Question

What are associative for rational numbers?
$A)$ Multiplication and addition
$B)$ Multiplication and subtraction
$C)$ Division and subtraction
$D)$ Division and subtraction

Answer 1

Hint: First we have to define the rational numbers and then explain the examples for the associative law property one by one. On doing some calculation we conclude the required answer.

Complete step-by-step solution:
Rational numbers are the numbers that can be written in the form of a fraction where numerator and denominator are integers.
Mathematically, we can write it as, rational numbers definition is given as the numbers $\dfrac{a}{b}$ if $a$ and $b$ are Co-prime, and $b \ne 0$.
Rational numbers follow the associative property for the mathematical operations of addition, multiplication, subtraction and division.
Let us say that we have three numbers, $a,b$ and $c$, where $a,b,c$ are rational numbers.
The associative law for rational numbers states that: $(a * b) * c = a * (b * c)$
Associative property of addition of the rational number is given by following.
$(a + b) + c = a + (b + c)$, here $a,b,c$ are rational numbers.
For example we have three rational numbers, $5, - 7$ and $\dfrac{2}{7}$
Let us see how the associative law works on the addition of the rational numbers.
So we can write it as, $\left( {5 + \left( { - 7} \right)} \right) + \dfrac{2}{7} = 5 + \left( {\left( { - 7} \right) + \dfrac{2}{7}} \right)$
Let’s take L.H.S, $(5 + ( - 7)) + \dfrac{2}{7}$
Let us add the bracket term and we get
$ \Rightarrow - 2 + \dfrac{2}{7}$
Taking LCM on both sides we get,
$ \Rightarrow \dfrac{{ - 12}}{7}$
Now, R.H.S $5 + ( - 7 + \dfrac{2}{7})$
Taking LCM on bracket term and we get
$ \Rightarrow 5 + \left( {\dfrac{{ - 49 + 2}}{7}} \right)$
On adding the numerator term and we get
$ \Rightarrow 5 - \dfrac{{47}}{7}$
Again taking LCM we get,
$ \Rightarrow \dfrac{{35 - 47}}{7}$
On subtracting the numerator term and we get,
$ \Rightarrow - \dfrac{{12}}{7}$
Hence, L.H.S = R.H.S
Therefore addition operation holds for the associativity.
Now let’s check for the multiplication operation
Associative on multiplication of the rational numbers
Let’s take an example taking $a,b,c$ as $2,3,4$.
To prove: $(a \times b) \times c = a \times (b \times c)$
Now taking LHS, $(2 \times 3) \times 4$
Let us multiply the bracket term,
$ \Rightarrow 6 \times 4$
On multiply we get,
$ \Rightarrow 24$
Also, we take R.H.S $2 \times (3 \times 4)$
Let us multiply the bracket term and we get,
$ \Rightarrow 2 \times 12$
On multiply the term and we get,
$ \Rightarrow 24$
∴L.H.S=R.H.S
Hence, multiplication operation holds for the associativity.
Now let’s check for the subtraction operation
Associative on Subtraction of the rational numbers:
Let’s take an example for subtraction $a,b,c$ are $2,6,4$
To prove: $(a - b) - c = a - (b - c)$
Taking L.H.S, $(2 - 6) - 4$
Let us subtract the bracket term and we get,
$ \Rightarrow - 4 - 4$
On adding the term because it has same sign,
$ \Rightarrow - 8$
Also, we take R.H.S $2 - (6 - 4)$
On subtract the term,
$ \Rightarrow 2 - 2$
Again subtract it
$ \Rightarrow 0$
Thus$ - 8 \ne 0$
Hence L.H.S≠R.H.S
Thus, Subtraction does not hold for associativity.
Let’s check division on the rational numbers:
Let’s take an example for the division of the $a,b$ and $c$ are $\dfrac{1}{2},\dfrac{3}{2},\dfrac{5}{2}$.
To prove: $(a \div b) \div c = a \div (b \div c)$
Let us take L.H.S $\left( {\dfrac{1}{2} \div \dfrac{3}{2}} \right) \div \dfrac{5}{2}$
Taking reciprocal of the second term in the bracket term and we get,
$ \Rightarrow \left( {\dfrac{1}{2} \times \dfrac{2}{3}} \right) \div \dfrac{5}{2}$
On cancel the bracket term
$ \Rightarrow \left( {\dfrac{1}{3}} \right) \div \dfrac{5}{2}$
Taking reciprocal of the second term in the bracket term and we get,
$ \Rightarrow \left( {\dfrac{1}{3} \times \dfrac{2}{5}} \right)$
Let us multiply the term and we get,
$ \Rightarrow \dfrac{2}{{15}}$
Also, we take, R.H.S $\dfrac{1}{2} \div \left( {\dfrac{3}{2} \div \dfrac{5}{2}} \right)$
Taking reciprocal of the second term in the bracket term and we get,
$ \Rightarrow \dfrac{1}{2} \div \left( {\dfrac{3}{2} \times \dfrac{2}{5}} \right)$
On cancel the term and we get,
$ \Rightarrow \dfrac{1}{2} \div \dfrac{3}{5}$
Taking reciprocal of the second term and we get,
$ \Rightarrow \dfrac{1}{2} \times \dfrac{5}{3}$
On multiply the terms and we get,
$ \Rightarrow \dfrac{5}{6}$
Thus $\dfrac{2}{{15}} \ne \dfrac{5}{6}$
Therefore L.H.S $ \ne $ R.H.S
Hence division operation does not hold for associativity.
Thus the associative property holds on addition and multiplication for rational numbers.

Hence the correct option is $(A)$ that is multiplication and addition.

Note: Rational numbers include numbers that are finite or recurring in nature, where recurring numbers or recurring decimal is $0.3333333.....,0.22222....$ etc. Perfect numbers are always rational numbers like $\sqrt {49} = 7$.