Call Alice's age: a

        Betty's age: b

        Clara's age: c

    (a,b,c\(\in\)N*)

According to the theme we have:
\(a+b+c=29\)(1)

\(b-a=4\Rightarrow b=a+4\)(2)

\(c-b=6\Rightarrow c=b+6\Rightarrow c=a+4+6=a+10\)(3)

Change (2)(3) to (1) we have:
\(\left(1\right)\Leftrightarrow a+b+c=29\)

\(a+a+4+a+10=29\)

\(3a+14=29\)

\(3a=29-14=15\)

\(a=5\)(satisfy)

Alice's age: 5 years old

The height of Paul:

\(\left(94+8\right):2=51\left(inches\right)\)

The height of Rachel:

\(51-8=43\left(inches\right)\)

\(\left(5\dfrac{1}{3}-2\dfrac{1}{2}\right)+\left(5\dfrac{1}{2}-3\dfrac{1}{3}\right)\)

\(=5\dfrac{1}{3}-2\dfrac{1}{2}+5\dfrac{1}{2}-3\dfrac{1}{3}\)

\(=\left(5\dfrac{1}{3}-3\dfrac{1}{3}\right)-\left(2\dfrac{1}{2}-5\dfrac{1}{2}\right)\)

\(=2\dfrac{1}{3}-\left(-3\dfrac{1}{2}\right)=2\dfrac{1}{3}+3\dfrac{1}{2}=5\dfrac{5}{6}\)