Question

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consider the rational numbers 2 / 3 , 1 / 2 and 3 / 4 .

check whether associative property holds subtraction and division.

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answer with explanation ✌​

Answer 1

Sᴏʟᴜᴛɪᴏɴ :-

Associative property for subtraction :-

→ (a - b) - c = a - ( b - c)

Given That :-

→ a = (2/3)

→ b = (1/2)

→ c = (3/4)

Putting values we get :-

→ [ (2/3) - (1/2) ] - (3/4) = (2/3) - [ (1/2) - (3/4) ]

→ [ (4 - 3)/6 ] - (3/4) = (2/3) - [ (2 - 3)/4 ]

→ (1/6) - (3/4) = (2/3) - (-1/4)

→ (2 - 9)/12 = (8 - (-3))/12

→ (-7/12) = (8+3)/12

→ (-7) ≠ (11)

Hence, we can Conclude That, Associative property for subtraction will Not Hold .

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Associative property for Division :-

→ (a ÷ b) ÷ c = a ÷ ( b ÷ c)

Given That :-

→ a = (2/3)

→ b = (1/2)

→ c = (3/4)

Putting values we get :-

→ [ (2/3) ÷ (1/2) ] ÷ (3/4) = (2/3) ÷ [ (1/2) ÷ (3/4) ]

→ [ (2/3) * (2/1) ] ÷ (3/4) = (2/3) ÷ [ (1/2) * (4/3) ]

→ (4/3) ÷ (3/4) = (2/3) ÷ (2/3)

→ (4/3) * (4/3) = (2/3) * (3/2)

→ (16/9) ≠ 1

Hence, we can Conclude That, Associative property for Division will also Not Hold .

Answer 2

Aɴꜱᴡᴇʀ

✭ Both of them does not hold true.

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Gɪᴠᴇɴ

$\mapsto{} \sf a = \dfrac{2}{3} \\ \\ \mapsto{} \sf{}b = \dfrac{1}{2} \\ \\ \mapsto \sf{}c = \frac{3}{4}$

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Tᴏ ꜰɪɴᴅ

➤ If they the associative property holds subtraction and divition

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Sᴛᴇᴘꜱ

➳ So the associative property of subtraction is,

$\large { \underline{ \boxed{ \sf{(a - b) - c = a - (b - c)}}}}$

❍ So substituting the given values,

$\large \tt \leadsto{}( \frac{2}{3} - \frac{1}{2} ) - \frac{3}{4} = \frac{2}{3} - ( \frac{1}{2} - \frac{3}{4}) \\ \\ \large \tt \leadsto{} \frac{4 - 3}{6} - \frac{3}{4} = \frac{2}{3} \: - \frac{2 - 3}{4} \\ \\ \large \tt \leadsto{} \frac{1}{6} - \frac{3}{4} = \frac{2}{3} - \frac{ - 1}{4} \\ \\ \large \tt \leadsto \frac{2 - 9}{12} = \frac{8 - ( - 3)}{12} \\ \\ \large \tt{ \pink{ \leadsto \frac{ - 7}{12} ≠ \frac{11}{12} }}$

☞ Therefore associative property of subtraction does not hold true.

➳ Associative property of division is,

$\large { \underline{ \boxed{ \sf{(a \div b) \div c = a \div (b \div c)}}}}$

❍ Substituting the given values,

$\large \tt \dashrightarrow (\dfrac{ \frac{2}{3} }{ \frac{1}{2} }) \div \dfrac{3}{4} = \dfrac{2}{3} \div ( \dfrac{ \frac{1}{2} }{ \frac{3}{4} } ) \\ \\ \large \tt \dashrightarrow( \frac{2}{3} \times \frac{2}{1} ) \div \frac{3}{4} = \frac{2} {3 } \div ( \frac{ 1 }{2} \times \frac{4}{3} ) \\ \\ \large \tt \dashrightarrow \dfrac{ \dfrac{4}{3} }{ \dfrac{3}{4} } = \dfrac{ \dfrac{2}{3} }{ \dfrac{2}{3} } \\ \\ \large \tt \dashrightarrow{} \frac{4}{3} \times \frac{4}{3} = \frac{2}{ 3} \times \frac{3}{2} \\ \\ \large \tt \pink{ \dashrightarrow \frac{16}{9}≠1 }$

☞ Therefore associative property of division also does not hold true.

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