State, true or false:

(i) $\dfrac{7}{9} = \dfrac{7 + 5}{9 + 5}$

(ii) $\dfrac{7}{9} = \dfrac{7 - 5}{9 - 5}$

(iii) $\dfrac{7}{9} = \dfrac{7 \times 5}{9 \times 5}$

(iv) $\dfrac{7}{9} = \dfrac{7 ÷ 5}{9 ÷ 5}$

(v) $\dfrac{-5}{-12}$ is a negative rational number.

(vi) $\dfrac{-13}{25}$ is smaller than $\dfrac{-25}{13}$

(i) False.

Reason:

$\dfrac{7 + 5}{9 + 5} = \dfrac{12}{14}$ ≠ $\dfrac{7}{9}$

(ii) False

Reason:

$\dfrac{7 - 5}{9 - 5} = \dfrac{2}{4}$ ≠ $\dfrac{7}{9}$

(iii) True

Reason:

$\dfrac{7 \times 5}{9 \times 5} = \dfrac{7 \times \cancel{5}}{9 \times \cancel{5}} = \dfrac{7}{9}$

(iv) True

Reason:

$\dfrac{7 ÷ 5}{9 ÷ 5} \\[1em] = \dfrac{7 \times \dfrac{1}{5}}{9 \times \dfrac{1}{5}} \\[1em] = \dfrac{7 \times \cancel{\dfrac{1}{5}}}{9 \times \cancel{\dfrac{1}{5}}} \\[1em] = \dfrac{7}{9}$

(v) False

Reason:

$\dfrac{-5}{-12} = \dfrac{5}{12}$ is a positive rational number.

(vi) False

Reason:

We need to check if $\dfrac{-13}{25}$ is smaller than $\dfrac{-25}{13}$

LCM of 25 and 13 is 325

$\dfrac{-13}{25} = \dfrac{-13 \times 13}{25 \times 13} = \dfrac{-169}{325}$

$\dfrac{-25}{13} = \dfrac{-25 \times 25}{13 \times 25} = \dfrac{-625}{325}$

And, $\dfrac{-169}{325}$ > $\dfrac{-625}{325}$

Hence, $\dfrac{-13}{25}$ > $\dfrac{-25}{13}$