**Answer:** Daniel is 7 years old and Jessica is 4 years old.

The first half of the question involves Daniel (D) and Alan (A).

Daniel is currently one fifth of Alan's age, so:

A = D x 5 (1)

In 21 years time, Alan will be twice his age, so:

A + 21 = (D + 21) x 2 (2)

Using (1) in (2) gives:

A + 21 = (D + 21) x 2

5D + 21 = (D + 21) x 2

5D + 21 = 2D + 42

3D = 21

D = 7

So Daniel is 7 (and Alan is 35).

The second half of the question involves Jessica (J) and Betty (B).

Betty's is exactly seven times the age of Jessica:

B = J x 7 (3)

In 8 years time, Betty will be three times the age of Jessica, so:

B + 8 = (J + 8) x 3 (4)

Using (3) in (4) gives:

B + 8 = (J + 8) x 3

7J + 8 = (J + 8) x 3

7J + 8 = 3J + 24

4J = 16

J = 4

So Jessica is 4 (and Betty is 28). QED.