Emily fills a six-ounce cup with three ounces of tea and another six-ounce cup with three ounces of milk. After pouring half of the tea from the first cup into the second, she stirs the mixture well and then returns half of the liquid from the second cup back to the first. What fraction of the liquid in the first cup is now milk?

Step 1: Initial Setup

- Cup 1: 6-ounce cup with $3$ ounces of tea.


- Cup 2: 6-ounce cup with $3$ ounces of milk.

Step 2: Pour Half of the Tea from Cup 1 into Cup 2

Half of the tea in Cup 1 is $\frac{3}{2} = 1.5$ ounces.


- Cup 1: $3 - 1.5 = 1.5$ ounces of tea.


- Cup 2: $0 + 1.5 = 1.5$ ounces of tea and $3$ ounces of milk.

Step 3: Stir the Mixture in Cup 2 Well

The mixture in Cup 2 is uniform. Total liquid in Cup 2 is $1.5 + 3 = 4.5$ ounces.


The ratio in Cup 2 is:


- Tea: $\frac{1.5}{4.5} = \frac{1}{3}$


- Milk: $\frac{3}{4.5} = \frac{2}{3}$

Step 4: Pour Half of the Liquid from Cup 2 Back into Cup 1

Half of the liquid in Cup 2 is $\frac{4.5}{2} = 2.25$ ounces.


Since the mixture is uniform, the poured liquid has the same ratio:


- Tea: $\frac{1}{3} \times 2.25 = 0.75$ ounces
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