Question

An illustration of the associative law for multiplication is given by
(a) $\left( \dfrac{1}{3}\times 5 \right)\times 8=\dfrac{1}{3}\times \left( 5\times 8 \right)$
(b) $\dfrac{1}{3}\times 5\times 8=\dfrac{1}{3}\times 8\times 5$
(c) $\dfrac{1}{3}\times 5+\dfrac{1}{3}\times 8=\dfrac{1}{3}\times 13$
(d) $\dfrac{1}{3}\times 5\times 8=\left( \dfrac{1}{3}\times 5 \right)\times \left( \dfrac{1}{3}\times 8 \right)$
(e) None of these

Answer 1

Hint: There are three laws in mathematics: Commutative law, Associative law and Distributive law. Here, associative law for multiplication is to be found out, so it states that “it doesn’t matter how we group numbers (i.e. which we calculate first) when we add or when we multiply.” For example: if we have to add 3 numbers 5, 2, 7 then here it doesn’t matter we add first 5+2 and then 7 to the result or first adding 2+7 and then 5 to the result. Answer will remain the same. Same way we will do for multiplication.

Complete step-by-step answer:
Here, we have to find the option which satisfies the rule of associative law i.e. “it doesn’t matter how we group numbers (i.e. which we calculate first) when we add or when we multiply.” In mathematical term it is easily understood as $a+\left( b+c \right)=\left( a+b \right)+c$ for addition and for multiplication it is given as $a\left( bc \right)=\left( ab \right)c$ where a, b, c are the numbers.
We will take here option (a): $\left( \dfrac{1}{3}\times 5 \right)\times 8=\dfrac{1}{3}\times \left( 5\times 8 \right)$ here, we can take that $a=\dfrac{1}{3},b=5,c=8$ .Now using the multiplication rule which is $a\left( bc \right)=\left( ab \right)c$, we can see that values satisfy the rule and so, this is the correct answer.
Taking option (b): $\dfrac{1}{3}\times 5\times 8=\dfrac{1}{3}\times 8\times 5$ here, $a=\dfrac{1}{3},b=5,c=8$. As in multiplication, there is no difference when multiplying the numbers in any order. So, LHS and RHS will have the same value and thus it satisfies the Associative rule. So, this is also the correct answer.
Taking option(c): $\dfrac{1}{3}\times 5+\dfrac{1}{3}\times 8=\dfrac{1}{3}\times 13$ here $a=\dfrac{1}{3},b=5,c=8$ , but there is plus sign in between. So, we cannot solve from left to right also, it does not satisfy multiplication or addition rules properly. So, this is not the correct answer.
Taking option (d): $\dfrac{1}{3}\times 5\times 8=\left( \dfrac{1}{3}\times 5 \right)\times \left( \dfrac{1}{3}\times 8 \right)$ here $a=\dfrac{1}{3},b=5,c=8$ , on RHS side it is as $\left( ab \right)\left( ac \right)$ which is not associative rule of multiplication. So, this is not correct.
Hence, option (a) and (b) are the correct answer.


Note: Another approach to solving this question is taking separately LHS and RHS values and comparing them. As the rule states $a\left( bc \right)=\left( ab \right)c$ so, if values are equal that is the correct answer. For example taking option (a): $\left( \dfrac{1}{3}\times 5 \right)\times 8=\dfrac{1}{3}\times \left( 5\times 8 \right)$ where $a=\dfrac{1}{3},b=5,c=8$
So, LHS $=a\left( bc \right)=\left( \dfrac{1}{3}\times 5 \right)\times 8=\dfrac{40}{3}$ and RHS$=\left( ab \right)c=\dfrac{1}{3}\times \left( 5\times 8 \right)=\dfrac{40}{3}$ .
Thus, both the values are equal, so it is the correct answer. Similarly, we can check for every option.