Carolyn, Julie and Roberta share $\$77$ in a ratio of 4:2:1,  respectively.  How many dollars did Carolyn receive?
The girls essentially divided the $\$77$ into $4 + 2 + 1 = 7$ equal portions and gave 4 portions to Carolyn, 2 to Julie, and 1 to Roberta.  So each portion is worth $\frac{\$77}{7} = \$ 11$.  Hence Carolyn received $11\cdot 4 = \boxed{44}$ dollars.