Strawberry-Banana Daiquiris

1 (3 ounce) can frozen daiquiri mix
1/2 cup plus 1/8 cup light rum
10 ounces strawberries, partially thawed
1 fully ripe banana

Put all ingredients into a blender and add ice cubes to within 1 inch of the top of ingredients. Blend. Add more cubes and blend until thick and smooth.