Express each of the following ratios in simplest form :

(i) 20 : 35

(ii) 95 : 57

(iii) 2.1 : 1.2

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

(v) 6a : 9b

(vi) 8ab² : 6a²b

(i) 20 : 35

The given ratio is 20 : 35.

Let us find H.C.F. of 20 and 35:

$\begin{array}{r} 1 \ 20 \overline{) 35} \ \underline{-20} \ 15 \overline{) 20} ( 1 \ \underline{-15} \ 5 \overline{) 15} ( 3 \ \underline{-15} \ 0 \end{array}$

H.C.F. = 5

$∴ 20 : 35 = \dfrac{20 \div 5}{35 \div 5} = \dfrac{4}{7} = 4 : 7$

Hence, the answer is 4 : 7

(ii) 95 : 57

The given ratio is 95 : 57.

Let us find H.C.F. of 95 and 57:

$\begin{array}{r} 1 \ 57 \overline{) 95} \ \underline{-57} \ 38 \overline{) 57} ( 1 \ \underline{-38} \ 19 \overline{) 38} ( 2 \ \underline{-38} \ 0 \end{array}$

H.C.F. = 19

$∴ 95 : 57 = \dfrac{95 \div 19}{57 \div 19} = \dfrac{5}{3} = 5 : 3$

Hence, the answer is 5 : 3

(iii) 2.1 : 1.2

First, multiply both by 10 to remove decimals:

(2.1 x 10 : 1.2 x 10) = 21 : 12

Let us find H.C.F. of 21 and 12:

$\begin{array}{r} 1 \ 12 \overline{) 21} \ \underline{-12} \ 9 \overline{) 12} ( 1 \ \underline{-9} \ 3 \overline{) 9} ( 3 \ \underline{-9} \ 0 \end{array}$

H.C.F. = 3

$∴ 21 : 12 = \dfrac{21 \div 3}{12 \div 3} = \dfrac{7}{4} = 7 : 4$

Hence, the answer is 7 : 4

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

Convert mixed fractions to improper fractions: $\dfrac{13}{4} : \dfrac{13}{2}$

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

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

L.C.M. = 2 x 2 = 4

Multiply both terms by 4 to convert fractions to whole number:

$\Big(\dfrac{13}{4} \times 4\Big) : \Big(\dfrac{13}{2} \times 4\Big) = 13 : 26$

Let us find H.C.F of 13 and 26:

$\begin{array}{r} 2 \ 13 \overline{) 26} \ \underline{-26} \ 0 \end{array}$

H.C.F = 13

$\dfrac{13 \div 13}{26 \div 13}$ = 1 : 2

Hence, the answer is 1 : 2

(v) 6a : 9b

Given ratio is 6a : 9b

Let us find H.C.F. of 6 and 9:

$\begin{array}{r} 1 \ 6 \overline{) 9} \ \underline{-6} \ 3 \overline{) 6} ( 2 \ \underline{-6} \ 0 \end{array}$

H.C.F. = 3

$\dfrac{6a}{9b} = \dfrac{6a \div 3}{9b \div 3} = \dfrac{2a}{3b} = 2a : 3b$

Hence, the answer is 2a : 3b

(vi) 8ab² : 6a²b

Find the common factors in the coefficients and variables:

Let us find H.C.F. of 8 and 6:

$\begin{array}{r} 1 \ 6 \overline{) 8} \ \underline{-6} \ 2 \overline{) 6} ( 3 \ \underline{-6} \ 0 \end{array}$

H.C.F. = 2

Common variables: ab.

Divide both terms by 2ab:

$\dfrac{8ab^2}{2ab} : \dfrac{6a^2b}{2ab} = 4b : 3a$

Hence, the answer is 4b : 3a