If p : q = $1\dfrac{1}{3} : 1\dfrac{1}{2}$ and q : r = $\dfrac{1}{2} : \dfrac{1}{3}$, find p : r.

Given:

p : q = $1\dfrac{1}{3} : 1\dfrac{1}{2}$

q : r = $\dfrac{1}{2} : \dfrac{1}{3}$

Convert p : q to improper fractions:

p : q = $\dfrac{4}{3} : \dfrac{3}{2}$

L.C.M. of 3 and 2 is 6.

Multiply p : q and q : r with 6 to convert fractions to whole numbers:

$p : q = (\dfrac{4}{3} \times 6) : (\dfrac{3}{2} \times 6) = 8 : 9$

And

$q : r = (\dfrac{1}{2} \times 6) : (\dfrac{1}{3} \times 6) = 3 : 2$

$\dfrac{p}{q} = \dfrac{8}{9} \\[1em] \dfrac{q}{r} = \dfrac{3}{2}$

Now,

$p : r = \dfrac{p}{r} = \left( \dfrac{p}{q} \times \dfrac{q}{r} \right) \\[1em] = \left( \dfrac{8}{9} \times \dfrac{3}{2} \right) \\[1em] = \left( \dfrac{8}{3} \times \dfrac{1}{2} \right) \\[1em] = \dfrac{8}{6} \\[1em] = \dfrac{4}{3} \\[1em] = 4 : 3$

∴ p : r = 4 : 3