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?

K:P=2:5
P+75 it is 4:15
at first, P=5 units
so we have
2:5 and 4:15
since the K units didn't change we conver them to the same
2:5 times 2:2=4:10
so from 4:10 to 4:15 is a change of 75 in the right side
therefor 75=5 units so 75/5=15/1= 1 unit
so at first K=4 units=4 times 1 unit=4 times 15=60
P=10 units so 10 times 15=150

at first
Karen=60
Patricia=150

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The decimal expansion would be .27 repeated