1979 AMC 12 Problem 15

Question

Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being

p:1

in one jar and

q:1

in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is

\frac{p+q}{2}

\frac{p^2+q^2}{p+q}

\frac{2pq}{p+q}

\frac{2(p^2+pq+q^2)}{3(p+q)}

\frac{p+q+2pq}{p+q+2}

Answer: E

Concepts:mixture ratio and proportion fraction

Difficulty rating: 1860

Small hint:

Assign the same total volume to each jar and convert each ratio to alcohol and water fractions

Big hint:

Add the two alcohol amounts and the two water amounts before forming their ratio

Solution:

For equal total volumes, the alcohol fractions are

\frac{p}{p+1}

and

\frac{q}{q+1},

while the water fractions are

\frac{1}{p+1}

and

\frac{1}{q+1}.

Their ratio is

\frac{\frac p{p+1}+\frac q{q+1}} {\frac1{p+1}+\frac1{q+1}} =\frac{p+q+2pq}{p+q+2}.

Therefore, the correct answer is E.