If $$P = \frac{5}{6}Q$$ and $$Q = \frac{4}{5}R$$, what is 2P : 3Q : 4R?

If $$P = \frac{5}{6}Q$$

$$\frac{P}{Q}=\frac{5}{6}$$

So let's assume P = 5y and Q  = 6y.    Eq.(i)

If $$Q = \frac{4}{5}R$$

$$\frac{Q}{R}=\frac{4}{5}$$

So let's assume Q = 4z and R = 5z.    Eq.(ii)

By Eq.(i) and Eq.(ii), 6y = 4z

3y = 2z

z = 1.5y    Eq.(iii)

Value of 2P : 3Q : 4R = $$2\times5y : 3\times6y : 4\times5z$$     [From Eq.(i) and Eq.(ii)]

= $$10y : 18y : 20z$$

Put the value of 'z' from Eq.(iii) in the above equation.

= $$10y : 18y : 20\times1.5y$$

= $$10y : 18y : 30y$$

= 5 : 9 : 15