Let

r = the age of Rosa

e = the age of Elena

f = the age of Felicia

r < e < f

Felicia is the oldest: 10 years older than Rosa means:

f = r + 10

The sum of all ages of the three Perez sisters is 50:

r + e + f = 50

plug f = r + 10

r + e + r + 10 = 50

2r + e + 10 = 50

e = 40 - 2r

Now we can write:

The age of Rosa is: r

The age of Elena is: 40 - 2r

The are of Felicia is: r + 10

since the ages need to be positive numbers we have: 40 - 2r >0

r<20

Five years ago, their ages were prime numbers, that is:

r - 5 is prime

40 - 2r - 5 = 35 - 2r is prime

r + 10 - 5 = r + 5 is prime

Also

r - 5 must be positive so we have r - 5 >0 that is r > 5

35 - 2r must be a positive so we have 35 - 2r > 0 that is 17.5 > r

r + 5 >= 35 - 2r (since Felicia is the oldest)

3r >= 30

r >= 10

We will solve using the trail and error method:

For r = 10 we have

10 - 5 = 5 is prime

35 - 2*10 = 15 is not a prime

For r = 12 we have

12 - 5 = 7 is prime

35 - 2*12 = 11 is prime

12 + 5 = 17 is a prime

Thus one solution is:

Rosa is 12 years old.

Elena is 40 - 2*12 = 16 years old.

Felicia is 12 + 10 = 22 years old.

Lets check if this solution if unique.

For r = 16 we have

16 - 5 = 11 is prime

35 - 2*16 = 3 is prime

16 + 5 = 21 is not a prime

For r = 17 we have

17 - 5 = 12 is not a prime.

This means we tried all the possibilities for r. Thus the unique solution of the problem is:

Rosa is 12 years old.

Elena is 16 years old.

Felicia is 22 years old.

Indeed 12 + 16 + 22 = 60.

Let

r = the age of Rosa

e = the age of Elena

f = the age of Felicia

r < e < f

Felicia is the oldest: 10 years older than Rosa means:

f = r + 10

The sum of all ages of the three Perez sisters is 50:

r + e + f = 50

plug f = r + 10

r + e + r + 10 = 50

2r + e + 10 = 50

e = 40 - 2r

Now we can write:

The age of Rosa is: r

The age of Elena is: 40 - 2r

The are of Felicia is: r + 10

since the ages need to be positive numbers we have: 40 - 2r >0

r<20

Five years ago, their ages were prime numbers, that is:

r - 5 is prime

40 - 2r - 5 = 35 - 2r is prime

r + 10 - 5 = r + 5 is prime

Also

r - 5 must be positive so we have r - 5 >0 that is r > 5

35 - 2r must be a positive so we have 35 - 2r > 0 that is 17.5 > r

r + 5 >= 35 - 2r (since Felicia is the oldest)

3r >= 30

r >= 10

We will solve using the trail and error method:

For r = 10 we have

10 - 5 = 5 is prime

35 - 2*10 = 15 is not a prime

For r = 12 we have

12 - 5 = 7 is prime

35 - 2*12 = 11 is prime

12 + 5 = 17 is a prime

Thus one solution is:

Rosa is 12 years old.

Elena is 40 - 2*12 = 16 years old.

Felicia is 12 + 10 = 22 years old.

Lets check if this solution if unique.

For r = 16 we have

16 - 5 = 11 is prime

35 - 2*16 = 3 is prime

16 + 5 = 21 is not a prime

For r = 17 we have

17 - 5 = 12 is not a prime.

This means we tried all the possibilities for r. Thus the unique solution of the problem is:

Rosa is 12 years old.

Elena is 16 years old.

Felicia is 22 years old.

Indeed 12 + 16 + 22 = 50.