q plus 1). The total water is V multiplied by the quantity 1/(p+1) plus 1/(q+1), which simplifies to V multiplied by (p plus q plus 2) divided by (pq plus p plus q plus 1). The final ratio of the volume of alcohol to the volume of water is (p plus q plus 2pq) divided by (p plus q plus 2).

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