This dilution problem uses the equation

##M_aV_a = M_bV_b##

##M_a## = 6.77M – the initial (concentration)
##V_a## = 15.00 mL – the initial volume
##M_b## = 1.50 M – the desired molarity (concentration)
##V_b## = (15.00 + x mL) – the volume of the desired solution

(6.77 M) (15.00 mL) = (1.50 M)(15.00 mL + x )
101.55 M mL= 22.5 M mL + 1.50x M
101.55 M mL – 22.5 M mL = 1.50x M
79.05 M mL = 1.50 M
79.05 M mL / 1.50 M = x
52.7 mL = x

59.7 mL needs to be added to the original 15.00 mL solution in order to dilute it from 6.77 M to 1.50 M.

I hope this was helpful.
SMARTERTEACHER