Express each of the following ratios in simplest form :

(i) $\dfrac{1}{8} : \dfrac{1}{12}$

(ii) $\dfrac{1}{9} : \dfrac{1}{6}$

(iii) $\dfrac{2}{3} : \dfrac{5}{6}$

(iv) $\dfrac{1}{4} : \dfrac{1}{6} : \dfrac{1}{8}$

(v) $1\dfrac{2}{3} : 2\dfrac{1}{2} : 1\dfrac{3}{4}$

(vi) $2\dfrac{1}{3} : 3\dfrac{1}{4} : 1\dfrac{1}{6}$

(i) $\dfrac{1}{8} : \dfrac{1}{12}$

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

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

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

Multiplying both terms by 24 to convert fractions to whole number:

$\left(\dfrac{1}{8} \times 24\right) : \left(\dfrac{1}{12} \times 24\right) = 3 : 2$

Hence, the answer is 3 : 2

(ii) $\dfrac{1}{9} : \dfrac{1}{6}$

Let us find L.C.M of 9 and 6:

$\begin{array}{l|l} 3 & 9, 6 \ \hline 3 & 3, 2 \ \hline 2 & 1, 2 \ \hline & 1, 1 \end{array}$

L.C.M. = 3 x 3 x 2 = 18.

Multiplying both terms by 18 to convert fractions to whole number:

$\left(\dfrac{1}{9} \times 18\right) : \left(\dfrac{1}{6} \times 18\right) = 2 : 3$

Hence, the answer is 2 : 3

(iii) $\dfrac{2}{3} : \dfrac{5}{6}$

Let us find L.C.M of 3 and 6:

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

L.C.M. = 3 x 2 = 6.

Multiplying both terms by 6 to convert fractions to whole number:

$\left(\dfrac{2}{3} \times 6\right) : \left(\dfrac{5}{6} \times 6\right) = 4 : 5$

Hence, the answer is 4 : 5

(iv) $\dfrac{1}{4} : \dfrac{1}{6} : \dfrac{1}{8}$

Let us find L.C.M of 4, 6 and 8:

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

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

Multiplying each term by 24 to convert fractions to whole number:

$\left(\dfrac{1}{4} \times 24\right) : \left(\dfrac{1}{6} \times 24\right) : \left(\dfrac{1}{8} \times 24\right) = 6 : 4 : 3$

Hence, the answer is 6 : 4 : 3

(v) $1\dfrac{2}{3} : 2\dfrac{1}{2} : 1\dfrac{3}{4}$

Convert to improper fractions: $\dfrac{5}{3} : \dfrac{5}{2} : \dfrac{7}{4}$.

Let us find L.C.M of 3, 2 and 4:

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

L.C.M. = 2 x 2 x 3 = 12.

Multiplying each term by 12 to convert fractions to whole number:

$\left(\dfrac{5}{3} \times 12\right) : \left(\dfrac{5}{2} \times 12\right) : \left(\dfrac{7}{4} \times 12\right) = 20 : 30 : 21$

Hence, the answer is 20 : 30 : 21

(vi) $2\dfrac{1}{3} : 3\dfrac{1}{4} : 1\dfrac{1}{6}$

Convert to improper fractions: $\dfrac{7}{3} : \dfrac{13}{4} : \dfrac{7}{6}$.

Let us find L.C.M of 3, 4 and 6:

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

L.C.M. = 2 x 2 x 3 = 12.

Multiplying each term by 12 to convert fractions to whole number:

$\left(\dfrac{7}{3} \times 12\right) : \left(\dfrac{13}{4} \times 12\right) : \left(\dfrac{7}{6} \times 12\right) = 28 : 39 : 14$

Hence, the answer is 28 : 39 : 14