2006 AMC 12B Problem 11

Question

Joe and JoAnn each bought

12

ounces of coffee in a

16

-ounce cup. Joe drank

2

ounces of his coffee and then added

2

ounces of cream. JoAnn added

2

ounces of cream, stirred the coffee well, and then drank

2

ounces. What is the resulting ratio of the amount of cream in Joe’s coffee to that in JoAnn’s coffee?

\dfrac{6}{7}

\dfrac{13}{14}

1

\dfrac{14}{13}

\dfrac{7}{6}

Joe adds cream after drinking, so all

2

ounces of his cream remain

JoAnn drinks from a well-mixed

14

-ounce cup, removing cream in proportion

Joe adds the cream last, so his cup holds all

2

ounces of cream.

JoAnn’s cup has

14

ounces of mixture containing

2

ounces of cream. Drinking

2

ounces removes a fraction

\tfrac{2}{14}

of everything, leaving

2 \cdot \frac{12}{14} = \frac{12}{7}

ounces of cream.

The ratio is

\frac{2}{\,\frac{12}{7}\,} = \frac{14}{12} = \frac{7}{6}.

Thus, the correct answer is E.

Problem 11 in other years