Which ratio is greater?

(i) (3 : 4) or (5 : 7)

The given ratios are 3 : 4 and 5 : 7, which are equivalent to the fractions $\dfrac{3}{4}$ and $\dfrac{5}{7}$ respectively.

Let us find the L.C.M. of 4 and 7:

$\begin{array}{l|rr} 2 & 4, & 7 \ \hline 2 & 2, & 7 \ \hline 7 & 1, & 7 \ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 2 x 7 = 28.

$∴ 3 : 4 = \dfrac{3}{4} = \dfrac{3 \times 7}{4 \times 7} = \dfrac{21}{28}$

$5 : 7 = \dfrac{5}{7} = \dfrac{5 \times 4}{7 \times 4} = \dfrac{20}{28}$

Clearly, $\dfrac{21}{28} \gt \dfrac{20}{28}$.

Hence, (3 : 4) > (5 : 7).

(ii) (11 : 21) or (19 : 28)

The given ratios are 11 : 21 and 19 : 28, which are equivalent to the fractions $\dfrac{11}{21}$ and $\dfrac{19}{28}$ respectively.

Let us find L.C.M. of 21 and 28:

$\begin{array}{l|l} 7 & 21, 28 \ \hline 3 & 3, 4 \ \hline 4 & 1, 4 \ \hline & 1, 1 \end{array}$

L.C.M. = 7 x 3 x 4 = 84.

$\therefore 11 : 21 = \dfrac{11}{21} = \dfrac{11 \times 4}{21 \times 4} = \dfrac{44}{84}$

$19 : 28 = \dfrac{19}{28} = \dfrac{19 \times 3}{28 \times 3} = \dfrac{57}{84}$

Clearly, $\dfrac{57}{84} \gt \dfrac{44}{84}$.

Hence, (19 : 28) > (11 : 21).