Question

Check the commutative property of addition for $\dfrac{{ - 8}}{{13}},\dfrac{{23}}{{27}}$.

Answer 1

Hint: This sum is a bit complicated if the student is not aware of the formula and the property of commutative addition. The commutative property of addition states that even when you change the addends for a set of numbers, the sum would remain the same. According to the commutative property of addition if $a\& b$, are real numbers and $a + b = b + a$, then they satisfy the property of commutative addition. In this sum we have to add the given numbers in two different ways i.e. $ - \dfrac{8}{{13}} + \dfrac{{23}}{{27}}$ & $\dfrac{{23}}{{27}} + ( - \dfrac{8}{{13}})$. If the sum is the same we can say that it satisfies the commutative property of addition.

Complete step by step solution:
Let $a\& b$, be a set of two real numbers for which the check has to be carried out.
Let $a = \dfrac{{ - 8}}{{13}}\& b = \dfrac{{23}}{{27}}$
Let us first find the value of $a + b$:
$a + b = \dfrac{{ - 8}}{{13}} + \dfrac{{23}}{{27}}$
Taking LCM of $13\& 27$, we get
$ \Rightarrow a + b = \dfrac{{ - 8 \times 27 + 23 \times 13}}{{27 \times 13}}$
$\therefore a + b = \dfrac{{83}}{{351}}..........(1)$
Let us now find the value of $b + a$:
$b + a = \dfrac{{23}}{{27}} + (\dfrac{{ - 8}}{{13}})$
$ \Rightarrow b + a = \dfrac{{23 \times 13 + ( - 8 \times 27)}}{{27 \times 13}})$
$ \Rightarrow b + a = \dfrac{{83}}{{351}}...........(2)$
Thus, from equation$1\& 2$, we can see that the value of $a + b = b + a$.

Hence the given numbers $\dfrac{{ - 8}}{{13}},\dfrac{{23}}{{27}}$,satisfy the commutative property of addition.

Note: This is one of the most important properties of addition. Other than this there are $3$ other properties which a student should be aware of, i.e. Closure Property, Associative Property & additive Identity property. The student may have to apply the property when they are given a word problem for example A had $4$ bananas in her shopping cart. She met B who gave her $4$ oranges, How many fruits does A have in her cart. In this sum, the student will have to add the fruits in two different ways to confirm the number of fruits. Thus the student is using the commutative property of addition indirectly.