At a florist's shop, \( 60 \% \) of the flowers are roses and the rest are lilies and daisies. The number of daisies is \( \frac{2}{3} \) of the number of lilies. There are 270 roses. How many lilies are there?

To solve for the number of lilies, we can start by determining the total number of flowers in the shop. Since \( 60\% \) of the flowers are roses and there are 270 roses, we can set up the equation: \[ 0.60 \times \text{Total Flowers} = 270 \] To find the total number of flowers, we divide both sides of the equation by \( 0.60 \): \[ \text{Total Flowers} = \frac{270}{0.60} = 450 \] This means there are 450 flowers in total. Since \( 60\% \) of the flowers are roses, the remaining \( 40\% \) are lilies and daisies. Therefore, we can calculate the number of lilies and daisies: \[ 0.40 \times 450 = 180 \] Now, we know that there are 180 flowers that are either lilies or daisies. We are also told that the number of daisies is \( \frac{2}{3} \) of the number of lilies. Let \( L \) be the number of lilies, then the number of daisies can be expressed as \( \frac{2}{3}L \). Now we can set up the equation: \[ L + \frac{2}{3}L = 180 \] Combining the terms on the left side, we have: \[ \frac{5}{3}L = 180 \] To solve for \( L \), we multiply both sides by \( \frac{3}{5} \): \[ L = 180 \times \frac{3}{5} = 108 \] Thus, the number of lilies is: \[ \boxed{108} \]