Today the ratio of elinor's age to her mother's age is 3: 8. after 15 years, the ratio will become 6: 11. what is elinor's age today?

X=Elinor´s age today.
y=Mother´s age today.
(x+15)=Elinor´s age after 15 years.
(y+15)=Mohter´s age after 15 years.

We can suggest this systemen of equations:
x/y=3/8                                  ⇒x=3y/8
(x+15)/(y+15)=6/11              ⇒11(x+15)=6(y+15)

Finally, we´ll have the following system:
x=3y/8
11(x+15)=6(y+15)

We can solve this system of equations by substitution method:
11[(3y/8)+15)=6(y+15)
11[(3y+120)/8)=6y+90
33y+1320=8(6y+90)
33y+1320=48y+720
33y-48y=720-1320
-15y=-600
y=-600/-15
y=40

We find out the value of "x";
x=3y/8
x=3(40)/8
x=120/8
x=15

Elinor is 15 years old.