Divide 468 g of rice into three heaps containing the quantities in the ratio $\dfrac{1}{6} : \dfrac{1}{8} : \dfrac{1}{12}$.

Given:

Total Rice: 468 g

Ratio: $\dfrac{1}{6} : \dfrac{1}{8} : \dfrac{1}{12}$

Let us find L.C.M. of 6, 8, and 12:

$\begin{array}{l|rrr} 2 & 6, & 8, & 12 \ \hline 2 & 3, & 4, & 6 \ \hline 2 & 3, & 2, & 3 \ \hline 3 & 3, & 1, & 3 \ \hline & 1, & 1, & 1 \end{array}$

L.C.M. = 2 x 2 x 2 x 3 = 24

Multiply all three terms with 24 to convert fractions to whole numbers:

$\Big(\dfrac{1}{6} \times 24\Big) : \Big(\dfrac{1}{8} \times 24\Big) : \Big(\dfrac{1}{12} \times 24\Big)$

= 4 : 3 : 2

Total parts: 4 + 3 + 2 = 9 parts.

Value of 1 part: 468 g ÷ 9 = 52 g.

Calculate heaps by multiplying each part with 52 g:

First heap: 4 x 52 g = 208 g

Second heap: 3 x 52 g = 156 g

Third heap: 2 x 52 g = 104 g

The three heaps are 208 g, 156 g and 104 g.