Write each of the following rational numbers in standard form :

(i) $\dfrac{56}{32}$

(ii) $\dfrac{16}{-40}$

(iii) $\dfrac{-36}{54}$

(iv) $\dfrac{-22}{-77}$

(v) $\dfrac{78}{-65}$

(vi) $\dfrac{-95}{114}$

(vii) $\dfrac{-69}{115}$

(viii) $\dfrac{155}{-217}$

(i) $\dfrac{56}{32}$

The HCF of 56 and 32 is 8.

Divide the numerator and denominator by 8:

$\dfrac{56}{32} = \dfrac{56 \div 8}{32 \div 8} = \dfrac{7}{4}$

∴ The standard form is $\dfrac{7}{4}$.

(ii) $\dfrac{16}{-40}$

First, multiply the numerator and denominator by -1 to make the denominator positive:

$\dfrac{16 \times (-1)}{-40 \times (-1)} = \dfrac{-16}{40}$

The HCF of 16 and 40 is 8.

Divide the numerator and denominator by 8:

$\dfrac{-16 \div 8}{40 \div 8} = \dfrac{-2}{5}$

∴ The standard form is $\dfrac{-2}{5}$.

(iii) $\dfrac{-36}{54}$

The HCF of 36 and 54 is 18.

Divide the numerator and denominator by 18:

$\dfrac{-36 \div 18}{54 \div 18} = \dfrac{-2}{3}$

∴ The standard form is $\dfrac{-2}{3}$.

(iv) $\dfrac{-22}{-77}$

First, multiply the numerator and denominator by -1 to make the denominator positive:

$\dfrac{-22 \times (-1)}{-77 \times (-1)} = \dfrac{22}{77}$

The HCF of 22 and 77 is 11.

$\dfrac{22 \div 11}{77 \div 11} = \dfrac{2}{7}$

∴ The standard form is $\dfrac{2}{7}$.

(v) $\dfrac{78}{-65}$

First, multiply the numerator and denominator by -1 to make the denominator positive:

$\dfrac{78 \times (-1)}{-65 \times (-1)}$ = $\dfrac{-78}{65}$

The HCF of 78 and 65 is 13.

$\dfrac{-78 \div 13}{65 \div 13} = \dfrac{-6}{5}$

∴ The standard form is $\dfrac{-6}{5}$.

(vi) $\dfrac{-95}{114}$

The HCF of 95 and 114 is 19.

$\dfrac{-95 \div 19}{114 \div 19} = \dfrac{-5}{6}$

∴ The standard form is $\dfrac{-5}{6}$.

(vii) $\dfrac{-69}{115}$

The HCF of 69 and 115 is 23.

$\dfrac{-69 \div 23}{115 \div 23} = \dfrac{-3}{5}$

∴ The standard form is $\dfrac{-3}{5}$.

(viii) $\dfrac{155}{-217}$

First, multiply the numerator and denominator by -1 to make the denominator positive:

$\dfrac{155 \times (-1)}{-217 \times (-1)}$ = $\dfrac{-155}{217}$

The HCF of 155 and 217 is 31.

$\dfrac{-155 \div 31}{217 \div 31} = \dfrac{-5}{7}$

∴ The standard form is $\dfrac{-5}{7}$.