2022/23 AHSMC Part 1 Problem 1

Problem 1 of 16

Consider two identical beakers, beaker $A$ and beaker $B$ . Beaker $B$ is full of liquid and beaker $A$ is empty. A scientist pours one third of the liquid from beaker $B$ into beaker $A$ . She then pours one fourth of the liquid in beaker $A$ back into beaker $B$ . Finally, she pours half of the liquid in beaker $B$ back into beaker $A$ . After this process, what fraction of the liquid is in beaker $A$ ?

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Solution

If $\ell$ denotes the amount of liquid in $B$ , then after the first step, the amount of liquid in $A$ is $\dfrac{1}{3}\ell$ while in $B$ is $\bigl(1 - \dfrac{1}{3}\bigr)\ell = \dfrac{2}{3}\ell$ . After the second step, the amount of liquid in $A$ is $\bigl(\dfrac{1}{3} - \dfrac{1}{12}\bigr)\ell = \dfrac{1}{4}\ell$ and in $B$ is $\bigl(\dfrac{2}{3} + \dfrac{1}{12}\bigr)\ell = \dfrac{3}{4}\ell$ . Finally, after the third step, the amount of liquid in $A$ will be $\bigl(\dfrac{1}{4} + \dfrac{3}{8}\bigr)\ell = \dfrac{5}{8}\ell$ . The answer is (A).

University of Alberta

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