Name the property of multiplication of rational numbers illustrated by the following statements:

**(i)** \(\dfrac{-5}{16}× \dfrac{8}{15} = \dfrac{8}{15} × \dfrac{-5}{16}\)

**(ii)** \(\dfrac{-17}{5}×9 = 9 × \dfrac{-17}{5}\)

**(iii) **\(\dfrac{7}{4} × (\dfrac{-8}{3}+ \dfrac{-13}{12}) = \dfrac{7}{4} × \dfrac{-8}{3} + \dfrac{7}{4} × \dfrac{-13}{12}\)

**(iv) **\( \dfrac{-5}{9} × (\dfrac{4}{15} × \dfrac{-9}{8}) = ( \dfrac{-5}{9} × \dfrac{4}{15}) × \dfrac{-9}{8}\)

**(v)** \(\dfrac{ 13}{-17}× 1 = \dfrac{ 13}{-17} = 1 ×\dfrac{ 13}{-17}\)

**(vi)** \(\dfrac { -11}{16 }× \dfrac{16}{-11 }= 1\)

**(vii)** \( \dfrac{ 2}{13} × 0 = 0 = 0 ×\dfrac{ 2}{13}\)

**(viii) **\( \dfrac{-3}{2} × \dfrac{5}{4} + \dfrac{-3}{2} × \dfrac{-7}{6} = \dfrac{-3}{2} ×\dfrac{5}{4} + \dfrac{-7}{6}\)

Asked by Aaryan | 1 year ago | 107

** (i)** \(\dfrac{-5}{16}× \dfrac{8}{15} = \dfrac{8}{15} × \dfrac{-5}{16}\)

According to commutative law, \(\dfrac{ a}{b} × \dfrac{ c}{d} = \dfrac{ c}{d} × \dfrac{ a}{b}\)

The above rational number satisfies commutative property.

**(ii)** \(\dfrac{-17}{5}×9 = 9 × \dfrac{-17}{5}\)

According to commutative law, \( \dfrac{ a}{b} × \dfrac{ c}{d} = \dfrac{ c}{d} × \dfrac{ a}{b}\)

The above rational number satisfies commutative property.

**(iii) **\(\dfrac{7}{4} × (\dfrac{-8}{3}+ \dfrac{-13}{12}) = \dfrac{7}{4} × \dfrac{-8}{3} + \dfrac{7}{4} × \dfrac{-13}{12}\)

According to given rational number,

\( \dfrac{ a}{b}× ( \dfrac{ c}{d} + \dfrac{ e}{f}) = ( \dfrac{ a}{b} × \dfrac{ c}{d}) + ( \dfrac{ a}{b}× \dfrac{e}{f})\)

Distributivity of multiplication over addition satisfies.

**(iv) **\( \dfrac{-5}{9} × (\dfrac{4}{15} × \dfrac{-9}{8}) = ( \dfrac{-5}{9} × \dfrac{4}{15}) × \dfrac{-9}{8}\)

According to associative law,

\( \dfrac{ a}{b}× ( \dfrac{ c}{d} + \dfrac{ e}{f}) = ( \dfrac{ a}{b} × \dfrac{ c}{d}) + ( \dfrac{ a}{b}× \dfrac{e}{f})\)

The above rational number satisfies associativity of multiplication.

**(v)** \(\dfrac{ 13}{-17}× 1 = \dfrac{ 13}{-17} = 1 ×\dfrac{ 13}{-17}\)

Existence of identity for multiplication satisfies for the given rational number.

**(vi)** \(\dfrac { -11}{16 }× \dfrac{16}{-11 }= 1\)

Existence of multiplication inverse satisfies for the given rational number.

**(vii)** \( \dfrac{ 2}{13} × 0 = 0 = 0 ×\dfrac{ 2}{13}\)

By using \( \dfrac{ a}{b}× 0 = 0 × \dfrac{ a}{b}\)

Multiplication of zero satisfies for the given rational number.

**(viii) **\( \dfrac{-3}{2} × \dfrac{5}{4} + \dfrac{-3}{2} × \dfrac{-7}{6} = \dfrac{-3}{2} ×\dfrac{5}{4} + \dfrac{-7}{6}\)

According to distributive law,

\( \dfrac{ a}{b}× ( \dfrac{ c}{d} + \dfrac{ e}{f}) = ( \dfrac{ a}{b} × \dfrac{ c}{d}) + ( \dfrac{ a}{b}× \dfrac{e}{f})\)

The above rational number satisfies distributive law.

Answered by Sakshi | 1 year agoBy what number should 1365 be divided to get 31 as quotient and 32 as remainder?

Which of the following statement is true / false?

**(i)** \(\dfrac{ 2 }{ 3} – \dfrac{4 }{ 5}\) is not a rational number.

**(ii) **\( \dfrac{ -5 }{ 7}\) is the additive inverse of \( \dfrac{ 5 }{ 7}\)

**(iii)** 0 is the additive inverse of its own.

**(iv) **Commutative property holds for subtraction of rational numbers.

**(v) **Associative property does not hold for subtraction of rational numbers.

**(vi)** 0 is the identity element for subtraction of rational numbers.

If x = \( \dfrac{4 }{ 9}\), y =\( \dfrac{-7 }{ 12}\) and z = \( \dfrac{-2 }{ 3}\), then verify that x – (y – z) ≠ (x – y) – z

If x = \(\dfrac{ – 4 }{ 7}\) and y = \( \dfrac{2 }{ 5}\), then verify that x – y ≠ y – x

Subtract the sum of \(\dfrac{ – 5 }{ 7} and\dfrac{ – 8 }{ 3}\) from the sum of \(\dfrac{5 }{ 2} and \dfrac{– 11 }{ 12}\)