First, let's find out how much lip balm Kate has in total after she makes more. She starts with \( \frac{3}{4} \) ounce and makes \( 1 \frac{1}{2} \) ounces more.
Converting \( 1 \frac{1}{2} \) to an improper fraction:
\[ 1 \frac{1}{2} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]
Now, we combine this amount with what she already has:
\[ \frac{3}{4} + \frac{3}{2} \]
To add these fractions, we need a common denominator. The common denominator between 4 and 2 is 4.
Convert \( \frac{3}{2} \) into a fraction with a denominator of 4:
\[ \frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4} \]
Now, add \( \frac{3}{4} \) and \( \frac{6}{4} \):
\[ \frac{3}{4} + \frac{6}{4} = \frac{3 + 6}{4} = \frac{9}{4} \]
Now Kate has \( \frac{9}{4} \) ounces of lip balm.
Next, we need to determine how many \( \frac{1}{2} \) ounce portions she can make. We divide \( \frac{9}{4} \) by \( \frac{1}{2} \):
\[ \frac{9}{4} \div \frac{1}{2} = \frac{9}{4} \times 2 = \frac{9 \times 2}{4 \times 1} = \frac{18}{4} \]
Simplifying \( \frac{18}{4} \):
\[ \frac{18}{4} = \frac{9}{2} \]
Now, \( \frac{9}{2} \) is equal to \( 4 \frac{1}{2} \), which means Kate can fill 4 full containers and has \( \frac{1}{2} \) ounces left over.
Therefore, Kate will use 4 containers, and she will have 1/2 ounce remaining.