let

r = the age of Rosa

e = the age of Elena

f = the age of Felicia

r + e + f = 50

r < e < f

f = 10 + r plug into r + e + f = 50

r + e + 10 + r = 50

2r + e = 40

Five years ago, their ages were prime numbers

5 years ago the sum of their ages was

2(r - 5) + e - 5 = 40 - 3*5

2(r-5) + (e - 5) = 25

r - 5 is prime

e - 5 is prime

we will use trail and error:

For r - 5 = 2 we have r = 7

e - 5 = 25 - 2*2

e - 5 = 21 is not a prime

For r-5 = 3 we have r = 8

e - 5 = 25 - 2*3

e - 5 = 19 is a prime

e = 24

f = 10 + r

f = 10 + 8

f = 18

but 18 < 24 thus this is not the solution

For r-5 = 5 we have r = 10

e - 5 = 25 - 10

e - 5 = 15 is not a prime

For r-5 = 7 we have r = 12

e - 5 = 25 - 2*7

e - 5 = 11

e = 16

f = 10 + r

f = 10 + 12

f = 22

indeed 12 + 16 + 22 = 50

thus one solution is Rosa is 12, Elena is 16 and Felicia is 22 years of age.

For r - 5 = 11 we have r = 16

e - 5 = 25 - 2*11

e - 5 = 3

e = 8 < r but from the conditions r > e thus this is not a solution and further on by increasing the value for e, the value for r is decreasing follows we have checked all the cases and finally we can say that the only solution is:

Rosa is 12, Elena is 16 and Felicia is 22 years of age.