Question

Which property of addition is satisfied by: $\left( {7 + 6} \right) \times 10 = 7 \times 10 + 6 \times 10$

Answer 1

Hint: Here we are asked to find the property that has been used in the given equation to get satisfied. To solve this problem, we must know the various properties of addition like closure, associative, inverse, identity, commutative and distributive. We will check whether the given equation gets satisfied by any of these properties. Then we can conclude our answer.

Complete step-by-step answer:
It is given that $\left( {7 + 6} \right) \times 10 = 7 \times 10 + 6 \times 10$ we aim to find the property that has been used here to satisfy this equation. Before getting into this problem let us first see the properties under addition.
Closure: If $a$ and $b$ are any integer then $a + b = c$ where $c$ is also an integer.
Associative: If $a,b$ and $c$are any three integers then $\left( {a + b} \right) + c = a + \left( {b + c} \right)$
Inverse: If $a$ is an integer such that $a + a' = 0$ then $a'$ is an inverse of $a$ .
Identity: If $a$ is an integer such that $a + e = e + a = a'$ then $e$ is an identity element.
Distributive: If $a,b$ and $c$ then $a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$ and $\left( {a + b} \right) \times c = \left( {a \times c} \right) + \left( {b \times c} \right)$
Now let us check whether the given equation get satisfied or not.
Consider the left-hand side of the given equation and simplify it.
$\left( {7 + 6} \right) \times 10 = 13 \times 10$
$ \Rightarrow \left( {7 + 6} \right) \times 10 = 130$
Thus, we got the answer for the left-hand side.
Now let us find the value of the right-hand side.
$\left( {7 \times 10} \right) + \left( {6 \times 10} \right) = 70 + 60$
$ \Rightarrow \left( {7 \times 10} \right) + \left( {6 \times 10} \right) = 130$
We got the same answer for the expressions on both sides.
We can see that here in the given equation the distributive property under addition is used.
We know that by the distributive property we have $\left( {a + b} \right) \times c = \left( {a \times c} \right) + \left( {b \times c} \right)$ here from the given equation $a = 7,b = 6$ and $c = 10$.

Note: The above type of problem can only be solved, only if we are aware of all the properties which are all under addition. Also, we can see that there is another form of distributive property that is $a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$ so if the given question is in the form $7 \times \left( {6 + 10} \right) = \left( {7 \times 6} \right) + \left( {7 \times 10} \right)$ it does not get satisfies so we cannot say that the given equation gets satisfied by the distributive property under addition.