To find out how much milk you should use when increasing the recipe by a factor of \(2 \frac{1}{2}\), we first need to convert both the quantities to improper fractions.
- Convert \(2 \frac{2}{3}\) cups of milk:
\[ 2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \]
- Convert \(2 \frac{1}{2}\) to an improper fraction:
\[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \]
- Multiply the amount of milk by the factor:
\[ \text{Amount of milk} = \frac{8}{3} \times \frac{5}{2} = \frac{8 \times 5}{3 \times 2} = \frac{40}{6} \]
- Simplify \(\frac{40}{6}\):
\[ \frac{40}{6} = \frac{20}{3} \]
- Convert back to a mixed number:
\[ \frac{20}{3} = 6 \frac{2}{3} \]
So, you should use 6 \(\frac{2}{3}\) cups of milk.