The ratio of the number of beads Karen had to the number of beads Patricia had was 2:5. After Patricia bought another 75 beads, the ratio became 4:15. How many beads did each girl have at first?

We can bring the ratios to a similar form, because 2:5 is the same as 4:10 (we can mupltiply rations by the same number)

so one ratio is 4:10 and the other is 4:15 and the difference is is 75 beads

in this ratio, we can see each number corresponding to a "unit" of beads: that means that what changed is that Patricia now has 5 more such groups (15-10=5), and since she has 75 beads more each unit must be 75/5=15.

so afterwards (and before) Karen had 4 such units, that is 60, and Patricia before had 10 such units, that is 150.